Evaluation of Procedures for 

Quality Assurance Specifications

PUBLICATION NO. FHWA-HRT-04-046 OCTOBER 2004

Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296
Foreword

Much has been written to guide highway agencies in the development, implementation, and use of 

quality assurance specifications. Unfortunately, the guidance is scattered and piecemeal. In some 

cases, it is out-of-date, inconsistent, or even contradicts statistical principles. Further, agencies’ 

negative experiences with quality assurance specifications have often not been recorded, and common 

mistakes are repeated by other agencies. 

This report is a companion to FHWA-RD-02-095, Optimal Procedures for Quality Assurance 

Specifications. While FHWA-RD-02-095 is a manual intended to provide guidance to highway 

agencies, this report summarizes the research work that was performed and contains the analyses to 

explain and justify the provided guidance. This report will be of interest to those materials, 

construction, specifications, and research engineers who wish to gain a better understanding of any 

specific procedures recommended in the manual. 

Sufficient copies of this report are being distributed to provide three copies to each FHWA Resource 

Center, a minimum of one copy to each FHWA Division, and a minimum of two copies to each State 

highway agency. Direct distribution is being made to the division offices. Additional copies for the 

public are available from the National Technical Information Services (NTIS), 5285 Port Royal Road, 

Springfield, VA 22161. 

 T. Paul Teng, P.E. 

 Director, Office of Infrastructure 

 Research and Development 

Notice 

This document is disseminated under the sponsorship of the U.S. Department of Transportation in the 

interest of information exchange. The U.S. Government assumes no liability for the use of the 

information contained in this document. 

The U.S. Government does not endorse products or manufacturers. Trademarks or manufacturers’ 

names appear in this report only because they are considered essential to the object of the document.

Quality Assurance Statement 

The Federal Highway Administration (FHWA) provides high-quality information to serve 

Government, industry, and the public in a manner that promotes public understanding. Standards and 

policies are used to ensure and maximize the quality, objectivity, utility, and integrity of its 

information. FHWA periodically reviews quality issues and adjusts its programs and processes to 

ensure continuous quality improvement. 
 Technical Report Documentation Page 

1. Report No. 

FHWA-HRT-04-046 

2. Government Accession No. 3. Recipient’s Catalog No.

5. Report Date

 October 2004 

4. Title and Subtitle

EVALUATION OF PROCEDURES FOR QUALITY 

ASSURANCE SPECIFICATIONS 6. Performing Organization Code 

7. Author(s)

J.L. Burati, R.M. Weed, C.S. Hughes, and H.S. Hill 

8. Performing Organization Report No.

9. Performing Organization Name and Address

Department of Civil Engineering 

10. Work Unit No. (TRAIS)

Clemson University 

Clemson, SC 29634-0911 

11. Contract or Grant No.

DTFH61-98-C-00069 

13. Type of Report and Period Covered

Final Report 

Sept. 1998 – Oct. 2003 

12. Sponsoring Agency’s Name and Address 

Office of Research, Development, and Technology 

Federal Highway Administration 

6300 Georgetown Pike 

McLean, VA 22101-2296 

14. Sponsoring Agency’s Code

15. Supplementary Notes

Contracting Officer’s Technical Representative (COTR): Peter A. Kopac, HRDI-11 

This study was conducted under State Planning and Research Pooled Fund Study No. 2 (199) and 

was administered by the Federal Highway Administration (FHWA). 

16. Abstract

The objective of this project was to develop a comprehensive quality assurance (QA) manual, 

supported by scientific evidence and statistical theory, which provides step-by-step procedures and 

instructions for developing effective and efficient QA specifications. 

This technical report summarizes the steps taken to accomplish this goal, along with the analyses 

that were conducted to support the recommendations made in the QA manual (FHWA-RD-02-095). 

The analytical techniques used depended on the decision that needed to be made. Both analytical and 

computer simulation approaches were used. 

Percent within limits (PWL) (or its complement, percent defective (PD)) was selected as the best 

quality measure because it combines both the sample mean and standard deviation into a single 

measure of quality. An approach based on a single composite quality measure derived from a general 

performance model to predict expected pavement life was developed and is the recommended 

approach for determining payment factors when multiple quality characteristics are measured. A 

detailed discussion and analysis are also presented regarding the risks involved in the various 

approaches to verifying the contractor’s test results. The relatively high risks that are associated with 

typical agency verification testing frequencies are highlighted.

17. Key Words

Quality assurance, quality control, specifications, 

statistical specifications, QA, QC, payment 

adjustments. 

18. Distribution Statement

No restrictions. This document is available to the 

public through the National Technical 

Information Service, Springfield, VA 22161. 

19. Security Classif. (of this report)

Unclassified 

20. Security Classif. (of this page)

Unclassified 

21. No. of Pages

 414 

22. Price

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized
 ii 

SI* (MODERN METRIC) CONVERSION FACTORS 

APPROXIMATE CONVERSIONS TO SI UNITS

Symbol When You Know Multiply By To Find Symbol 

LENGTH 

in inches 25.4 millimeters mm 

ft feet 0.305 meters m 

yd yards 0.914 meters m 

mi miles 1.61 kilometers km 

AREA 

in2 square inches 645.2 square millimeters mm2

ft2 square feet 0.093 square meters m2

yd2 square yard 0.836 square meters m2

ac acres 0.405 hectares ha 

mi2 square miles 2.59 square kilometers km2

VOLUME 

fl oz fluid ounces 29.57 milliliters mL 

gal gallons 3.785 liters L 

ft3 cubic feet 0.028 cubic meters m3 

yd3 cubic yards 0.765 cubic meters m3 

NOTE: volumes greater than 1000 L shall be shown in m3

MASS 

oz ounces 28.35 grams g

lb pounds 0.454 kilograms kg

T short tons (2000 lb) 0.907 megagrams (or "metric ton") Mg (or "t") 

TEMPERATURE (exact degrees) o

F Fahrenheit 5 (F-32)/9 Celsius oC 

or (F-32)/1.8 

ILLUMINATION 

fc foot-candles 10.76 lux lx 

fl foot-Lamberts 3.426 candela/m2 cd/m2

FORCE and PRESSURE or STRESS 

lbf poundforce 4.45 newtons N 

lbf/in2 poundforce per square inch 6.89 kilopascals kPa 

APPROXIMATE CONVERSIONS FROM SI UNITS 

Symbol When You Know Multiply By To Find Symbol 

LENGTH

mm millimeters 0.039 inches in 

m meters 3.28 feet ft 

m meters 1.09 yards yd 

km kilometers 0.621 miles mi 

AREA mm2

 square millimeters 0.0016 square inches in2 

m2 square meters 10.764 square feet ft2 

m2 square meters 1.195 square yards yd2 

ha hectares 2.47 acres ac 

km2 square kilometers 0.386 square miles mi2 

VOLUME 

mL milliliters 0.034 fluid ounces fl oz 

L liters 0.264 gallons gal 

m3 cubic meters 35.314 cubic feet ft3 

m3 cubic meters 1.307 cubic yards yd3 

MASS 

g grams 0.035 ounces oz

kg kilograms 2.202 pounds lb

Mg (or "t") megagrams (or "metric ton") 1.103 short tons (2000 lb) T 

TEMPERATURE (exact degrees) o

C Celsius 1.8C+32 Fahrenheit oF 

ILLUMINATION 

lx lux 0.0929 foot-candles fc 

cd/m2 candela/m2 0.2919 foot-Lamberts fl

FORCE and PRESSURE or STRESS 

N newtons 0.225 poundforce lbf 

kPa kilopascals 0.145 poundforce per square inch lbf/in2

*SI is the symbol for th International System of Units. Appropriate rounding should be made to comply with Section 4 of ASTM E e 380. 

(Revised March 2003)
 iii 

Preface

It is important to note that two documents have been prepared for this project—a manual for use 

by State highway administrations (SHAs), and this technical report, which summarizes the 

procedures and findings of the project. The manual is intended to be a comprehensive guide that 

a SHA can use when developing new or modifying existing acceptance plans and quality 

assurance (QA) specifications. While the focus and objectives of these documents are quite 

different, they are not entirely stand-alone documents. In preparing the two documents, an 

attempt has been made to minimize duplication of the contents. As such, this technical report 

should be read in conjunction with and as a companion to the QA specifications manual, Optimal 

Procedures for Quality Assurance Specifications (Report No. FHWA-RD-02-095), which also 

resulted from this project. 

The focus of the manual is on what should be done when developing QA specifications. The 

reasons for the various steps and possible decisions are explained and easy-to-follow examples 

are included to assist in understanding the process that is involved. The manual does not explain 

what was done during the project, nor what analytical and simulation analyses were conducted, 

unless it was necessary to clarify why certain steps in the process were necessary. This technical 

report contains the detailed descriptions and summaries of the results for the analyses that were 

conducted to arrive at the decisions and recommendations included in the manual. 
 

iv 

Table of Contents

Chapter Page 

1. INTRODUCTION......................................................................................................................1 

BACKGROUND .......................................................................................................................1 

PROJECT OBJECTIVE ............................................................................................................1 

METHODOLOGY ....................................................................................................................2 

Review of the Literature ......................................................................................................3

Development of the Process.................................................................................................4 

Analysis of Specific Options ...............................................................................................5 

Development of the Manual.................................................................................................5 

ORGANIZATION OF THIS REPORT.....................................................................................6

2. LITERATURE AND SPECIFICATION REVIEW...............................................................7 

LITERATURE REVIEW ..........................................................................................................7 

Search Methodology ............................................................................................................7

Results..................................................................................................................................8 

Summary..............................................................................................................................9 

SPECIFICATION REVIEW .....................................................................................................9 

HMAC Specifications..........................................................................................................9 

Superpave Specifications...................................................................................................11 

PCC Specifications ............................................................................................................12 

Summary............................................................................................................................12 

3. SPECIFICATIONS DEVELOPMENT PROCESS..............................................................13 

METHODOLOGY FOR DEVELOPING THE PROCESS ....................................................13 

THE PROCESS .......................................................................................................................13 

FLOWCHARTS AS A GUIDE TO REQUIRED ANALYSES..............................................14 

4. SELECTING TOPICS FOR DETAILED ANALYSES.......................................................23 

FIRST PANEL MEETING......................................................................................................23 

SURVEY OF TOPICS FOR DETAILED ANALYSES .........................................................23 

Survey Results ...................................................................................................................27 

TOPICS SELECTED FOR DETAILED ANALYSES ...........................................................30 

5. QUALITY MEASURES: ACCURACY AND PRECISION ...............................................33 

INTRODUCTION ...................................................................................................................33 

ACCURACY AND PRECISION............................................................................................33 

PWL EVALUATION ..............................................................................................................34

Simulation Analyses ..........................................................................................................34

AAD EVALUATION..............................................................................................................49 

Simulation Analyses ..........................................................................................................49

CI EVALUATION...................................................................................................................53 

Simulation Analyses ..........................................................................................................53

CONCLUSION........................................................................................................................56 
 

v 

6. QUALITY MEASURES: NORMALITY ASSUMPTION ..................................................59 

INTRODUCTION ...................................................................................................................59 

SKEWED DISTRIBUTIONS..................................................................................................59 

PWL Evaluation.................................................................................................................59 

AAD Evaluation.................................................................................................................93 

Comparison of PWL and AAD........................................................................................106 

BIMODAL DISTRIBUTIONS..............................................................................................106 

PWL Evaluation...............................................................................................................106 

AAD Evaluation...............................................................................................................119 

CONCLUSION......................................................................................................................119

7. VERIFICATION PROCEDURES.......................................................................................121 

INTRODUCTION .................................................................................................................121 

HYPOTHESIS TESTING AND LEVELS OF SIGNIFICANCE.........................................121 

TEST METHOD VERIFICATION.......................................................................................122 

Maximum Allowable Difference .....................................................................................122 

Paired t-Test.....................................................................................................................131 

PROCESS VERIFICATION .................................................................................................135 

AASHTO Appendix G Method .......................................................................................135 

AASHTO Appendix H Method .......................................................................................141 

CONCLUSIONS AND RECOMMENDATIONS ................................................................158 

Recommendation for Test Method Verification..............................................................158 

Recommendation for Process Verification ......................................................................158 

8. QUALITY MEASURES AND PAYMENT.........................................................................161 

CANDIDATE QUALITY MEASURES ...............................................................................161 

ANALYSES PERFORMED..................................................................................................161 

PAYMENT FOR A SINGLE QUALITY CHARACTERISTIC ..........................................162 

PWL Payment Factors .....................................................................................................162 

AAD Payment Factors .....................................................................................................167 

Matching PWL and AAD Payment Equations ................................................................167 

PAYMENT FOR MULTIPLE QUALITY CHARACTERISTICS ......................................178 

Simulating Correlated Variables......................................................................................178 

Correlated Quality Characteristics...................................................................................184 

SELECTION OF THE QUALITY MEASURE....................................................................197

9. EVALUATING RISKS .........................................................................................................199

INTRODUCTION .................................................................................................................199 

Types of Risk ...................................................................................................................199 

OC and EP Curves ...........................................................................................................200 

CALCULATING RISKS FOR A SINGLE QUALITY CHARACTERISTIC.....................201 

Example ...........................................................................................................................201 

Effect of Sample Size.......................................................................................................207

Distribution of Estimated PWL Values ...........................................................................207 

CALCULATING RISKS FOR MULTIPLE QUALITY CHARACTERISTICS.................212 

Calculating EP Contours..................................................................................................212 
 

vi 

EFFECT OF ADDED PROVISIONS ...................................................................................213 

CLOSING COMMENTS ON RISK......................................................................................220 

10. THE COMPOSITE PAYMENT FACTOR.......................................................................221 

TRADITIONAL COMPOSITE PAYMENT FACTORS .....................................................221 

ANALYSES PROCEDURES................................................................................................221 

ANALYSES RESULTS ........................................................................................................222 

Comparing the Methods...................................................................................................229 

CONCLUSION: TRADITIONAL COMPOSITE QUALITY MEASURES........................233

11. RELATING PAYMENT TO PERFORMANCE..............................................................235 

BACKGROUND ...................................................................................................................235 

PERFORMANCE-RELATED SPECIFICATIONS (PRS)...................................................236 

Decision Regarding PRS..................................................................................................238 

LCC BASIS FOR PAYMENT ADJUSTMENTS.................................................................238 

LCC Approach.................................................................................................................239 

Performance Relationships ..............................................................................................240 

Composite Quality Measure ............................................................................................242 

12. BRIEF SUMMARY AND RECOMMENDATIONS .......................................................245 

PROJECT OBJECTIVE ........................................................................................................245 

QA SPECIFICATIONS DEVELOPMENT PROCESS ........................................................245 

RESULTS OF DETAILED ANALYSES .............................................................................245 

APPENDIX A: ANNOTATED BIBLIOGRAPHY OF SELECTED ITEMS FROM 

 THE INITIAL LITERATURE SEARCH ....................................................249 

APPENDIX B: SUMMARY OF AGENCY HMAC SPECIFICATIONS RECEIVED......263 

APPENDIX C: SUMMARY OF AGENCY SUPERPAVE SPECIFICATIONS 

RECEIVED .....................................................................................................307 

APPENDIX D: SUMMARY OF AGENCY PCC SPECIFICATIONS RECEIVED..........329 

APPENDIX E: MINUTES FROM THE FIRST PANEL MEETING .................................345 

APPENDIX F: ILLUSTRATIONS OF POSSIBLE RANGES FOR PD OR PWL 

ESTIMATES ...................................................................................................349 

APPENDIX G: BIAS HISTOGRAMS FROM THE SIMULATION PROGRAM.............367 

APPENDIX H: DISTRIBUTION OF SAMPLE PWL ESTIMATES ..................................383 

REFERENCES...........................................................................................................................393 
 

vii 

List of Figures 

Figure Page 

1. States that provided funding for the study ..............................................................................2 

2. Flowchart for phase I—Initiation and planning....................................................................15 

3. Flowchart for phase II—Specifications development...........................................................16 

4. Flowchart for phase III—Implementation ............................................................................21 

5. Survey sent to panel members ..............................................................................................24

6. Graphical presentation of survey results for the first ranking method..................................29 

7. Graphical presentation of survey results for the second ranking method.............................29 

8. Plot of how the standard deviation of PWL estimates varies with the population PWL......39 

9a. Histogram illustrating the distribution of estimated PWL values for 1000 simulated 

 lots from a population with 90 PWL, sample 3 ....................................................................40 

9b. Histogram illustrating the distribution of estimated PWL values for 1000 simulated 

 lots from a population with 90 PWL, sample 5 ....................................................................40 

9c. Histogram illustrating the distribution of estimated PWL values for 1000 simulated 

 lots from a population with 90 PWL, sample 10 ..................................................................40 

10a. Histogram illustrating the distribution of estimated PWL values for 1000 simulated 

 lots from a population with 70 PWL, sample 3 ....................................................................41 

10b. Histogram illustrating the distribution of estimated PWL values for 1000 simulated 

 lots from a population with 70 PWL, sample 5 ....................................................................41 

10c. Histogram illustrating the distribution of estimated PWL values for 1000 simulated 

 lots from a population with 70 PWL, sample 10 ..................................................................41 

11a. Histogram illustrating the distribution of estimated PWL values for 1000 simulated 

 lots from a population with 50 PWL, sample 3 ....................................................................42 

11b. Histogram illustrating the distribution of estimated PWL values for 1000 simulated 

 lots from a population with 50 PWL, sample 5 ....................................................................42 

11c. Histogram illustrating the distribution of estimated PWL values for 1000 simulated 

 lots from a population with 50 PWL, sample 10 ..................................................................42 

12a. Illustration 1 of accuracy and precision of PWL estimates ..................................................43 

12b. Illustration 2 of accuracy and precision of PWL estimates ..................................................43 

12c. Illustration 3 of accuracy and precision of PWL estimates ..................................................43 

13. Plot of average difference of simulated PWL values versus actual PWL values for 

 sample sizes = 3, 5, and 10 ...................................................................................................47 

14a. Plots of the 95th percentile for the average estimated PWL minus the actual PWL at 50 

 versus the number of lots per project....................................................................................47 

14b. Plots of the 95th percentile for the average estimated PWL minus the actual PWL at 70 

 versus the number of lots per project....................................................................................48 

14c. Plots of the 95th percentile for the average estimated PWL minus the actual PWL at 90 

 versus the number of lots per project....................................................................................48 

15. Plot of average difference of simulated AAD values versus actual AAD values for 

 sample sizes = 3, 5, and 10 ...................................................................................................55 

16. Plot of how the standard deviation of AAD estimates varies with the population 

 AAD value ............................................................................................................................55 
 

viii 

17. Plot of average difference of simulated CI values versus actual CI values for 

 sample sizes = 3, 5, and 10 ...................................................................................................58 

18. Plot of how the standard deviation of CI estimates varies with the population 

CI value.................................................................................................................................58 

19a. Plots of bias versus actual PWL for 10,000 simulated lots with 3 tests per lot and 

 one-sided limits showing positive skewness.........................................................................65 

19b. Plots of bias versus actual PWL for 10,000 simulated lots with 3 tests per lot and 

 one-sided limits showing negative skewness........................................................................65 

20a. Plots of bias versus actual PWL for 10,000 simulated lots with 5 tests per lot and 

 one-sided limits showing positive skewness.........................................................................66 

20b. Plots of bias versus actual PWL for 10,000 simulated lots with 5 tests per lot and 

 one-sided limits showing negative skewness........................................................................66 

21a. Plots of bias versus actual PWL for 10,000 simulated lots with 10 tests per lot and 

 one-sided limits with positive skewness...............................................................................67 

21b. Plots of bias versus actual PWL for 10,000 simulated lots with 10 tests per lot and 

 one-sided limits with negative skewness ..............................................................................67 

22a. Plot of bias versus actual PWL for 10,000 simulated lots with various tests per lot 

 and one-sided limits with +1 skewness.................................................................................69 

22b. Plot of bias versus actual PWL for 10,000 simulated lots with various tests per lot 

 and one-sided limits with +2 skewness.................................................................................69 

22c. Plot of bias versus actual PWL for 10,000 simulated lots with various tests per lot 

 and one-sided limits with +3 skewness.................................................................................70 

23a. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the PWL 

 estimate for two-sided specification limits (skewness coefficient = +1.0, split=8/0)...........71 

23b. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the PWL 

 estimate for two-sided specification limits (skewness coefficient = +1.0, split=7/1)...........71 

23c. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the PWL 

 estimate for two-sided specification limits (skewness coefficient = +1.0, split=6/2)...........71 

23d. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the PWL 

 estimate for two-sided specification limits (skewness coefficient = +1.0, split=5/3)...........72 

23e. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the PWL 

 estimate for two-sided specification limits (skewness coefficient = +1.0, split=4/4)...........72 

23f. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the PWL 

 estimate for two-sided specification limits (skewness coefficient = +1.0, split=3/5)...........72 

23g. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the PWL 

 estimate for two-sided specification limits (skewness coefficient = +1.0, split=2/6)...........73 

23h. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the PWL 

 estimate for two-sided specification limits (skewness coefficient = +1.0, split=1/7)...........73 

23i. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the PWL 

 estimate for two-sided specification limits (skewness coefficient = +1.0, split=0/8)...........73 

24a. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots with PD = 10, sample=3, 

 and two-sided limits..............................................................................................................78 

24b. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots with PD = 10, sample=5, 

 and two-sided limits..............................................................................................................78 

24c. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots with PD = 10, sample=10, 

 and two-sided limits..............................................................................................................79 
 

ix 

25a. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots with PD = 30, 

 sample = 3, and two-sided limits ..........................................................................................80 

25b. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots with PD = 30, 

 sample = 5, and two-sided limits ..........................................................................................80 

25c. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots with PD = 30, 

 sample = 10, and two-sided limits ........................................................................................81 

26a. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots with PD = 50, 

 sample = 3, and two-sided limits ..........................................................................................82 

26b. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots with PD = 50, 

 sample = 5, and two-sided limits ..........................................................................................82 

26c. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots with PD = 50, 

 sample = 10, and two-sided limits ........................................................................................83 

27a. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots with PD = 10, 

 skewness = 1, and two-sided limits ......................................................................................84 

27b. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots with PD = 10, 

 skewness = 2, and two-sided limits ......................................................................................84 

28a. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots with PD = 30, 

 skewness = 1, and two-sided limits ......................................................................................85 

28b. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots with PD = 30, 

 skewness = 2, and two-sided limits ......................................................................................85 

29a. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots with PD = 50, 

 skewness = 1, and two-sided limits ......................................................................................86 

29b. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots with PD = 50, 

 skewness = 2, and two-sided limits ......................................................................................86 

30. Comparison of a normal population with a population with a skewness 

 coefficient = +1.0..................................................................................................................87 

31. Sample program output screen for a population with PD = 10, skewness 

coefficient = 0.00, and sample size = 5.................................................................................89 

32a. Portion of program output screens for PD = 10, skewness coefficient = 0.00, and 

 sample sizes = 3 ....................................................................................................................90 

32b. Portion of program output screens for PD = 10, skewness coefficient = 0.00, and 

 sample sizes = 5 ....................................................................................................................90 

32c. Portion of program output screens for PD = 10, skewness coefficient = 0.00, and 

 sample sizes = 10 ..................................................................................................................90 

33. Sample program output screen for a population with PD = 30, skewness 

 coefficient = 1.00, and sample size = 5.................................................................................91 

34a. Portion of program output screens for PD = 30, skewness coefficient = 1.00, and 

 sample sizes = 3 ....................................................................................................................92 

34b. Portion of program output screens for PD = 30, skewness coefficient = 1.00, and 

 sample sizes = 5 ....................................................................................................................92 

34c. Portion of program output screens for PD = 30, skewness coefficient = 1.00, and 

 sample sizes = 10 ..................................................................................................................92 

35a. Examples of shapes and actual AAD values for populations centered on the target 

 and skewness coefficient of 0 ...............................................................................................94

35b. Examples of shapes and actual AAD values for populations centered on the target 

 and with skewness coefficients between .5 and 1.5..............................................................94 
 

x 

35c. Examples of shapes and actual AAD values for populations centered on the target 

 and with skewness coefficients between 2.0 and 3.0............................................................94 

36a. Comparison of the shapes and spread of estimated AAD values for populations 

 centered on the target and with various skewness coefficients.............................................95 

36b. Comparison of the shapes and spread of estimated AAD values for populations 

 centered on the target and with various skewness coefficients, sample size = 3..................95 

36c. Comparison of the shapes and spread of estimated AAD values for populations 

 centered on the target and with various skewness coefficients, sample size = 5..................95 

36d. Comparison of the shapes and spread of estimated AAD values for populations 

 centered on the target and with various skewness coefficients, sample size = 10................96 

37. Comparison of the shapes and spread of estimated AAD values for normal 

 populations centered on and offset from the target...............................................................97 

38. Example 1 of populations that are very dissimilar in shape, but have approximately 

 the same AAD.......................................................................................................................98 

39a. Bias of the AAD sample estimates for populations centered on the 

 target, but with various levels of skewness.........................................................................101 

39b. Spread of the AAD sample estimates for populations centered on the 

 target, but with various levels of skewness.........................................................................101 

40a. Bias of the AAD sample estimates for normal populations with various 

 offsets from the target and n = 5 .........................................................................................103 

40b. Spread of the AAD sample estimates for normal populations with various 

 offsets from the target and n = 5 .........................................................................................103 

41a. Bias of the AAD sample estimates for normal populations centered on 

 the target, with various standard deviation values, and n = 5.............................................105 

41b. Spread of the AAD sample estimates for normal populations centered on 

 the target, with various standard deviation values, and n = 5.............................................105 

42. Sample output screen from the simulation program for bimodal distributions ..................108 

43a. Program output screens for sample size = 5 and mean offsets = 1.....................................109 

43b. Program output screens for sample size = 5 and mean offsets = 2.....................................109 

43c. Program output screens for sample size = 5 and mean offsets = 3.....................................110 

43d. Program output screens for sample size = 5 and mean offsets = 4.....................................110 

43e. Program output screens for sample size = 5 and mean offsets = 5.....................................111 

44a. Illustration 1 of program output screens when combining distributions with equal means for 

 sample size = 5....................................................................................................................114 

44b. Illustration 2 of program output screens when combining distributions with equal means for 

 sample size = 5....................................................................................................................114 

44c. Illustration 3 of program output screens when combining distributions with equal means for 

 sample size = 5....................................................................................................................115 

44d. Illustration 4 of program output screens when combining distributions with equal means for 

 sample size = 5....................................................................................................................115 

45. OC surface for the maximum allowable difference test method verification 

 method (assuming the smaller σ = σtest) .............................................................................125 

46. OC surface for the maximum allowable difference test method verification 

 method (assuming the smaller σ = 0.5σtest) ........................................................................125 

47. OC surface for the maximum allowable difference test method verification 

 method (assuming the smaller σ = 2σtest) ...........................................................................126 
 

xi 

48a. Example 1 of some of the cases considered in the average run length analysis for 

 the maximum allowable difference method........................................................................127 

48b. Example 2 of some of the cases considered in the average run length analysis for 

 the maximum allowable difference method........................................................................128 

48c. Example 3 of some of the cases considered in the average run length analysis for 

 the maximum allowable difference method........................................................................129 

49. OC curves for a two-sided t-test (α = 0.05)........................................................................133 

50. OC curves for a two-sided t-test (α = 0.01)........................................................................134 

51a. OC surfaces (also called power surfaces) for the appendix G method for 5 contractor 

 tests compared to a single agency test ................................................................................137 

51b. OC surfaces (also called power surfaces) for the appendix G method for 6 contractor 

 tests compared to a single agency test ................................................................................137 

51c. OC surfaces (also called power surfaces) for the appendix G method for 7 contractor 

 tests compared to a single agency test ................................................................................138 

51d. OC surfaces (also called power surfaces) for the appendix G method for 8 contractor 

 tests compared to a single agency test ................................................................................138 

51e. OC surfaces (also called power surfaces) for the appendix G method for 9 contractor 

 tests compared to a single agency test ................................................................................139 

51f. OC surfaces (also called power surfaces) for the appendix G method for 10 contractor 

 tests compared to a single agency test ................................................................................139 

52. OC curves for the two-sided F-test for level of significance α = 0.05...............................143 

53. OC curves for the two-sided F-test for level of significance α = 0.01...............................143 

54a. EP curves for PWL payment schedule with sample size = 3..............................................163 

54b. EP curves for PWL payment schedule with sample size = 5..............................................163 

54c. EP curves for PWL payment schedule with sample size = 10............................................163 

55a. Distribution of individual lot PWL = 90, sample = 3, and the 

 resulting payment factors....................................................................................................164 

55b. Distribution of individual lot PWL = 50, sample = 3, and the 

 resulting payment factors....................................................................................................164 

55c. Distribution of individual lot PWL = 90, sample = 5, and the 

 resulting payment factors....................................................................................................165 

55d. Distribution of individual lot PWL = 50, sample = 5, and the 

 resulting payment factors....................................................................................................165 

55e. Distribution of individual lot PWL = 90, sample = 10, and the 

 resulting payment factors....................................................................................................166 

55f. Distribution of individual lot PWL = 50, sample = 10, and the 

 resulting payment factors....................................................................................................166 

56. EP curve with the 90th and 10th payment percentiles for the AAD 

 payment schedule................................................................................................................168 

57. Illustration of measuring AAD in standard deviation units (ZTarg) from the mean............169 

58. Example illustrating the PWL specification limits and the offset in σ units 

 between the population mean and the target.......................................................................171 

59. EP curves for matched PWL and AAD payment equations for 

sample size = 5....................................................................................................................177 

60. Standard deviations for individual payment factors for matched PWL and AAD 

 payment equations for sample size = 5...............................................................................177 
 

xii 

61a. Illustration 1 of two normal variables with various values for the 

 correlation coefficient .........................................................................................................179 

61b. Illustration 2 of two normal variables with various values for the 

 correlation coefficient .........................................................................................................179 

61c. Illustration 3 of two normal variables with various values for the 

 correlation coefficient .........................................................................................................180 

61d. Illustration 4 of two normal variables with various values for the 

 correlation coefficient .........................................................................................................180 

61e. Illustration 5 of two normal variables with various values for the 

 correlation coefficient .........................................................................................................181 

61f. Illustration 6 of two normal variables with various values for the 

 correlation coefficient .........................................................................................................181 

61g. Illustration 7 of two normal variables with various values for the 

 correlation coefficient .........................................................................................................182 

61h. Illustration 8 of two normal variables with various values for the 

 correlation coefficient .........................................................................................................182 

62. Expected combined weighted average payment factors for various weights and 

 correlation coefficients based on PWL...............................................................................187 

63. Standard deviations of weighted average payment factors for various weights and 

 correlation coefficients based on PWL...............................................................................187 

64. Expected combined weighted average payment factors for various weights and 

 correlation coefficients based on AAD...............................................................................188 

65. Standard deviations of weighted average payment factors for various weights and 

 correlation coefficients based on AAD...............................................................................188 

66. Expected average payment factors for two populations with various correlation 

 coefficients based on PWL .................................................................................................191 

67. Expected average payment factors for two populations with various correlation 

 coefficients based on AAD .................................................................................................191 

68. Standard deviations of individual payment factors for two populations with various 

 correlation coefficients based on PWL...............................................................................192 

69. Standard deviations of individual payment factors for two populations with various 

 correlation coefficients based on AAD...............................................................................192 

70. Bias for the average payment for two populations with various actual PWL values .........195 

71. Bias for the average payment for two populations with various actual AAD values .........195 

72. Standard deviation for the individual average payment values for two populations 

 with various actual PWL values .........................................................................................196 

73. Standard deviation for the individual average payment values for two populations 

 with various actual AAD values .........................................................................................196 

74. OC curve for an acceptance plan that calls for rejection if the estimated PWL is less 

 than 60, for sample size = 4 ................................................................................................202 

75. OC curves for the probabilities of receiving at least some payment and at least 

 100-percent payment, sample size = 4................................................................................203 

76. OC curves for the probability of receiving various payments, sample size = 4 .................204 

77. EP curve for the payment relationship Pay = 55 + 0.5PWL, with an RQL provision, 

 sample size = 4....................................................................................................................205 
 

xiii 

78a. Distribution of estimated PWL values for an AQL population ..........................................206 

78b. Distribution of payment factors for an AQL population ....................................................206 

79. EP curve for the payment relationship Pay = 55 + 0.5PWL, sample size = 4 ....................206 

80. Distributions of sample PWL estimates for a population with 90 PWL.............................209 

81a. Distributions of sample PWL estimates for a population with 50 PWL and one-sided 

speculations.........................................................................................................................210 

81b. Distributions of sample PWL estimates for a population with 50 PWL and two-sided 

speculations.........................................................................................................................210 

82a. Distribution of sample standard deviations for a sample size, n = 3, based on 

 1000 simulated samples ......................................................................................................211 

82b. Distribution of sample standard deviations for a sample size, n = 5, based on 

 1000 simulated samples ......................................................................................................211 

82c. Distribution of sample standard deviations for a sample size, n = 10, based on 

 1000 simulated samples ......................................................................................................211 

83. EP contours for the values in table 53 ................................................................................215 

84. EP surface for the values in table 53...................................................................................216

85. EP curves for the values in table 53....................................................................................217

86. OC curve for an acceptance plan that calls for rejection if the estimated PWL 

 is less than 60, sample sizes = 5..........................................................................................218

87a. Simulation results of expected payment for the averaging method, combining 

 two populations with equal PWL values ............................................................................223 

87b. Simulation results of standard deviation values for the averaging method, combining 

 two populations with equal PWL values ............................................................................223 

88a. Simulation results of expected payment for the weighted average method, combining 

 two populations with equal PWL values ............................................................................224 

88b. Simulation results of standard deviation values for the weighted average method, 

combining two populations with equal PWL values ..........................................................224 

89a. Simulation results of expected payment for the multiplication method, 

 combining two populations with equal PWL values ..........................................................225 

89b. Simulation results of standard deviation values for the multiplication method, 

 combining two populations with equal PWL values ..........................................................225 

90a. Simulation results of expected payment for the summation method, 

 combining two populations with equal PWL values ..........................................................226 

90b. Simulation results of standard deviation values for the summation method, 

 combining two populations with equal PWL values ..........................................................226 

91a. Simulation results of expected payment for the maximum method, 

 combining two populations with equal PWL values ..........................................................227 

91b. Simulation results of standard deviation values for the maximum method, 

 combining two populations with equal PWL values ..........................................................227 

92a. Simulation results of expected payment for the minimum method, 

 combining two populations with equal PWL values ..........................................................228 

92b. Simulation results of standard deviation values for the minimum method, 

 combining two populations with equal PWL values ..........................................................228 

93a. Comparison of simulation results for various methods for combining individual 

 expected payment factors for two populations with PWL = 90..........................................230 
 

xiv 

93b. Comparison of simulation results for various methods for combining individual 

 standard deviation payment factors for two populations with PWL = 90 ..........................230 

94a. Comparison of simulation results for various methods for combining individual 

 expected payment factors for two populations with PWL = 70..........................................231 

94b. Comparison of simulation results for various methods for combining individual standard 

deviation factors for two populations with PWL = 70........................................................231 

95a. Comparison of simulation results for various methods for combining individual 

 expected payment factors for two populations with PWL = 50..........................................232 

95b. Comparison of simulation results for various methods for combining individual 

 standard deviation payment factors for two populations with PWL = 50 ..........................232 

96. Flowchart of the PRS process.............................................................................................236 

97. Illustration of the net impact of rescheduling an overlay 2 years earlier than 

 originally planned ...............................................................................................................240 

98. Spread of possible sample means for a normal distribution with 10 PD below the 

 lower specification limit and sample size = 3.....................................................................350 

99. Spread of possible sample means for a normal distribution with 10 PD below the 

 lower specification limit and sample size = 10...................................................................352 

100. Spread of possible sample means for a normal distribution with 5 PD outside 

each specification limit and sample size = 3.......................................................................353 

101. Spread of possible sample means for a normal distribution with 5 PD outside 

each specification limit and sample size = 10.....................................................................354 

102. Spread of possible sample means for a normal distribution with 50 PD below 

the lower specification limit and sample size = 3...............................................................355 

103. Spread of possible sample means for a normal distribution with 50 PD below 

the lower specification limit and sample size = 10.............................................................356 

104. Spread of possible sample means for a normal distribution with 25 PD outside 

each specification limit and sample size = 3.......................................................................357 

105. Spread of possible sample means for a normal distribution with 25 PD outside 

each specification limit and sample size = 10.....................................................................358 

106. Spread of possible sample means for a distribution with skewness = 1.0, 10 PD 

below the lower specification limit, and sample size = 3 ...................................................359 

107. Spread of possible sample means for a distribution with skewness = 1.0, 10 PD 

below the lower specification limit, and sample size = 10 .................................................360 

108. Spread of possible sample means for a distribution with skewness = 1.0, 5 PD 

outside each specification limit, and sample size = 3 .........................................................361 

109. Spread of possible sample means for a distribution with skewness = 1.0, 5 PD 

 outside each specification limit, and sample size = 10 .......................................................362 

110. Spread of possible sample means for a distribution with skewness = 1.0, 50 PD 

below the lower specification limit, and sample size = 3 ...................................................363 

111. Spread of possible sample means for a distribution with skewness = 1.0, 50 PD 

below the lower specification limit, and sample size = 10 .................................................364 

112. Spread of possible sample means for a distribution with skewness = 1.0, 25 PD 

 outside each specification limit, and sample size = 3 .........................................................365 

113. Spread of possible sample means for a distribution with skewness = 1.0, 25 PD 

 outside each specification limit, and sample size = 10 .......................................................366 
 

xv 

114. Sample output screen for a population with PD = 10, skewness coefficient = 0.00, 

 and sample size = 5.............................................................................................................368 

115. Portions of output screens for PD = 10, skewness coefficient = 0.00, and sample 

sizes = 3, 5, and 10..............................................................................................................369 

116. Sample output screen for a population with PD = 10, skewness coefficient = 1.00, 

 and sample size = 5.............................................................................................................370 

117. Portions of output screens for PD = 10, skewness coefficient = 1.00, and sample 

sizes = 3, 5, and 10..............................................................................................................371 

118. Sample output screen for a population with PD = 10, skewness coefficient = 2.00, 

and sample size = 5.............................................................................................................372 

119. Portions of output screens for PD = 10, skewness coefficient = 2.00, and sample 

 sizes = 3, 5, and 10..............................................................................................................373 

120. Sample output screen for a population with PD = 30, skewness coefficient = 0.00, 

 and sample size = 5.............................................................................................................374 

121. Sample output screen for a population with PD = 50, skewness coefficient = 0.00, 

 and sample size = 5.............................................................................................................375 

122. Sample output screen for a population with PD = 30, skewness coefficient = 0.00, 

 and sample size = 10...........................................................................................................376 

123. Sample output screen for a population with PD = 50, skewness coefficient = 0.00, 

 and sample size = 10...........................................................................................................377 

124. Portions of output screens for PD = 50, sample size = 5, and skewness coefficients = 

 0.00, 1.00, 2.00, and 3.00....................................................................................................378 

125. Sample output screen for a population with PD = 10, skewness coefficient = 3.00, 

 and sample size = 5.............................................................................................................379 

126. Sample output screen for a population with PD = 30, skewness coefficient = 3.00, 

 and sample size = 5.............................................................................................................380 

127. Sample output screen for a population with PD = 50, skewness coefficient = 3.00, 

 and sample size = 5.............................................................................................................381 

128a. Distribution of sample means for 1000 samples from a normal population 

 with μ = 0.00, σ =1.00.........................................................................................................385 

128b.Distribution of standard deviations for 1000 samples from a normal population 

 with μ = 0.00, σ =1.00.........................................................................................................385 

129a. Distribution of sample PWL values for 1000 samples from a normal population 

 with PWL = 90, one-sided specifications ...........................................................................386 

129b.Distribution of sample PWL values for 1000 samples from a normal population 

 with PWL = 90, two-sided specifications...........................................................................386 

130a. Distribution of sample PWL values for 1000 samples from a normal population 

 with PWL = 80, one-sided specifications ...........................................................................387 

130b.Distribution of sample PWL values for 1000 samples from a normal population 

 with PWL = 80, two-sided specifications...........................................................................387 

131a. Distribution of sample PWL values for 1000 samples from a normal population 

 with PWL = 70, one-sided specifications ...........................................................................388 

131b.Distribution of sample PWL values for 1000 samples from a normal population 

 with PWL = 70, two-sided specifications...........................................................................388 

132a. Distribution of sample PWL values for 1000 samples from a normal population 

 with PWL = 60, one-sided specifications ...........................................................................389 
 

xvi 

132b.Distribution of sample PWL values for 1000 samples from a normal population 

 with PWL = 60, two-sided specifications...........................................................................389 

133a. Distribution of sample PWL values for 1000 samples from a normal population 

 with PWL = 50, one-sided specifications ...........................................................................390 

133b.Distribution of sample PWL values for 1000 samples from a normal population 

 with PWL = 50, two-sided specifications...........................................................................390 

134. Illustration of the populations for which the distributions of sample PWL values are 

 shown in figures 129 through 133 ......................................................................................391 
 

xvii 

List of Tables 

Table Page 

1. Project team ............................................................................................................................3 

2. Panel members........................................................................................................................4 

3. Agencies that provided copies of their specifications...........................................................10 

4. Survey results for the first ranking method...........................................................................27 

5. Survey results for the second ranking method......................................................................28 

6. Overall rankings of the survey topics ...................................................................................30 

7. Quality index values for estimating PWL.............................................................................35 

8. Accuracy and precision for PWL estimates (based on the results of 10,000 

simulated lots).......................................................................................................................38 

9. Results of simulation analyses with actual PWL = 90 (distribution of the results 

 from 1000 simulated projects) ..............................................................................................44 

10. Results of simulation analyses with actual PWL = 70 (distribution of the results 

 from 1000 simulated projects) ..............................................................................................45 

11. Results of simulation analyses with actual PWL = 50 (distribution of the results 

 from 1000 simulated projects) ..............................................................................................46 

12. Distributions of sample AAD values for a population centered on the target and 

 for sample sizes = 1, 3, 5, and 10..........................................................................................51

13. Distributions of sample AAD values for sample size = 3 and population means 

offset from the target by 0.50, 1.00, 1.50, 2.00, and 2.50 standard deviations.....................52 

14. Accuracy and precision for AAD estimates (based on the results of 10,000 

simulated lots).......................................................................................................................54 

15. Accuracy and precision for CI estimates (based on the results of 10,000 

simulated lots).......................................................................................................................57 

16. Distributions with various levels of skewness......................................................................60 

17. Bias in estimating PWL for various skewness coefficients and one-sided limits 

 (3 tests per lot and 10,000 simulated lots) ............................................................................62 

18. Bias in estimating PWL for various skewness coefficients and one-sided limits 

 (5 tests per lot and 10,000 simulated lots) ............................................................................63 

19. Bias in estimating PWL for various skewness coefficients and one-sided limits 

 (10 tests per lot and 10,000 simulated lots) ..........................................................................64 

20. Bias in estimating PWL for various skewness coefficients, sample sizes, and 

 one-sided limits (10,000 simulated lots)...............................................................................68 

21. Results of simulations with PD = 10, 10,000 simulated lots, sample sizes = 

 3, 5, and 10, and two-sided limits .........................................................................................75 

22. Results of simulations with PD = 30, 10,000 simulated lots, sample sizes = 

 3, 5, and 10, and two-sided limits .........................................................................................76 

23. Results of simulations with PD = 50, 10,000 simulated lots, sample sizes = 

 3, 5, and 10, and two-sided limits .........................................................................................77 

24. Bias and spread of the AAD sample estimates for populations centered on the target, 

 but with various levels of skewness....................................................................................100 

25. Bias and spread of the AAD sample estimates for normal populations with various 

 offsets from the target and n = 5 .........................................................................................102 
 

xviii 

26. Bias and spread of the AAD sample estimates for normal populations with various 

 standard deviation values and n = 5....................................................................................104 

27. Bias results for combining two populations with equal σ with one-sided limits................112 

28. Bias results for combining two populations with equal σ with two-sided limits 

 equidistant from the mean...................................................................................................113 

29. Shapes of combined distributions when the combined distributions have different 

 means and standard deviations............................................................................................117 

30. Bias in estimating PD when two populations are combined (sample size = 5, 10,000 

simulated lots).....................................................................................................................118 

31. Average run length results for the single split-sample method (5000 simulated lots)........130 

32. Allowable intervals for the AASHTO appendix G method................................................135 

33. Average run length results for the appendix G method (5000 simulated lots) ...................140 

34. F-test power values for n = 3-10 and s-ratio λ = 1-5..........................................................144 

35. F-test power values for n = 3-10 and s-ratio λ = 0-1..........................................................147 

36. F-test power values for n = 5-100 and s-ratio λ = 1-3........................................................150 

37. Average run length results for the appendix H method (5 contractor tests and 

 1 agency test per lot) for 1000 simulated lots.....................................................................157 

38. Simulated EP factors and correct payment factors based on AAD ....................................168 

39. Relationship between ZTarg, AAD, and PWL for a normal population when the PWL 

specification limits are set at 1.645σ ..................................................................................172 

40. Relationship between AAD, ZTarg, and PWL for a normal population when the PWL 

specification limits are set at 1.645σ ..................................................................................173 

41. Relationship between PWL, AAD, and ZTarg for a normal population when the PWL 

specification limits are set at 1.645σ ..................................................................................174 

42. AAD values equivalent to the corresponding PWL payment factors.................................175 

43. Results of the simulation of matched PWL and AAD payment equations for 

 sample size = 5....................................................................................................................176 

44. Demonstration of the simulation of three correlated normal variables with selected 

 values for correlation coefficients.......................................................................................183 

45. Demonstration of the simulation of four correlated normal variables with selected 

 values for correlation coefficients.......................................................................................184 

46. Results of PWL simulation analyses for two correlated normal variables .........................186 

47. Results of AAD simulation analyses for two correlated normal variables.........................186 

48. Bias in EP for two populations with equal PWL values and various correlation 

 coefficients..........................................................................................................................193 

49. Bias in EP for two populations with equal AAD values and various correlation 

 coefficients..........................................................................................................................193 

50. Standard deviations of individual payment factors for two populations with equal 

 PWL values and various correlation coefficients ...............................................................194 

51. Standard deviations of individual payment factors for two populations with equal 

AAD values and various correlation coefficients ...............................................................194 

52. EP values using Pay = 55 + 0.5PWL for two individual payment factors and then 

 averaging them, sample size = 5.........................................................................................214 

53. EP values using Pay = 55 + 0.5PWL for two individual payment factors and then 

multiplying them, sample size = 5, correlation coefficient = +0.5 .....................................215 
 

xix 

54. Probabilities that populations with various quality levels would require removal 

and replacement for the example in figure 86 ....................................................................219 

55. Completed data matrix for the example of an exponential model......................................242 

56. Examples of computed PD* values for selected individual PD values..............................243 

57. Illustration of the problem with separate RQL provisions..................................................244 

58. Alaska weight factors..........................................................................................................264 

59. Alaska QC/QA tests............................................................................................................264 

60. Arkansas QC/QA tests ........................................................................................................266 

61. Colorado pay factor equations ............................................................................................268 

62. Colorado QC/QA tests ........................................................................................................268 

63. Idaho QC/QA tests..............................................................................................................270 

64. Illinois smoothness pay factors...........................................................................................272 

65. Illinois QC/QA tests............................................................................................................272 

66. Iowa pay factors for density................................................................................................273 

67. Iowa pay factors for thickness ............................................................................................273 

68. Iowa QC/QA tests...............................................................................................................274 

69. Maryland profile index adjustment (normal projects) ........................................................276 

70. Maryland profile index adjustment (incentive projects).....................................................276 

71. Maryland QC/QA tests .......................................................................................................276 

72. Michigan QC/QA tests........................................................................................................277 

73. Minnesota payment schedule..............................................................................................278

74. Minnesota determination of lots for density .......................................................................278 

75. Minnesota adjusted payment schedule for maximum density (disincentive) .....................279 

76. Minnesota adjusted payment schedule for maximum density (incentive)..........................279 

77. Minnesota payment schedule for smoothness.....................................................................279 

78. Minnesota QC/QA tests ......................................................................................................280 

79. Montana price reduction factors .........................................................................................281

80. Montana QC/QA tests.........................................................................................................282 

81. Nebraska density of asphalt concrete (first lot) ..................................................................283 

82. Nebraska density of asphalt concrete (subsequent lots) .....................................................283 

83. Nebraska QC/QA tests........................................................................................................284 

84. Nevada pay factors for profile index ..................................................................................285 

85. Nevada pay factors for gradation........................................................................................286

86. Nevada pay factors for asphalt content...............................................................................286 

87. Nevada pay factors for in-place air voids ...........................................................................286 

88. Nevada QC/QA tests...........................................................................................................287 

89. North Dakota QC/QA tests .................................................................................................288 

90. Ohio QC/QA tests...............................................................................................................289 

91. Ontario QC/QA tests...........................................................................................................290 

92. Oregon weighting factors....................................................................................................291 

93. Oregon QC/QA tests...........................................................................................................292 

94. Pennsylvania adjustment of contract price relative to the specification limits...................293 

95. Pennsylvania QC/QA tests..................................................................................................294 

96. South Carolina required QC tests and verifications............................................................295 

97. South Carolina required acceptance tests ...........................................................................296 

98. Texas pay factors for flexural strength ...............................................................................298 
 

xx 

99. Texas pay factors for thickness...........................................................................................298 

100. Texas QC/QA tests .............................................................................................................299 

101. Virginia QC/QA tests..........................................................................................................300 

102. Washington QC/QA tests....................................................................................................301 

103. Wisconsin sampling frequencies.........................................................................................302

104. Wisconsin percent payment for mixture.............................................................................302 

105. Wisconsin QC/QA tests ......................................................................................................303 

106. Wyoming QC/QA tests.......................................................................................................305 

107. Connecticut M.A.D. from job-mix formula for consecutive tests ......................................308 

108. Connecticut pay factors for joint and mat density ..............................................................308 

109. Connecticut QC/QA tests....................................................................................................309 

110. Kansas pay factors for specified density.............................................................................311 

111. Kansas pay factors for air voids (lot size of four tests).......................................................312 

112. Kansas pay factors for air voids (lot size of three tests) .....................................................312 

113. Kansas pay factors for air voids (lot size of five tests).......................................................313 

114. Kansas pay factors for air voids (lot size of six tests).........................................................313 

115. Kansas QC/QA tests ...........................................................................................................314 

116. Louisiana payment adjustment schedule ............................................................................316 

117. Louisiana QC/QA tests .......................................................................................................316 

118. Maine QC/QA tests.............................................................................................................318 

119. Minnesota payment schedule..............................................................................................319 

120. Minnesota determination of lots for density .......................................................................320 

121. Minnesota payment schedule for density............................................................................320 

122. Minnesota payment schedule for smoothness.....................................................................320 

123. Minnesota QC/QA tests ......................................................................................................321 

124. Mississippi sampling frequency..........................................................................................322 

125. Mississippi lot determination for density............................................................................322 

126. Mississippi pay factors for mixture quality ........................................................................323 

127. Mississippi pay factors for density .....................................................................................323 

128. Mississippi pay factors for smoothness ..............................................................................323 

129. Mississippi QC/QA tests.....................................................................................................324 

130. New York QC/QA tests ......................................................................................................326 

131. North Carolina testing frequency........................................................................................327 

132. North Carolina payment for mix produced in warning bands ............................................327 

133. North Carolina pay factor ...................................................................................................328 

134. North Carolina QC/QA tests...............................................................................................328 

135. Kansas concrete thickness and comprehensive strength pay adjustment ...........................329 

136. Kansas QC/QA tests ...........................................................................................................330 

137. Illinois QC/QA tests............................................................................................................332 

138. Iowa pay factors for thickness ............................................................................................333 

139. Iowa pay factors for flexural strength.................................................................................334

140. Iowa QC/QA tests...............................................................................................................334 

141. North Carolina pay factors for thickness ............................................................................336 

142. North Carolina pay factors for flexural strength.................................................................336 

143. North Carolina QC/QA tests...............................................................................................337 

144. Oregon pay factors for thickness ........................................................................................338 
 

xxi 

145. Oregon QC/QA tests...........................................................................................................339 

146. Pennsylvania QC/QA tests..................................................................................................340 

147. Texas pay factors for flexural strength ...............................................................................341 

148. Texas pay factors for thickness...........................................................................................341 

149. Texas QC/QA tests .............................................................................................................342 

150. Wisconsin pay factors for compressive strength ................................................................343 

151. Wisconsin pay factors for the profile index........................................................................344 

152. Wisconsin pay factors for thickness ...................................................................................344 

153. Wisconsin QC/QA tests ......................................................................................................344 

 1 

1. INTRODUCTION 

BACKGROUND 

The majority of State highway administrations (SHAs) now employ statistical quality assurance 

(QA) specifications to some degree. These specifications contain statistical acceptance plans that 

serve to inform a contractor of at least three items: (1) the quality level that the agency desires, 

(2) how the contractor’s submitted quality level will be determined (i.e., estimated), and (3) the 

consequences for the contractor when the submitted quality level estimate is below or above the 

desired level. Whether the acceptance plan leads to simple pass/fail decisions or adjustments in 

contract payment, its proper development is critical for the plan to be effective. 

The development of statistical acceptance plans for highway construction requires a good 

understanding of statistics, materials and construction variability, and the product 

quality/performance/cost interrelationship. Currently, however, acceptance plan designers must 

make some design decisions relying more on intuition than on established engineering and 

economic principles. One problem, of course, is that there is much that is still unknown about 

highway quality, variability, performance, and cost. Some long-term research into these topics is 

currently underway. Nonetheless, designers must make do with whatever information, 

instructions, and resources are currently available when developing new, or revising existing, 

acceptance plans. 

Although much good information exists for acceptance plan design, there is also much 

misinformation and confusion. This has resulted in a general lack of uniformity among highway 

agency acceptance plans; use of acceptance plans that range from totally ineffective to 

impractically severe; difficulties in evaluating the effectiveness of some nonstandard acceptance 

plans; and, presumably, general dissatisfaction as evidenced by frequent revisions in agency 

acceptance plans (these major revisions often have a significant impact on construction). It 

would help greatly if the acceptance plan designer had clear, specific, supported, and 

comprehensive guidance for developing acceptance plans based on best practices and on what is 

currently known. The guidance should also, where possible, take the subjectivity out of 

acceptance plan design and replace it with rational and defensible scientific procedures. 

As part of a pooled fund study, 18 SHAs and 1 Canadian Province provided funds for this 

project. The agencies that provided funding for this study are shown in figure 1. 

PROJECT OBJECTIVE 

The objective of the project was to develop a comprehensive manual, supported by scientific 

evidence and statistical theory, that provides step-by-step procedures and instructions for 

developing effective and efficient QA specifications. 
 2 

Figure 1. States that provided funding for the study. 

METHODOLOGY 

For the project, a team was assembled that had extensive experience in the area of QA and 

statistical applications to highway materials. The team included individuals with extensive 

experience in the development and analysis of QA and acceptance procedures. A statistician was 

also included to assist in the statistical analyses of the various procedures to be considered. The 

members of the project team are identified in table 1. 

To oversee the project, a panel (subsequently referred to as the panel) was formed that consisted 

of one representative from each of the agencies that contributed funding for the study. The 

contracting officer’s technical representative (COTR) for the Federal Highway Administration 

(FHWA) was also a member of the panel. The panel member for a few of the States changed 

over the course of the project. Table 2 shows the members comprising the panel that approved 

the QA specifications manual that was developed. 

A number of tasks were completed in the process of accomplishing the goal of a step-by-step 

manual for developing QA specifications. The methodologies for accomplishing these various 

tasks are discussed in the following sections. 

Ontario

WA

IA

KS

NC

SC

TX LA

OR

NV IL

WI

MN

NJ

CT

PA

NY

ID

VA
 3 

Table 1. Project team.

Member Experience Areas of Expertise 

James L. Burati, Jr., Ph.D. 

Project Principal 

Investigator 

25 years 

QA Specifications, Evaluation of Asphalt 

Concrete (AC) Mixtures, AC Pavement 

Construction, Total Quality Management 

(TQM), Statistical Applications in Highway 

Construction, Computer Simulation 

Hoke S. Hill, Jr., Ph.D. 

Project Statistical Consultant 23 years 

Sampling Theory, Statistical Analysis, 

Statistical Graphics, Statistical Computing, 

Experiment Design 

Richard M. Weed, P.E. 

Project QA Consultant 40 years 

Statistical QA, Computer Simulation, 

Construction Specifications Development, 

Applications of Statistical Theory to 

Engineering Problems, Portland Cement 

Concrete (PCC) Specifications 

Charles S. Hughes, P.E. 

Project QA Consultant 40 years 

AC Mix Design, QA Specifications, 

Pavement Performance, Risk Assignment 

and Assessment, Asphalt Pavement 

Construction 

Review of the Literature 

The project team members have written a good deal of the highway materials and construction 

QA literature. So, the resumes of the project team were reviewed and a list of their publications 

relating to the proposed research was compiled. In addition, a number of computer database 

searches were conducted. The computer search capability of Clemson University’s Cooper 

Library and Web-based search approaches were employed. 

The abstracts obtained from the database searches were reviewed and the full publications were 

obtained for those abstracts that appeared to be most appropriate for further study for the current 

project. 

In addition, all SHAs were contacted to request copies of their specifications for hot-mix asphalt 

concrete (HMAC) and portland cement concrete (PCC). Specifications were obtained from 31 

agencies, including most of the pooled fund States for this project. 
 4 

Table 2. Panel members.

Panel Member Representing 

Chris Abadie Louisiana Department of Transportation and 

Development 

Roger Apple Pennsylvania Department of Transportation 

(DOT) 

Ataur Bacchus Ontario Ministry of Transportation (MOT) 

Kevin Dayton Washington State DOT 

Steve DeWitt North Carolina DOT 

Doug Dirks Illinois DOT 

Milton Fletcher South Carolina DOT 

Steve Gage Connecticut DOT 

Jeff Hale Nevada DOT 

Kurt Johnson Wisconsin DOT 

James Klessig Minnesota DOT 

Peter Kopac FHWA 

Rick Kreider Kansas DOT 

Bill Maupin Virginia Transportation Research Council 

Garth Newman Idaho Transportation Department 

Thomas Reis Iowa DOT 

Deniz Sandhu New York State DOT 

Jeffrey Seiders Texas DOT 

Ken Stoneman Oregon DOT 

Richard Weed New Jersey DOT 

Development of the Process 

One of the most important tasks for the project was to identify the necessary steps (and the 

specific options available at each step) for a highway agency to comprehensively develop a QA 

specification. To help accomplish this task, the information collected during the literature search 

was sorted and synthesized. The project team was also able to call on their many collective years 

of experience in the development and analysis of QA specifications. 

The entire project team met in Clemson, SC, to develop the structure for the QA specifications 

development, implementation, and monitoring process. At this meeting, a detailed flowchart was 

developed and the major discussion points to accompany the flowchart were identified. 
 5 

Subsequently, Charles Hughes developed the initial draft of an executive summary to accompany 

and describe the elements of the process flowchart. The executive summary and flowchart were 

distributed to the panel for review. The flowchart was then reviewed and discussed at a meeting 

attended by the panel members and members of the project team. After a great deal of 

discussion, some revisions were agreed upon and were subsequently incorporated into the final 

flowchart. 

Analysis of Specific Options 

To a great extent, the approved QA specifications development process flowchart identified the 

various analyses that needed to be conducted. A number of decisions needed to be made to 

progress through the flowchart. Each of these decisions had several possible options from which 

to select. These options were analyzed to determine which one was the best approach to 

recommend. 

The analysis techniques used depended on the decision that needed to be made. Both analytical 

and computer simulation approaches were used. In some cases, such as with the risks associated 

with certain hypothesis tests, an analytical evaluation was relatively simple to use. In many 

cases, however, a computer simulation approach was the best (and, in some cases, the only) 

analysis method to use. This is particularly true for issues that are related to multiple quality 

characteristics. 

The specific analyses that were conducted, and the decisions that were made based on these 

analyses, are covered in detail in the various chapters of this report. 

Development of the Manual 

The principal product of this project, as stated in the initial request for proposal (RFP), is 

... a comprehensive manual that a highway agency can use when developing new, or 

modifying existing, acceptance plans. The draft manual shall provide all necessary 

instructions and illustrative examples to clearly lead the agency through the entire process of 

acceptance plan development, including: 

a. Setting up the initial data collection/experimentation to determine typical parameters of 

current construction. 

b. Establishing the desired level of quality to be specified. 

c. Designing the actual acceptance plan itself, including selecting the statistical quality 

measure, buyer’s and seller’s risk, lot size, number of samples, specification limits, and payadjustment provision. 

d. Monitoring how the acceptance plan is performing. 

e. Making necessary adjustments. 

It is important to note that two documents have been prepared for the project—a manual for 

SHAs and this technical report, which summarizes the procedures and findings of the project.(1)

While the focus and objectives of these documents are quite different, they are not entirely standalone documents. In preparing the two documents, an attempt has been made to minimize 
 6 

duplication of coverage. The technical report should be read in conjunction with and as a 

companion to the QA specifications manual that also resulted from the project. 

The QA specifications development manual is directed toward SHAs that have a need to develop 

new QA specifications, or revise or update existing specifications. The focus of the manual is on 

what to do when developing QA specifications. The reasons for the various steps and possible 

decisions are explained, and easy-to-follow examples are included to assist in understanding the 

process that is involved. The manual does not explain what was done during the project, nor 

what analytical and simulation analyses were conducted unless it was necessary to clarify why 

certain steps in the process are necessary. 

Individual members of the project team were assigned the task of preparing the first draft of 

specific portions of the manual. The principal investigator then collected the individual drafts 

and edited them into a cohesive draft manual for review by the COTR and the panel members. A 

panel meeting was then held to review and discuss the final draft. Comments and suggestions 

resulting from the panel meeting were incorporated into the final manual. 

ORGANIZATION OF THIS REPORT 

Chapter 2 presents a brief overview of the literature review, with primary emphasis placed on the 

specifications obtained from the various agencies that responded to the request for information. 

Chapter 3 presents the flowchart of the QA specifications development process that guided the 

specific analyses that were conducted to determine and support the recommendations that are 

made in the QA specifications manual. The chapters that follow the flowchart present the results 

and conclusions from the various analyses. 
 7 

2. LITERATURE AND SPECIFICATION REVIEW 

LITERATURE REVIEW 

One of the initial tasks associated with the project was to conduct a literature search to determine 

what information was currently available on the topic of QA specifications and to determine its 

potential relevance to the project. This task was an iterative process requiring multiple searches 

as the project moved forward. To encompass all of the potentially pertinent information 

available, a methodical approach to the literature search was adopted. 

Search Methodology 

Since the project team members have written extensively in the area of statistically based and 

QA specifications, the initial step in the literature search was to locate and review all of the 

information previously published by members of the research team. This yielded many published 

documents on such topics as sampling schemes, statistical specifications, incentive and 

disincentive payment schedules, acceptance procedures, statistical methods for analyzing data, 

and the construction of operating characteristic (OC) curves. Upon completion of this phase of 

the search, efforts were then directed toward outside sources of information. 

The next source of information that was explored was all of the publications contained in 

Clemson University’s Cooper Library databases. The databases were explored using various 

combinations of keywords, as well as subject searches. This generated numerous potential 

sources of information. Each possible source was reviewed and either added to the list of source 

documents or discarded based on the relevance of the information to the project. Upon 

completion of this phase, efforts were then directed toward national and international databases. 

Searching these databases by keyword and subject yielded hundreds of potential titles. Among 

the databases searched were: 

• TRIS (Transportation Research Information Service). 

• Ei Compendex (engineering information). 

• Dissertation Abstracts (online database of doctoral dissertations). 

• ICONDA (International Construction Database). 

• NTIS (National Technical Information Service). 

• MathSci (American Mathematical Society). 

• Engineering Materials Abstracts (engineering materials). 

• Federal Research in Progress (ongoing Federal research projects). 

The MathSci database was included to identify related articles on topics such as sampling plans, 

acceptance plans, etc., from mathematical and statistical sources such as the Journal of Quality 

Technology and the Journal of the American Statistical Association. 

All of the titles generated from this search were examined to determine if the publications should 

be located and obtained. Where the titles did not provide a clear indication as to the relevance of 
 8 

the publication, abstracts were requested to determine if the publication was an appropriate 

source document. 

The final phase of the literature search was to review the bibliographic contents of the collected 

articles to determine if publications cited in the current source documents would also be relevant 

to the research project. This phase yielded some additional sources. 

Results 

The information contained in the bibliographic sources can be separated into three major 

categories: 

• Sampling plans. 

• Price- or payment-adjustment plans. 

• QA specifications and acceptance plans. 

Sampling Plans: The bibliographic sources contain several different types of sampling plans. 

Some of the plans are currently in use for highway construction, while others were evaluated to 

determine whether there is an appropriate use for the plans in QA specifications. Sampling plans 

for which literature was reviewed included: 

• Stratified random sampling. 

• Random sampling. 

• Zero acceptance number sampling plans. 

• Variables sampling plans. 

• Attributes sampling plans. 

• Attributes single sampling plans. 

• Attributes double sampling plans. 

• Attributes multiple sampling plans. 

• Attributes proportion sampling plans. 

• Inverse Gaussian acceptance sampling plans. 

• Conditional sampling plans. 

• Bayesian acceptance sampling schemes. 

Price- or Payment-Adjustment Plans: The bibliographic sources also contain a number of 

articles discussing price- and payment-adjustment plans. Again, some of the practices discussed 

are currently in use, while others were examined for potential relevance to the project. Several 

articles discuss adjusted payment schedules as they pertain to the quality of work and expected 

performance, and whether current practices are fair or can be improved. The development of 

price-adjustment systems is also addressed from serviceability, cost of production, value concept, 

and OC curve approaches. The concept of composite payment equations and their development 

and use is addressed in several of the bibliographic sources. 

QA Specifications and Acceptance Plans: The last category of bibliographic sources is related 

to specifications development, acceptance procedures, and QA programs. Several articles 

investigate the older prescriptive highway specifications and discuss the shift toward end-result 
 9 

and performance-related specifications (PRS). Included is an examination of several statistical 

measures and how each has been used. 

Summary 

Not much new information was learned from the literature review. The project team members 

had written much of the relevant literature from the highway materials and construction industry. 

Most of the material from outside the highway industry was directed toward industrial 

applications, where more stable processes and much larger sample sizes are involved. Some of 

the relevant information from the literature search is discussed in subsequent chapters related to 

the specific analysis topics, and some is presented in the QA specifications manual that was 

developed for the project. 

A summary of some of the documents that were reviewed during the course of the search is 

presented as an annotated bibliography in appendix A. 

SPECIFICATION REVIEW 

At the beginning of the project, all SHAs were requested to provide copies of their current 

specifications for HMAC and PCC pavements. In response, 23 provided copies of their 

specifications for HMAC, 8 provided specifications for Superpave® AC mixes, and 9 provided 

specifications for PCC (see table 3). Many States indicated that Superpave specifications were 

under development at that time. The reasons for collecting and reviewing the various agency 

specifications were to: 

• Determine what properties were being evaluated for quality control (QC) purposes. 

• Determine what properties were being evaluated for acceptance purposes. 

• Determine what properties were being evaluated for payment determination. 

• Determine what statistical methods were being used to determine a composite payment 

factor when there are multiple quality characteristics. 

HMAC Specifications 

A total of 23 agencies (see table 3) provided copies of their current HMAC specifications. A 

summary of each agency’s HMAC specifications is provided in appendix B. While current at the 

time they were obtained, these specifications may no longer be current as of the writing of this 

report. Each of the agency’s specifications was carefully reviewed to determine the similarities 

and differences in the properties evaluated for QC, acceptance, and payment determination. 

Additionally, the statistical measures for determining compliance and the calculations for 

payment factors were reviewed. This task was complicated and required many iterations because 

of the differing terminologies used by the various agencies. For example, some agencies use 

assurance testing for the same purpose as tests that are referred to as acceptance testing by other 

agencies. 
 10 

Table 3. Agencies that provided copies of their specifications. 

Agency HMAC 

Specification 

Superpave 

Specification PCC Specification 

Alaska X 

Arkansas X 

Colorado X 

Connecticut X 

Idaho X 

Illinois X X 

Iowa X X 

Kansas X X 

Louisiana X 

Maine X 

Maryland X 

Michigan X 

Minnesota X X 

Mississippi X 

Montana X 

Nebraska X 

New Jersey X 

New York X 

North Carolina X X 

North Dakota X 

Nevada X 

Ohio X 

Ontario X 

Oregon X X 

Pennsylvania X X 

South Carolina X 

Texas X X 

Virginia X 

Washington X 

Wisconsin X X 

Wyoming X 

Total 23 8 9 
 11 

Examination of the specifications revealed that the majority of the agencies use the Marshall mix 

design and therefore the quality characteristics evaluated for QC and acceptance are similar. The 

most significant difference is in the number of quality characteristics that the contractor is 

responsible for controlling. Some agencies require the contractor to control a few common 

characteristics, such as gradation, asphalt content, density, voids in the mineral aggregate 

(VMA), voids filled with asphalt (VFA), and total air voids. Almost all of the agencies use these 

characteristics for QC or acceptance. However, some agencies require control over many more 

characteristics, including Hveem stability, Marshall stability, Marshall flow, dust-to-asphalt 

ratio, maximum specific gravity (MSG), bulk specific gravity (BSG), moisture content, binder 

temperature, liquid limit, plastic index, fractured faces, absorption, indirect tensile strength 

(ITS), and tensile strength ratio (TSR). Additionally, differences in the lot sizes for testing varied 

widely from agency to agency. Testing frequencies are also significantly different for the various 

agencies. This review indicated that with the exception of a few commonly measured 

characteristics, the QC and acceptance procedures varied widely among the responding agencies. 

The methods for determining acceptance were also investigated. The method for acceptance of 

material varies from agency to agency, but can be grouped into four general categories: 

• Acceptance testing by the department. 

• Verification of the contractor’s QC tests by the department’s assurance tests. 

• Acceptance testing by the contractor under departmental supervision. 

• Some combination of contractor and departmental testing. 

The final aspects of the specifications examined were the properties evaluated and the methods 

for determining payment factors. Most of the responding agencies evaluate only a few properties 

for determining payment factors. The most common properties used are gradation, in-place 

density, asphalt content, VMA, and air voids. Additional properties evaluated by some agencies 

include Marshall stability, crushed particle count, thickness, moisture content, theoretical 

maximum density (TMD), laboratory-molded density, and smoothness. 

Superpave Specifications 

The eight States that responded with copies of their Superpave specifications are indicated in 

table 3. The response for this specification was low at the time because of its recent introduction 

as a method of mix design. A summary of each agency’s Superpave specifications is provided in 

appendix C. The Superpave specifications were reviewed for QC, acceptance, and payment 

factor information. 

Examination of the specifications indicated that verification of the mix design is similar for all of 

the agencies. Additionally, the quality characteristics evaluated for QC and acceptance do not 

differ substantially from agency to agency. However, there is a significant difference in the 

quality characteristics evaluated for acceptance. Most of the agencies evaluate the following 

characteristics: asphalt content, gradation, air voids, VMA, and in-place density. In addition to 

these commonly evaluated characteristics, three agencies evaluate mix moisture, VFA, and BSG, 

and two agencies evaluate TMD, dust-to-asphalt ratio, and Gmb @ Ndes. At least one agency 

evaluates a number of other quality characteristics, such as TSR, sand equivalent, percent 
 12 

crushed aggregate, Nini, Ndes, and Nmax. Four of the eight agencies use smoothness as an 

acceptance quality characteristic. 

The methods used to determine acceptance can be grouped into two categories: acceptance 

testing by the agency and verification of the contractor’s tests by the agency’s verification tests. 

The most common quality characteristics used in determining payment factors are gradation, 

asphalt content, air voids, in-place density, and smoothness. 

PCC Specifications 

The nine agencies that provided copies of their PCC specifications are indicated in table 3. A 

summary of each agency’s PCC specifications is provided in appendix D. The review of the PCC 

specifications revealed similarities in the quality characteristics that the contractor is responsible 

for controlling. Each of the agencies requires the contractor to conduct QC tests for aggregate 

gradation, air content, slump, unit weight/yield, and compressive or flexural strength. A majority 

of the agencies also require the contractor to control thickness, temperature, and smoothness. 

Additional characteristics evaluated by at least one agency include water-cement ratio, percent 

passing the 75-micrometer (μ m) sieve, moisture content of the aggregate, fineness modulus, 

sand equivalent, fine aggregate organic impurity, and admixture dosage. The QC testing 

frequency requirements vary widely from agency to agency. 

The quality characteristics evaluated for the acceptance of material vary only slightly from 

agency to agency. Most of the agencies evaluate aggregate gradation, slump, temperature, 

smoothness, unit weight/yield, thickness, air content, and compressive or flexural strength. The 

method for determining the acceptance of material varies from agency to agency, with either 

agency verification testing of the contractor’s tests or acceptance testing by the agency. The 

responding agencies assigned payment factors for one or more of the following quality 

characteristics: smoothness, thickness, air content, compressive strength, and flexural strength.

Summary 

While the review of agency specifications provided background information, not a great deal of 

new and useful information resulted from the review. The project team members were already 

familiar with many agency specifications. Also, at the panel meeting to approve the QA 

specification process flowchart (see next chapter), the panel decided that the QA manual would 

not be prescriptive with regard to which quality characteristics should be measured for QC, 

acceptance, or payment purposes. 
 13 

3. SPECIFICATIONS DEVELOPMENT PROCESS 

METHODOLOGY FOR DEVELOPING THE PROCESS 

As stated in chapter 1, the objective of the project was to develop a comprehensive manual, 

supported by scientific evidence and statistical theory, which provides step-by-step procedures 

and instructions for developing effective and efficient QA specifications. Therefore, one of the 

most important tasks for the project was to identify the necessary steps, along with the specific 

options available at each step, for a SHA to comprehensively develop a QA specification. 

The first step toward identifying what should be considered in the specification process was a 

meeting between the COTR for the project and the principal investigator. This meeting took 

place in Clemson, SC, and was also attended by the statistical consultant member of the project 

team, Hoke Hill, and the graduate research assistant who had primary responsibility for the 

literature search and specification review activities on the project. 

The full research team then met in Clemson, SC, to develop the structure for the QA 

specifications development, implementation, and monitoring process. At this meeting, it was 

decided that the best way to present the specifications development process was in a flowchart. A 

detailed flowchart was developed and the major discussion points to accompany the flowchart 

were identified. Subsequently, one of the project team members, Charles Hughes, developed the 

initial draft of an executive summary to accompany and describe the elements of the process 

flowchart. The flowchart and the executive summary were then distributed to the members of the 

panel for their review. 

The initial flowchart and executive summary were then reviewed and discussed at a meeting of 

the panel that was held in Washington, DC. After a great deal of discussion, some revisions were 

agreed upon. A few minor revisions to the flowchart also resulted from issues that were 

identified while writing the QA specifications manual. 

THE PROCESS 

The overall specifications development and implementation process can be divided into three 

primary phases: 

• Phase I: Initiation and Planning. 

• Phase II: Specifications Development. 

• Phase III: Implementation. 

The steps in each of these phases can be presented in a flowchart for each phase. The steps in 

each of the three phases of the overall specifications development and implementation process 

are presented and discussed in detail in the QA specifications manual that was developed during 

the project. The steps in the process are therefore not discussed in detail in this technical report. 

Phase I of the specifications development and implementation process is initiation and planning. 

The steps that are involved in this process are identified in the flowchart in figure 2. Phase II is 
 14 

specifications development (the steps are identified in the flowchart in figure 3). Phase III is 

implementation (the steps are identified in the flowchart in figure 4). 

FLOWCHARTS AS A GUIDE TO REQUIRED ANALYSES 

The process flowcharts for each of the three phases of the specifications development and 

implementation process are presented in this technical report because the flowcharts were used to 

identify some of the major questions that a SHA must answer when developing a new, or 

modifying an existing, QA specification. Each of these questions has several possible answers. 

The flowcharts, therefore, helped lead to the analyses that needed to be performed on the current 

project to determine the recommended answers to these major questions. 

There are many decisions that are required as an agency progresses through the specifications 

development and implementation process outlined in figures 2 through 4. Many of these 

decisions are subjective and depend on the specific circumstances of the SHA. Each of the steps 

in the process is discussed in the manual. Some of the questions, however, were best answered 

by further analysis of the possible options. 

The following questions, identified from the specifications development and implementation 

process flowcharts, supported detailed analyses to be conducted during the project: 

• What quality measure should be used for individual quality characteristics? 

• What payment relationships should be used for individual quality characteristics? 

• How should multiple quality characteristics be combined into a single payment factor? 

• What procedures should be used to verify the contractor’s test results if they are to be 

used in the acceptance and payment decisions? 
 15 

Figure 2. Flowchart for phase I—Initiation and planning.

3. Reach agency 

consensus

1. Identify need for 

the specification(s) 

within the agency

6. Establish industry 

contact

7. Hold first joint 

agency–industry 

task force meeting

Proceed to Phase II: 

Specification Development

1.1. Problems that need to be solved? 

1.2. Innovative or progressive ideas? 

1.3. Industry suggested need to improve? 

3.1. Obtain/verify top management 

 commitment and support 

3.2. Choose task force leaders and agency members

3.3. Build consensus within task force 

3.4. Set target date for initial draft specification

6.1. Present concepts to selected industry leaders, 

 present potential benefits to industry and

 agency 

6.2. Select industry representatives for task force

7.1. Present concepts

7.2. Build consensus among members 

7.3. Establish short–term and long –term goals 

7.4. Establish deadline for trial specification

7.5. Establish schedule for the initiative 

7.6. Determine if outside expertise is needed

7.7. Set frequency of task force meetings 

7.8. Make specific work assignments 

2. Define goal and 

expectations

2.1. Identify benefits to agency and industry 

2.2. What is expected of the final product?

2.3. Define criteria for success 

 

4. Understand 

“Best Practices”

4.1. Review the literature

4.2. Learn other States’ experiences 

5. Confirm interest 

and commitment

5.1. Reconfirm top management 

 commitment and support 
 16 

Figure 3. Flowchart for phase II—Specifications development. 

2. Procure outside assistance, if required (see Phase I, Item 7.6)

1. Select material(s) and/or construction specifications to develop 

3. Recognize need to identify current practices

6. Develop outline for the QA specification e.g., General Information, Definitions, Quality 

 Assurance, Quality Control, Acceptance, 

 Payment, Conflict Resolution 

4. Search the literature 

(for specific material(s) selected)

5. Contact/interview other 

 agencies and associations 

(for specific material(s) selected)

7. Develop introductory information for the QA specification e.g., Responsibilities of agency and contractor, 

 Requirements for technician and laboratory qualification

8. Begin to develop procedures

To 

9. Develop QC procedures

and requirements 

To 

19. Develop acceptance procedures

and requirements 
 17 

Figure 3. Flowchart for phase II—Specifications development (continued). 

From 

8. Begin to develop procedures

9. Develop QC procedures

 and requirements

11. Determine quality characteristics

 to measure

10. Establish QC requirements e.g., QC Plan, Qualified 

 Technicians, Qualified 

 Laboratories, Control 

 Charts, Action Criteria

16. Determine sampling and testing 

 procedures and test frequency

To 

42. Finalized initial draft specification

18. QC procedures and 

 requirements completed

13. Evaluate available data

14. Are there 

 sufficient valid 

 data? 

15. Obtain

 data

Yes

No

17. Use for 

 QC testing? 

No

Yes

Eliminate or consider 

for acceptance

12. For each QC quality 

 characteristic, follow steps 13–17
 18 

Figure 3. Flowchart for phase II—Specifications development (continued). 

 

19. Develop acceptance 

 procedures and 

 requirements 

From

8. Begin to develop procedures

23. Determine quality 

 characteristics to measure 

24. For EACH acceptance 

 quality characteristic, 

 follow steps 25–36

25. Evaluate 

 available data 

20. Will contractor 

 tests be used in 

 the acceptance 

 decision? 

21. Use con tractor to do 

 acceptance testing 

 with agency doing 

 verification testing 

22. Develop verification 

 procedures 

 e.g., -test and -test, 

 allowable 

 between tests, 

 

26. Are there 

 sufficient valid 

 data? 

27. Obtain 

 data

Yes

Yes

No

No

To 

42. Finalized initial

 draft specification

To

30. Use for payment 

determination?

29. Is quality 

characteristic valid 

 for acceptance? 

Yes

No

28. Analyze data for statistical 

 parameters and distribution 

Consider for QC 

or eliminate 
 19 

Figure 3. Flowchart for phase II—Specifications development (continued). 

From 

29. Is quality characteristic valid for acceptance? 

31. Use as a screening 

 (pass/fail) test

32. Determine the quality 

 measure to use 

 e.g., PWL, PD, AAD

33. Determine the quality 

 measure to use

34. Determine specification

 limits, decide on AQL 

 and RQL

37. Decide pay relationships e.g., Performance-related pay, 

 Incentive/disincentive, 

 Minimum pay provisions, 

 Remove/replace provisions,

 Retest provisions

36. Determine acceptance/ 

 rejection procedures, 

 including rework 

 provisions

To 

39. Develop OC curve and evaluate risks

30. Use for pay 

 determination?

Yes No

35. Determine specification 

 limits, decide on AQL 

 and RQL

38. Determine sample size, 

 lot size, sublot size
 20 

Figure 3. Flowchart for phase II—Specifications development (continued). 

39. Develop OC curves, 

 EP curve, and 

 evaluate risks

From 

38. Determine sample size, 

lot size, sublot size 

40. Are risks 

 acceptable? 

41. Modify specification

 limits, acceptance 

 limits, pay schedule,

 sample size, and/or 

 lot size 

Yes 

No

From 

22. Develop validation/ 

verification procedures

42. Finalized initial draft 

 specification

From 

18. QC procedures and 

requirements completed

Proceed to Phase III: Implementation
 21 

Figure 4. Flowchart for phase III—Implementation.

2. Begin/continue technician 

 qualification training 

1. Simulate specification 

3. Try specification on a 

 limited number of pilot 

 projects 

4. Analyze pilot 

 project results 

1.1. Use several projects under construction or 

 recently completed 

1.2. Be sure random sampling was used 

1.3. Use same sampling and testing procedures as

 decided upon in Phase II 

1.4. Analyze simulated payment factor data 

1.5. Revise or fine–tune as necessary for a 

 Draft Special Provision 

2.1. Include training in sampling and testing procedures

2.2. Include training in basic statistical procedures

3.1. Let projects using Draft Special Provision (Item 1.5)

3.2. Apply only a percentage of the disincentives 

3.3. Bid prices may not reflect future project prices

5. Need 

major 

revisions?

6. Prepare new Draft 

 Special Provision 

7. Phase in projects 

 agency–wide 

7.1. Phase in until all major 

 projects are included 

7.2. Phase in payment factors 

8. Ongoing monitoring of 

 specification performance 

Yes

No

8.1. Look at quality levels achieved each year 

8.2. Look for administrative problems 

8.3. Consider contractor concerns 

8.4. Identify technology changes 

8.5. Tie results into the pavement management system 

8.6. Compare with established criteria for success 
 
 23 

4. SELECTING TOPICS FOR DETAILED ANALYSES 

FIRST PANEL MEETING 

In accordance with contract provisions, once the specifications development flowchart was 

developed (see chapter 3), members of the research team met with the panel to seek its approval 

of the process and to identify topics for detailed analyses. The meeting took place on March 1, 

1999, at the Turner-Fairbank Highway Research Center (TFHRC) in McLean, VA. The minutes 

of that meeting are included in appendix E. 

Two major goals were planned for the meeting. The first goal was to present the preliminary 

specifications development flowchart to the panel members, discuss their comments and input, 

and obtain approval from the panel to proceed with a final process flowchart. The second goal 

was to determine the specific topics that the panel wanted to include for detailed analyses in the 

project. The minutes in appendix E indicate the process that was followed during the meeting. 

With regard to the first goal, the researchers obtained input from the panel members and it was 

agreed that some modifications would be made to the initial flowchart. These changes were 

included in the final flowcharts shown in chapter 3. Concerning the second goal, there was 

discussion on a number of potential topics; however, there was not sufficient time for the panel 

to select the most desirable items for further study. It was therefore agreed that the principal 

investigator would distribute a survey form to the panel members to solicit their rankings of the 

various topics to be analyzed during the project. 

SURVEY OF TOPICS FOR DETAILED ANALYSES 

The principal investigator distributed a survey form to the panel members to determine a priority 

ranking for the various topics that were candidates for detailed analyses. The survey form that 

was distributed is shown in figure 5. 

Of the 20 survey forms distributed (19 State representatives plus the COTR), 18 were returned. 

Two ranking methods were used. The first asked the respondents to group the topics into three 

categories—highest priority, next highest priority, and lowest priority. In summarizing these 

results, 5 points, 3 points, and 1 point were assigned to the topics in each category, respectively. 

The second ranking method asked the respondents to rank the topics in decreasing order from 

highest to lowest priority. In summarizing these results, 10 points were assigned to the highest 

priority topic, with the points decreasing to 9, 8, 7, ... 3, 2, 1. Zero points were assigned to any 

topics that were not on the list of the top 10. 
 24 

FAX TO: [contact] FROM: _______________________ 

Optimal Acceptance Procedures for Statistical Specifications 

Complete the tables below using two different ranking methods. Keep in mind that all items in the 

flowcharts will be addressed in the manual and the report. Some will just be addressed in general 

conceptual terms, while others will need to include detailed analyses to support recommendations.

First: Rank the 4 highest priority numbered items in the table shown, along with the 4 items with 

second highest priority, and, finally, the 4 items with lowest priority. You may include write-in 

items in your priority rankings. 

Priority Numbered Items From the List 

Highest (list 4) 

Next Highest (list 4) 

Lowest (list 4) 

Second: Rank the 10 highest priority numbered items in decreasing order from most important, 1, 

to less important, 10. You may include write-in items in your priority rankings. 

Priority Ranking (1-10) Numbered Item From the List 

1 (highest) 

2 

3 

4 

5 

6 

7 

8 

9 

10 (lowest) 

Third: Cross out any of the bulleted items that you do not feel need to be included. 

Fourth: Fax your ratings (and pages with crossed-out bullets) to Jim Burati at 864–656–2670. 

Figure 5. Survey sent to panel members. 
 25 

List of Possible Topics for Further Analysis 

1) Analysis of the Percent Within Limits (PWL) approach, including: 

• Bias and precision of the PWL estimates versus sample size 

• Precision in OC curves for PWL versus sample size 

• Precision of average project PWL versus number of project lots 

• Precision of individual payments based on PWL 

• Precision of average project payment versus number of project lots 

• Effects of non-normal populations (bimodal and skewed) 

2) Analysis of the Average Absolute Deviation (AAD) approach, including: 

• Bias and precision of the AAD estimates versus sample size 

• Methodology for developing and presenting AAD OC curves 

• Precision in OC curves for AAD versus sample size 

• Precision of individual payments based on AAD 

• Effects of non-normal populations (bimodal and skewed) 

3) Analysis of the sample mean (mean) acceptance approach, including: 

• Bias and precision of the mean estimates versus sample size 

• Precision in OC curves for mean versus sample size 

• Precision of average project mean versus number of project lots 

• Precision of individual payments based on mean 

• Precision of average project payment versus number of project lots 

• Effects of non-normal populations (bimodal and skewed) 

4) Analysis of the Conformal Index (CI) approach, including: 

• Bias and precision of the CI estimates versus sample size 

• Methodology for developing and presenting CI OC curves 

• Precision in OC curves for CI versus sample size 

• Precision of individual payments based on CI 

• Effects of non-normal populations (bimodal and skewed) 

5) Analysis of the single sample variability ( X ± ks) approach, including: 

• Bias and precision of the X ± ks estimates versus sample size 

• Methodology for developing and presenting X ± ks OC curves 

• Precision in OC curves for X ± ks versus sample size 

• Precision of individual payments based on X ± ks

• Effects of non-normal populations (bimodal and skewed) 

Figure 5. Survey sent to panel members (continued). 
 26 

6) Analysis of the moving average (m) approach for acceptance, including: 

• Bias and precision of the m estimates versus sample size 

• Investigation of the possibility of developing and presenting m OC curves 

• Methods for applying price adjustments when using m

• Precision of individual payments based on m

• Precision of average project payment versus number of project lots 

• Effects of non-normal populations (bimodal and skewed) 

Note: Some of the bulleted items for moving averages may not be possible to determine. 

7) Analysis of methods for determining lot pay factors for individual acceptance properties 

8) Analysis of methods for determining composite lot pay factors when multiple acceptance 

properties are used 

9) Analysis of the use of Bayesian procedures that incorporate information from prior lots or prior 

projects into the acceptance decision for the current lot 

10) Analysis of procedures for verifying or validating contractor and agency test results, including: 

• Use of the F-test and t-test (AASHTO QA Guide Spec.) 

• Use of a single agency test and the mean and range of contractor tests (AASHTO QA 

Guide Spec.) 

• Use of a maximum allowable difference between individual agency and contractor tests 

11) Analysis of various individual “bells and whistles,” that is, additional provisions that are used in 

conjunction with the traditional acceptance approaches, for example: 

• Use of payment based on PWL but with no price reductions applied if all individual tests 

are within the limits 

• Use of sample mean for acceptance, but also placing wider limits on individual test results 

• Use of limits on sample range or standard deviation in addition to limits on the sample 

average 

• Other provisions: _______________________________________________________ 

 _____________________________________________________________________ 

12) Other major items for analyses: 

 __________________________________________________________________________ 

 __________________________________________________________________________ 

Figure 5. Survey sent to panel members (continued). 
 27 

Survey Results

A summary of the survey responses is provided in table 4 for the first ranking method and in 

table 5 for the second ranking method. The same results are shown in graphical form, from 

highest to lowest priority, in figures 6 and 7 for the first and second ranking methods, 

respectively. 

Table 4. Survey results for the first ranking method. 

Agency 1 2 3 4 5 6 7 8 9 10 11 12 

FHWA 1 3 1 3 1 3 5 5 5 3 1 5* 

CT 3 1 3 5 1 3 5 5 1 5 3 

ID 5 1 3 3 3 1 5 5 1 5 3 

IL 5 3 5 3 3 5 1 1 1 5 3 

KS 5 5 3 5 3 5 1 1 1 3 1 3? 

LA 5 1 1 1 3 3 5 5 5 3 3 

MN 5 1 1 3 5 5 3 5 1 3 3 

NV 3 5 3 1 3 3 5 5 1 5 1 

NJ 3 3 1 3 1 3 5 5 1 5 1 5+ 

NY 3 3 1 1 3 1 5 5 3 5 5 

ON 5 1 3 1 1 3 5 5 3 5 3 

OR 5 5 3 1 1 3 5 3 1 5 3 

PA 5 5 3 5 3 1 3 3 1 5 1 

SC 5 3 1 1 1 3 5 5 3 5 3 

TX 5 5 3 3 3 5 1 0 5 3 0 

VA 0 0 0 0 0 0 5 5 0 5 0 

WA 5 3 3 3 1 5 5 3 1 5 1 

WI 5 3 1 3 5 3 5 1 1 5 3 

Total 73 51 39 45 41 55 74 67 35 80 38 13 

* Procedures for determining acceptable alpha and beta risks 

? Listed an item 12 in the ranking, but did not identify it 

+ Establishment of the relationship between quality/performance/value 
 28 

Table 5. Survey results for the second ranking method. 

Agency 1 2 3 4 5 6 7 8 9 10 11 12 

FHWA 2 5 0 6 0 3 9 7 8 4 1 10* 

CT 6 2 3 10 0 5 8 7 1 9 4 

ID 10 2 5 4 3 1 8 7 0 9 6 

IL 9 5 7 4 3 10 2 2 0 8 6 

KS 10 9 5 7 4 8 2 1 0 6 0 3? 

LA 10 2 1 0 6 3 8 9 7 5 4 

MN 9 2 1 3 7 10 5 8 0 6 4 

NV 5 8 6 0 3 4 10 9 2 7 1 

NJ 5 3 0 4 0 6 9 8 1 7 2 10+ 

NY 6 5 1 2 4 0 9 10 3 7 8 

ON 10 0 3 2 1 6 8 7 5 9 4 

OR 10 7 3 2 1 4 9 6 0 8 5 

PA 10 9 5 8 4 2 6 3 1 7 0 

SC 10 4 2 0 1 3 7 8 5 9 6 

TX 9 10 5 3 4 8 2 0 7 6 0 

VA 0 0 0 0 0 0 9 9 0 9 0 

WA 7.5 5 5 5 2.5 7.5 9.5 2.5 1 9.5 0 

WI 10 5 1 6 9 4 7 2 0 8 3 

Total 138.5 83 53 66 52.5 84.5 127.5 105.5 41 133.5 54 23 

* Procedures for determining acceptable alpha and beta risks 

? Listed an item 12 in the ranking, but did not identify it 

+ Establishment of the relationship between quality/performance/value
 29 

Figure 6. Graphical presentation of survey results for the first ranking method. 

Figure 7. Graphical presentation of survey results for the second ranking method. 

80

74 73

67

55 51

45 41 39 38 35

13

0

30

60

90

10 7 1 8 6 2 4 5 3 11 9 12

Topic Number

138.5 133.5 127

105

84.5 83

66

54 53 52.5

41

23

0

50

100

150

1 10 7 8 6 2 4 11 3 5 9 12

Topic Number
 30 

Table 6 shows the rankings from the two different methods and the overall ranking, which is the 

average of the rankings from the two ranking methods. 

Table 6. Overall rankings of the survey topics. 

Topic: Analysis of ... 

First 

Ranking 

Method 

Second 

Ranking 

Method 

Overall 

Ranking 

10 Procedures for verifying or validating 

contractor’s and agency’s test results 1 2 1 

1 PWL approach 3 1 2 

7 Methods for determining lot pay factors for 

individual acceptance properties 2 3 3 

8 Methods for determining composite pay factors 

when multiple properties are used 4 4 4 

6 Moving average approach 5 5 5 

2 AAD approach 6 6 6 

4 CI approach 7 7 7 

3 Sample mean approach 9 9 8 (tie) 

5 Sample variability approach 8 10 8 (tie) 

11 Various “bells and whistles” 10 8 8 (tie) 

9 Use of Bayesian procedures 11 11 11 

As would be expected, the two ranking methods had very similar results. The clear winners were 

the topics related to verifying or validating the contractor’s results, the percent within limits 

(PWL) approach, and the determination of payment factors. There was a considerable dropoff 

between this group and the moving average, average absolute deviation (AAD), and conformal 

index (CI) approaches. 

Two additional topics were proposed (each by one responder). These were “procedures for 

determining acceptable α and β risks” and “establishment of the relationship between quality, 

performance, and value.” Each of these additional proposed topics would require considerable 

effort and, indeed, would constitute major research projects in their own right. It was not possible 

to tackle these topics with the time and resources that were allocated for the current project. 

TOPICS SELECTED FOR DETAILED ANALYSES 

Table 6 identifies the priority topics that, in the opinion of the panel, required detailed analyses 

during the current project. The priority items selected by the panel can be reiterated as: 

• Analysis of the procedures for verifying or validating the contractor’s and agency’s test 

results. 

• Analysis of the use of PWL as the quality measure. 

• Analysis of the methods for determining lot pay factors for individual acceptance 

properties. 
 31 

• Analysis of the methods for determining the composite payment factor when multiple 

acceptance properties are used. 

These are essentially the same topics that were identified from the process flowcharts in 

chapter 3. Those topics were: 

• What quality measure should be used for individual quality characteristics? 

• What payment relationships should be used for individual quality characteristics? 

• How should multiple quality characteristics be combined into a single payment factor? 

• What procedures should be used to verify the contractor’s test results if they are to be 

used in the acceptance and payment decision? 

The only difference is that the panel members were interested primarily in the PWL quality 

measure, while the flowcharts indicate that a quality measure must be selected but do not imply 

that it must be PWL. Therefore, it was decided to conduct initial analyses on several potential 

quality measures, but to concentrate the detailed analyses on the PWL measure as long as the 

initial analyses indicated that it was the recommended quality measure. 

Each of the bulleted items listed above is presented in depth in subsequent chapters. 
 
33 

5. QUALITY MEASURES: ACCURACY AND PRECISION 

INTRODUCTION 

There are several quality measures that can be used in an acceptance plan. In early QA 

specifications in the late 1960s and 1970s, the mean (or average), or the average deviation from a 

target value, was often used as the quality measure. However, the use of the average alone 

provides no measure of variability, and it is now recognized that variability is often an important 

predictor of performance. The problems associated with basing acceptance on only the mean are 

well documented. Therefore, there was no need to consider this approach in the analyses that 

were performed. 

Several quality measures, including percent defective (PD) and PWL, have been preferred in 

recent years because they simultaneously measure both the average level and the variability in a 

statistically efficient way. Other measures that have been used by some agencies include AAD 

and the moving average. An additional measure that may be considered by some agencies is the 

CI. 

PD and PWL are, in reality, the same quality measure since they are directly related by the 

simple relationship PWL = 100 – PD. Therefore, analyses reported using PWL also apply 

equally to PD, and vice versa. 

The quality measures that were evaluated during the project include: 

• PWL. 

• AAD. 

• CI. 

ACCURACY AND PRECISION 

Since each of the above measures provides an estimate for a true population value, it is important 

to know that the estimator provides an unbiased estimate for the population parameter. It is 

known that this is true for the procedure for estimating PWL. Computer simulation was used to 

evaluate whether there was bias in any of the quality measures. A lack of bias means that the 

estimator is accurate. Another important factor to be considered is the variability associated with 

the quality measure (i.e., how much do individual estimated values vary about the long-term 

average value for the quality measure?) This, too, was investigated by computer simulation 

studies. Low variability for the estimated values is a sign that the estimator is precise. 

Computer simulation studies were performed to evaluate the accuracy and precision of the 

estimated quality measures for various sample sizes. The results of these studies are presented 

for each measure in the following sections. 
34 

PWL EVALUATION 

The PWL for a lot can be estimated by using the quality index, Q. The Q-statistic is used with a 

PWL table to determine the estimated PWL for the lot. A PWL table is shown in table 7. 

Conceptually, the Q-statistic, or quality index, performs the same function as the Z-statistic, 

except that the reference point is the mean of an individual sample, X , instead of the population 

mean, μ, and the points of interest with regard to areas under the curve are the specification 

limits.

s

X LSL QL

− = (1) 

and 

s

USL X QU

− = (2) 

where: QL = quality index for the lower specification limit 

QU = quality index for the upper specification limit 

LSL = lower specification limit 

USL = upper specification limit 

X = sample mean for the lot 

s = sample standard deviation for the lot 

QL is used when there is a one-sided lower specification limit, while QU is used when there is a 

one-sided upper specification limit. For two-sided specification limits, the PWL value is 

estimated as: 

PWLT = PWLU + PWLL – 100 (3) 

where: PWLU = percent below the upper specification limit (based on QU) 

PWLL = percent above the lower specification limit (based on QL) 

PWLT = percent within the upper and lower specification limits 

Simulation Analyses 

Computer simulation was used to generate samples of various sizes from known populations. 

These sample results were then used to estimate the PWL for the population. The mean and 

standard deviation for the estimated PWL values were then calculated. The difference between 

the mean estimated PWL was subtracted from the known population PWL to provide a measure 

of accuracy for the PWL estimates. The standard deviation of the estimated PWL values was 

used as a measure of precision (i.e., the amount of variability) for the PWL estimates. 
35 

Table 7. Quality index values for estimating PWL. 

PWL n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 

to 11 

100 1.16 1.50 1.79 2.03 2.23 2.39 2.53 2.65 

99 – 1.47 1.67 1.80 1.89 1.95 2.00 2.04 

98 1.15 1.44 1.60 1.70 1.76 1.81 1.84 1.86 

97 – 1.41 1.54 1.62 1.67 1.70 1.72 1.74 

96 1.14 1.38 1.49 1.55 1.59 1.61 1.63 1.65 

95 – 1.35 1.44 1.49 1.52 1.54 1.55 1.56 

94 1.13 1.32 1.39 1.43 1.46 1.47 1.48 1.49 

93 – 1.29 1.35 1.38 1.40 1.41 1.42 1.43 

92 1.12 1.26 1.31 1.33 1.35 1.36 1.36 1.37 

91 1.11 1.23 1.27 1.29 1.30 1.30 1.31 1.31 

90 1.10 1.20 1.23 1.24 1.25 1.25 1.26 1.26 

89 1.09 1.17 1.19 1.20 1.20 1.21 1.21 1.21 

88 1.07 1.14 1.15 1.16 1.16 1.16 1.16 1.17 

87 1.06 1.11 1.12 1.12 1.12 1.12 1.12 1.12 

86 1.04 1.08 1.08 1.08 1.08 1.08 1.08 1.08 

85 1.03 1.05 1.05 1.04 1.04 1.04 1.04 1.04 

84 1.01 1.02 1.01 1.01 1.00 1.00 1.00 1.00 

83 1.00 0.99 0.98 0.97 0.97 0.96 0.96 0.96 

82 0.97 0.96 0.95 0.94 0.93 0.93 0.93 0.92 

81 0.96 0.93 0.91 0.90 0.90 0.89 0.89 0.89 

80 0.93 0.90 0.88 0.87 0.86 0.86 0.86 0.85 

79 0.91 0.87 0.85 0.84 0.83 0.82 0.82 0.82 

78 0.89 0.84 0.82 0.80 0.80 0.79 0.79 0.79 

77 0.87 0.81 0.78 0.77 0.76 0.76 0.76 0.75 

76 0.84 0.78 0.75 0.74 0.73 0.73 0.72 0.72 

75 0.82 0.75 0.72 0.71 0.70 0.70 0.69 0.69 

74 0.79 0.72 0.69 0.68 0.67 0.66 0.66 0.66 

73 0.76 0.69 0.66 0.65 0.64 0.63 0.63 0.63 

72 0.74 0.66 0.63 0.62 0.61 0.60 0.60 0.60 

71 0.71 0.63 0.60 0.59 0.58 0.57 0.57 0.57 

70 0.68 0.60 0.57 0.56 0.55 0.55 0.54 0.54 

69 0.65 0.57 0.54 0.53 0.52 0.52 0.51 0.51 

68 0.62 0.54 0.51 0.50 0.49 0.49 0.48 0.48 

67 0.59 0.51 0.47 0.47 0.46 0.46 0.46 0.45 

66 0.56 0.48 0.45 0.44 0.44 0.43 0.43 0.43 

65 0.52 0.45 0.43 0.41 0.41 0.40 0.40 0.40 

64 0.49 0.42 0.40 0.39 0.38 0.38 0.37 0.37 

63 0.46 0.39 0.37 0.36 0.35 0.35 0.35 0.34 

62 0.43 0.36 0.34 0.33 0.32 0.32 0.32 0.32 

61 0.39 0.33 0.31 0.30 0.30 0.29 0.29 0.29 

60 0.36 0.30 0.28 0.27 0.27 0.27 0.26 0.26 

59 0.32 0.27 0.25 0.25 0.24 0.24 0.24 0.24 

58 0.29 0.24 0.23 0.22 0.21 0.21 0.21 0.21 

57 0.25 0.21 0.20 0.19 0.19 0.19 0.18 0.18 

56 0.22 0.18 0.17 0.16 0.16 0.16 0.16 0.16 

55 0.18 0.15 0.14 0.14 0.13 0.13 0.13 0.13 

54 0.14 0.12 0.11 0.11 0.11 0.11 0.10 0.10 

53 0.11 0.09 0.08 0.08 0.08 0.08 0.08 0.08 

52 0.07 0.06 0.06 0.05 0.05 0.05 0.05 0.05 

51 0.04 0.03 0.03 0.03 0.03 0.03 0.03 0.03 

50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 
36 

Table 7. Quality index values for estimating PWL (continued). 

PWL n = 12 

to 14 

n = 15 

to 18 

n = 19 

to 25 

n = 26 

to 37 

n = 38 

to 69 

n = 70 

to 200 

n = 201 

to ∞

100 2.83 3.03 3.20 3.38 3.54 3.70 3.83 

99 2.09 2.14 2.18 2.22 2.26 2.29 2.31 

98 1.91 1.93 1.96 1.99 2.01 2.03 2.05 

97 1.77 1.79 1.81 1.83 1.85 1.86 1.87 

96 1.67 1.68 1.70 1.71 1.73 1.74 1.75 

95 1.58 1.59 1.61 1.62 1.63 1.63 1.64 

94 1.50 1.51 1.52 1.53 1.54 1.55 1.55 

93 1.44 1.44 1.45 1.46 1.46 1.47 1.47 

92 1.37 1.38 1.39 1.39 1.40 1.40 1.40 

91 1.32 1.32 1.33 1.33 1.33 1.34 1.34 

90 1.26 1.27 1.27 1.27 1.28 1.28 1.28 

89 1.21 1.22 1.22 1.22 1.22 1.22 1.23 

88 1.17 1.17 1.17 1.17 1.17 1.17 1.17 

87 1.12 1.12 1.12 1.12 1.12 1.13 1.13 

86 1.08 1.08 1.08 1.08 1.08 1.08 1.08 

85 1.04 1.04 1.04 1.04 1.04 1.04 1.04 

84 1.00 1.00 1.00 1.00 0.99 0.99 0.99 

83 0.96 0.96 0.96 0.96 0.95 0.95 0.95 

82 0.92 0.92 0.92 0.92 0.92 0.92 0.92 

81 0.89 0.88 0.88 0.88 0.88 0.88 0.88 

80 0.85 0.85 0.85 0.84 0.84 0.84 0.84 

79 0.82 0.81 0.81 0.81 0.81 0.81 0.81 

78 0.78 0.78 0.78 0.78 0.77 0.77 0.77 

77 0.75 0.75 0.75 0.74 0.74 0.74 0.74 

76 0.72 0.71 0.71 0.71 0.71 0.71 0.71 

75 0.69 0.68 0.68 0.68 0.68 0.68 0.67 

74 0.66 0.65 0.65 0.65 0.65 0.64 0.64 

73 0.62 0.62 0.62 0.62 0.62 0.61 0.61 

72 0.59 0.59 0.59 0.59 0.59 0.58 0.58 

71 0.57 0.56 0.56 0.56 0.56 0.55 0.55 

70 0.54 0.53 0.53 0.53 0.53 0.53 0.52 

69 0.51 0.50 0.50 0.50 0.50 0.50 0.50 

68 0.48 0.48 0.47 0.47 0.47 0.47 0.47 

67 0.45 0.45 0.45 0.44 0.44 0.44 0.44 

66 0.42 0.42 0.42 0.42 0.41 0.41 0.41 

65 0.40 0.39 0.39 0.39 0.39 0.39 0.39 

64 0.37 0.36 0.36 0.36 0.36 0.36 0.36 

63 0.34 0.34 0.34 0.34 0.33 0.33 0.33 

62 0.31 0.31 0.31 0.31 0.31 0.31 0.31 

61 0.29 0.29 0.28 0.28 0.28 0.28 0.28 

60 0.26 0.26 0.26 0.26 0.26 0.25 0.25 

59 0.23 0.23 0.23 0.23 0.23 0.23 0.23 

58 0.21 0.21 0.20 0.20 0.20 0.20 0.20 

57 0.18 0.18 0.18 0.18 0.18 0.18 0.18 

56 0.16 0.15 0.15 0.15 0.15 0.15 0.15 

55 0.13 0.13 0.13 0.13 0.13 0.13 0.13 

54 0.10 0.10 0.10 0.10 0.10 0.10 0.10 

53 0.08 0.08 0.08 0.08 0.08 0.08 0.08 

52 0.05 0.05 0.05 0.05 0.05 0.05 0.05 

51 0.03 0.03 0.03 0.03 0.03 0.03 0.02 

50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 

Source: “Specification Conformity Analysis,” FHWA Technical Advisory T5080.12, June 23, 1989 
37 

As was known in advance, the quality index provided an unbiased estimate of the population 

mean, as evidenced by the lack of bias in the mean of the estimated PWL values. Also as 

expected, the amount of variability in the PWL estimates decreased as the sample size increased. 

This is shown in table 8. Also shown in the table is the fact that the standard deviation of the 

estimated PWL values is a maximum at an actual PWL of 50, and decreases to a minimum as the 

actual PWL approaches 100 or zero. This is also shown in the plot in figure 8. 

The reason for this is illustrated in figures 9 through 11, which plot histograms of the distribution 

of estimated PWL values for sample sizes = 3, 5, and 10, and for actual PWL values = 90, 70, 

and 50. As shown in the figures, as the actual PWL approaches the natural boundary imposed by 

100 PWL, the spread of estimated values decreases since the estimated values cannot exceed 100 

(see figure 9). As the actual PWL value moves away from the natural boundary of 100 to a value 

of 70, there is a greater spread available for the estimated values (see figure 10). When the actual 

PWL value is 50, there is the maximum opportunity for the estimated values to spread out before 

hitting the natural boundaries of 100 and 0 PWL (see figure 11). 

To further illustrate the lack of bias and how the variability of PWL estimates varies with sample 

size, figure 12 plots the average estimated PWL values from the simulation results versus the 

actual PWL values. The straight line at a 45-degree angle, which indicates that the estimated 

values vary only very slightly from the actual PWL values, shows the lack of bias in the 

estimator. The other lines on each plot represent the 10th and 90th percentiles (i.e., the values that 

10 percent of the estimates are either below or exceed, respectively). As the sample size 

increases, these bounds become narrower, indicating a reduction in variability that is also 

reflected in the standard deviation values in table 7. Also, figure 13 plots the average differences 

versus the actual PWL value to show that the differences are centered on zero and show no 

consistent positive or negative bias. 

To investigate the effect of project size, for sample sizes = 3, 5, and 10, various numbers of lots 

per project, ranging from 1 to 90, were also considered. As would be expected, since it 

represented an increase in the total number of tests, the amount of variability in the average 

estimated PWL decreased as the number of lots in the project increased. This is shown in tables 9 

through 11 and in figure 14. 
38 

Table 8. Accuracy and precision for PWL estimates 

 (based on the results of 10,000 simulated lots). 

Tests per Lot, 

n

Actual PWL 

of Population 

Mean of Estimated 

PWL Values Minus 

Actual PWL 

Standard Deviation 

of Estimated PWL 

Values 

95 –0.08 10.93 

90 +0.29 14.78 

85 –0.16 18.48 

80 –0.21 20.89 

70 –0.21 23.51 

60 +0.28 25.31 

50 –0.26 26.52 

40 –0.60 25.50 

30 –0.05 24.23 

20 +0.24 20.67 

15 –0.01 18.13 

10 –0.30 15.31 

3 

 5 +0.10 10.79 

95 –0.12 7.94 

90 –0.43 11.42 

85 –0.26 13.42 

80 +0.08 15.15 

70 –0.18 17.10 

60 +0.13 17.71 

50 +0.24 18.39 

40 –0.43 17.89 

30 +0.08 17.71 

20 +0.10 14.96 

15 –0.32 13.49 

10 +0.30 11.84 

5 

 5 +0.18 7.88 

95 –0.18 5.45 

90 +0.25 7.08 

85 +0.24 9.28 

80 –0.12 10.38 

70 +0.08 11.85 

60 +0.07 12.50 

50 –0.30 13.01 

40 +0.27 12.47 

30 –0.26 11.89 

20 –0.06 10.65 

15 –0.24 9.02 

10 +0.23 7.88 

10 

 5 –0.27 4.89 
39 

0

5

10

15

20

25

30

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Standard Deviation

n = 3

n = 5

n = 10

Figure 8. Plot of how the standard deviation of PWL estimates 

 varies with the population PWL. 
40 

Figure 9a. Histogram illustrating the distribution of estimated PWL values 

 for 1000 simulated lots from a population with 90 PWL, sample 3. 

Figure 9b. Histogram illustrating the distribution of estimated PWL values 

 for 1000 simulated lots from a population with 90 PWL, sample 5. 

Figure 9c. Histogram illustrating the distribution of estimated PWL values 

 for 1000 simulated lots from a population with 90 PWL, sample 10. 

Estimated PWL Values 

Average = 90.25 

Std. Dev. = 7.08 Sample Size, n =10 

Average = 89.57 

Std. Dev. = 11.42 Sample Size, n = 5 

Estimated PWL Values 

Sample Size, n = 3 Average = 90.29 

Std. Dev. = 14.78 

Estimated PWL Values 
41 

Figure 10a. Histogram illustrating the distribution of estimated PWL values 

for 1000 simulated lots from a population with 70 PWL, sample 3. 

Figure 10b. Histogram illustrating the distribution of estimated PWL values 

for 1000 simulated lots from a population with 70 PWL, sample 5. 

Figure 10c. Histogram illustrating the distribution of estimated PWL values 

for 1000 simulated lots from a population with 70 PWL, sample 10. 

Average = 70.08 

Std. Dev. = 11.85 

Sample Size, n =10 

Estimated PWL Values 

Average = 69.79 

Std. Dev. = 23.51 

Sample Size, n = 3 

Estimated PWL Values 

Average = 69.82 

Std. Dev. = 17.10 

Sample Size, n = 5 

Estimated PWL Values 
42 

Figure 11a. Histogram illustrating the distribution of estimated PWL values 

for 1000 simulated lots from a population with 50 PWL, sample 3. 

Figure 11b. Histogram illustrating the distribution of estimated PWL values 

 for 1000 simulated lots from a population with 50 PWL, sample 5. 

Figure 11c. Histogram illustrating the distribution of estimated PWL values 

 for 1000 simulated lots from a population with 50 PWL, sample 10.

Average = 49.70 

Std. Dev. = 13.01 

Sample Size, n =10 

Estimated PWL Values 

Average = 50.24 

Std. Dev. = 18.39 

Sample Size, n = 5 

Estimated PWL Values 

Average = 49.74 

Std. Dev. = 26.52 

Sample Size, n = 3 

Estimated PWL Values 
43 

Figure 12a. Illustration 1 of accuracy and precision of PWL estimates.

Figure 12b. Illustration 2 of accuracy and precision of PWL estimates. 

Figure 12c. Illustration 3 of accuracy and precision of PWL estimates. 

0

20

40

60

80

100

0 10 20 30 40 50 60 70 80 90 100

Actual Population PWL

Average Estimated PWL

Mean

90th

10th

0

20

40

60

80

100

0 10 20 30 40 50 60 70 80 90 100

Actual Population PWL

Average Estimated PWL

Mean

90th

10th

0

20

40

60

80

100

0 10 20 30 40 50 60 70 80 90 100

Actual Population PWL

Average Estimated PWL

Mean

90th

10th

n = 10 
44 

Table 9. Results of simulation analyses with actual PWL = 90 

 (distribution of the results from 1000 simulated projects). 

Tests/Lot Lots/Project Tests/Project Avg. Diff. 2.5% 97.5% 

1 3 +0.1 –10.0 +35.0 

3 9 +0.1 –10.0 +20.3 

5 15 –0.3 –10.0 +14.3 

10 30 –0.2 –9.0 +9.9 

20 60 +0.1 –6.5 +7.6 

30 90 –0.0 –5.1 +5.9 

40 120 –0.1 –4.7 +4.8 

50 150 +0.0 –4.1 +4.6 

60 180 –0.1 –3.6 +3.9 

70 210 +0.0 –3.4 +3.7 

80 240 –0.1 –3.4 +3.4 

3 

90 270 –0.0 –2.9 +3.1 

1 5 –0.2 –10.0 +25.5 

3 15 –0.0 –10.0 +15.5 

5 25 +0.0 –8.3 +10.5 

10 50 +0.2 –6.2 +7.5 

20 100 +0.0 –4.6 +5.6 

30 150 +0.1 –3.8 +4.2 

40 200 –0.1 –3.4 +3.7 

50 250 –0.0 –3.2 +3.1 

5 

60 300 +0.0 –2.8 +2.8 

1 10 +0.1 –10.0 +16.6 

3 30 +0.2 –7.6 +9.8 

5 50 +0.0 –5.7 +7.2 

10 100 –0.1 –4.4 +4.7 

20 200 –0.1 –3.3 +3.1 

30 300 +0.1 –2.7 +2.9 

40 400 –0.1 –2.4 +2.4 

10 

50 500 –0.0 –2.2 +2.1 

Avg. Diff. = average difference between the average estimated PWL and the actual PWL 

 for each simulated project 

2.5% = value for which 2.5 percent of the simulated average differences were less than 

or equal 

97.5% = value for which 97.5 percent of the simulated average differences were less than 

or equal 
45 

Table 10. Results of simulation analyses with actual PWL = 70 

(distribution of the results from 1000 simulated projects). 

Tests/Lot Lots/Project Tests/Project Avg. Diff. 2.5% 97.5% 

1 3 –1.0 –30.0 +44.0 

3 9 +0.3 –30.0 +29.6 

5 15 –0.5 –20.5 +21.7 

10 30 +0.2 –14.9 +14.5 

20 60 –0.0 –10.3 +10.8 

30 90 +0.1 –8.1 +8.7 

40 120 –0.1 –7.6 +6.9 

50 150 –0.1 –6.7 +6.4 

60 180 –0.1 –5.8 +6.0 

70 210 +0.0 –5.5 +5.6 

80 240 +0.2 –5.0 +5.5 

3 

90 270 –0.0 –4.9 +5.2 

1 5 –0.3 –30.0 +34.5 

3 15 –0.1 –19.2 +21.7 

5 25 +0.2 –15.1 +16.9 

10 50 +0.1 –10.3 +10.9 

20 100 –0.1 –7.7 +7.6 

30 150 +0.1 –6.2 +6.0 

40 200 –0.1 –5.5 +5.4 

50 250 +0.2 –4.8 +5.4 

5 

60 300 +0.1 –4.5 +4.7 

1 10 +0.5 –22.6 +27.3 

3 30 +0.2 –13.5 +13.7 

5 50 –0.0 –10.4 +10.5 

10 100 –0.0 –8.0 +7.5 

20 200 –0.0 –5.3 +5.1 

30 300 +0.0 –4.4 +4.5 

40 400 –0.0 –3.8 +3.5 

10 

50 500 +0.0 –3.2 +3.4 

Avg. Diff. = average difference between the average estimated PWL and the actual PWL 

 for each simulated project 

2.5% = value for which 2.5 percent of the simulated average differences were less than 

 or equal 

97.5% = value for which 97.5 percent of the simulated average differences were less than 

 or equal 
46 

Table 11. Results of simulation analyses with actual PWL = 50 

(distribution of the results from 1000 simulated projects). 

Tests/Lot Lots/Project Tests/Project Avg. Diff. 2.5% 97.5% 

1 3 +0.0 –50.0 +50.0 

3 9 +0.1 –31.0 +29.6 

5 15 –0.5 –24.8 +22.6 

10 30 –0.2 –16.3 +16.2 

20 60 +0.2 –12.1 +11.7 

30 90 +0.1 –9.1 +9.4 

40 120 +0.0 –7.7 +8.3 

50 150 –0.0 –6.9 +7.1 

60 180 –0.0 –6.9 +6.7 

70 210 –0.0 –6.0 +6.0 

80 240 +0.1 –5.7 +6.2 

3 

90 270 –0.0 –5.4 +5.6 

1 5 –0.3 –41.0 +40.8 

3 15 +0.5 –21.4 +20.2 

5 25 +0.2 –16.6 +17.7 

10 50 –0.2 –12.2 +12.5 

20 100 –0.0 –8.5 +8.8 

30 150 –0.0 –6.5 +6.6 

40 200 +0.0 –5.7 +6.0 

50 250 –0.1 –5.0 +5.0 

5 

60 300 +0.2 –4.9 +5.0 

1 10 –0.0 –28.1 +29.0 

3 30 +0.2 –15.0 +14.7 

5 50 +0.0 –11.5 +10.5 

10 100 +0.2 –8.1 +8.2 

20 200 +0.0 –5.7 +5.5 

30 300 –0.1 –4.8 +4.3 

40 400 –0.0 –4.1 +3.9 

10 

50 500 –0.1 –3.8 +3.6 

Avg. Diff. = average difference between the average estimated PWL and the actual PWL 

 for each simulated project 

2.5% = value for which 2.5 percent of the simulated average differences were less than 

 or equal 

97.5% = value for which 97.5 percent of the simulated average differences were less than 

 or equal 
47 

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Average Difference

n = 3

n = 5

n = 10

Figure 13. Plot of average difference of simulated PWL values versus 

 actual PWL values for sample sizes = 3, 5, and 10. 

Figure 14a. Plots of the 95th percentile for the average estimated PWL minus 

the actual PWL at 50 versus the number of lots per project.

Actual PWL = 50

-50

-40

-30

-20

-10

0

10

20

30

40

50

0 10 20 30 40 50 60 70 80 90

Lots per Project

Average Difference

3 tests/lot

5 tests/lot

10 tests/lot
48 

Actual PWL = 70

-50

-40

-30

-20

-10

0

10

20

30

40

50

0 10 20 30 40 50 60 70 80 90

Lots per Project

Average Difference

3 tests/lot

5 tests/lot

10 tests/lot

Figure 14b. Plots of the 95th percentile for the average estimated PWL minus 

the actual PWL at 70 versus the number of lots per project. 

Actual PWL = 90

-50

-40

-30

-20

-10

0

10

20

30

40

50

0 20 40 60 80 100

Lots per Project

Average Difference

3 tests/lot

5 tests/lot

10 tests/lot

Figure 14c. Plots of the 95th percentile for the average estimated PWL minus 

the actual PWL at 90 versus the number of lots per project. 
49 

AAD EVALUATION 

For specifications that have a target value, the average deviation from the target value has, in the 

past, sometimes been used as a means for determining the acceptability of the product. This 

approach can have the effect of encouraging the contractor to manipulate its process during the 

production of a lot. For example, if two test results in the morning are below the target value, 

there is a strong incentive for the contractor to increase the process mean in the afternoon in an 

effort to get two higher test results so that the average of the four tests for the lot will be near the 

target value. In essence, this acceptance approach encourages the contractor to increase process 

variability by making frequent adjustments to the process mean. To avoid the problem of 

overadjusting the process in response to early test results, the average absolute deviation from 

the target can be used for the acceptance decision. 

The equation for calculating AAD is as follows: 

n

X T

AAD ∑ i − = (4) 

where: Xi = individual test results 

T = target value 

n = number of tests per lot 

Simulation Analyses

A computer simulation program was developed to evaluate the accuracy and precision of 

estimated AAD values. The program generates 1000 simulated lots. The mean, or average, 

estimated AAD value and the standard deviation for the estimated AAD values are then 

calculated. 

The program can represent populations that are centered on the target value, but can also 

generate data for populations with means that are offset from the desired target. The actual AAD 

value for a population centered on the target is 0.798 multiplied by the standard deviation of the 

population. The actual AAD value increases as the population mean is offset from the target 

value. Without sacrificing generality, the simulation studies were performed for populations with 

standard deviation values of 1.00. For a population with a standard deviation other than 1.00, the 

simulated average AAD values would simply be multiplied by the standard deviation of the 

population in question. 

For the study of AAD accuracy and precision, populations were simulated with offsets from the 

target that ranged from 0.00 to 2.50 in increments of 0.25. The populations from which the 

samples were generated were normal distributions with a standard deviation of 1.00. 

The shape of the distribution of the sample AAD values varies with the sample size. For small 

mean offsets, the distribution is skewed to the right, with the degree of skewness determined by 

the sample size. For a sample size = 1, the peak of the distribution occurs at AAD values near 

zero. As the sample size increases, the value for the peak of the distribution increases and the 
50 

degree of skewness decreases, because as the sample size increases, it becomes more and more 

difficult to have average AAD values near zero. This relationship is shown in table 12. 

For a given sample size, the shape of the distribution of the sample AAD values also varies with 

the offset of the population mean from the target value. The skewness of the distribution stems 

from the fact that the deviations from the target cannot be below zero, thereby yielding 

distributions skewed away from zero. As the offset departs farther from the target, the 

distribution of the sample AAD values approaches symmetry since the deviation values are 

always greater than zero. This relationship is shown in table 13 for a sample size = 3 and for 

mean offsets = 0.50 to 2.50. 
51 

Table 12. Distributions of sample AAD values for a population centered on 

 the target and for sample sizes = 1, 3, 5, and 10. 

Sample Size, n Distribution of Sample AAD Values 

1 

3 

5 

10 
52 

Table 13. Distributions of sample AAD values for sample size = 3 and 

population means offset from the target by 0.50, 1.00, 1.50, 2.00, 

and 2.50 standard deviations. 

Mean Offset Distribution of Sample AAD Values 

0.50 

1.00 

1.50 

2.00 

2.50 
53 

Table 14 presents a summary of the simulation results for AAD. The table includes sample sizes 

= 3, 5, and 10, and populations with means offset from the target by 0.00 to 2.50 standard 

deviations in increments of 0.25. The table also includes the actual AAD values for each 

population, which are based on the mean offset values. Finally, the table includes the average 

differences and the standard deviations for the estimated AAD values. 

The results in table 14 show that the sample AAD is an unbiased estimator of the population 

AAD. It also shows that the standard deviation values decrease as the sample size increases and 

the standard deviation values increase as the offset from the target value increases. This is caused 

by the fact that as the offset increases, a deviation value of zero no longer presents a barrier, 

thereby allowing the deviations to spread evenly on both sides of the AAD value. These two 

trends are illustrated in figures 15 and 16. 

CI EVALUATION 

Conceptually, the CI is similar to AAD. While AAD uses the average of the absolute values of 

the individual deviations from the target value, the CI uses the squares of the individual 

deviations from the target value. The CI is also similar in concept to the standard deviation. The 

standard deviation is the root mean square of differences from the mean, whereas the CI is the 

root mean square of differences from a target. Like AAD, the CI discourages mid-lot process 

adjustments by not allowing positive and negative deviations from the target to cancel out one 

another. The CI is calculated as follows: 

n

X T

CI i

2 ( − ) = ∑ (5) 

where: Xi = individual test results 

T = target value 

n = number of tests per lot 

Simulation Analyses

A computer simulation program was developed to evaluate the accuracy and precision of 

estimated CI values. The program generated 10,000 simulated lots. The mean estimated CI value 

and the standard deviation for the estimated CI values are then calculated. 

The simulation studies present the CI in terms of population standard deviation. Without 

sacrificing generality, the simulation studies were performed for populations with standard 

deviations of 1.00. For a population with a standard deviation other than 1.00, the simulated 

average CI values would simply be multiplied by the standard deviation of the population in 

question. The program can represent populations that have various CI values. If the population 

mean is centered on the target, then the CI is the same as the standard deviation. Therefore, the 

CI cannot be less than zero. 
54 

Table 14. Accuracy and precision for AAD estimates 

 (based on the results of 10,000 simulated lots). 

Tests per Lot, 

n

Offset of 

Population 

Mean From 

Target 

Actual AAD of 

Population 

Mean of 

Estimated AAD 

Values Minus 

Actual AAD 

Standard 

Deviation of 

Estimated AAD 

Values 

0.00 0.798 –0.003 0.3495 

0.25 0.823 +0.002 0.3624 

0.50 0.896 +0.003 0.3863 

0.75 1.012 –0.001 0.4223 

1.00 1.167 +0.005 0.4649 

1.25 1.351 –0.005 0.4911 

1.50 1.559 –0.001 0.5200 

1.75 1.782 +0.008 0.5496 

2.00 2.017 +0.004 0.5529 

2.25 2.258 +0.008 0.5660 

3 

2.50 2.504 +0.003 0.5777 

0.00 0.798 +0.002 0.2717 

0.25 0.823 +0.000 0.2758 

0.50 0.896 +0.002 0.2964 

0.75 1.012 –0.002 0.3257 

1.00 1.167 –0.005 0.3575 

1.25 1.351 +0.004 0.3822 

1.50 1.559 +0.000 0.4049 

1.75 1.782 +0.006 0.4216 

2.00 2.017 +0.002 0.4220 

2.25 2.258 +0.002 0.4386 

5 

2.50 2.504 –0.004 0.4425 

0.00 0.798 –0.002 0.1905 

0.25 0.823 +0.000 0.1979 

0.50 0.896 –0.003 0.2101 

0.75 1.012 +0.003 0.2293 

1.00 1.167 +0.003 0.2561 

1.25 1.351 –0.001 0.2693 

1.50 1.559 –0.002 0.2848 

1.75 1.782 –0.002 0.2978 

2.00 2.017 +0.001 0.3033 

2.25 2.258 –0.003 0.3131 

10 

2.50 2.504 –0.004 0.3152 
55 

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Actual AAD

Average Difference

n= 3

n = 5

n =10

Figure 15. Plot of average difference of simulated AAD values versus 

 actual AAD values for sample sizes = 3, 5, and 10. 

 

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

Actual AAD

Standard Deviation

n = 3

n = 5

n = 10

Figure 16. Plot of how the standard deviation of AAD estimates varies 

 with the population AAD value. 
56 

For the study of CI accuracy and precision, populations were simulated with CI values ranging 

from 1.00 to 3.00. The populations from which the samples were generated were normal 

distributions with a standard deviation of 1.00. 

The shape of the distribution of the sample CI values varies with the sample size in a similar 

manner to that of AAD. For small CI values, the distribution is skewed to the right, with the 

degree of skewness determined by the sample size. As the sample size increases, the degree of 

skewness decreases, because as the sample size increases, it becomes more and more difficult to 

have average CI values near zero. 

Similar to that of AAD, for a given sample size, the shape of the distribution of the sample CI 

values varies with the CI of the population. For small values of CI, the skewness of the 

distribution stems from the fact that the squared deviations from the target cannot be below zero, 

thereby yielding distributions skewed away from zero. As the CI increases, meaning that the 

population mean is farther from the target, the distribution approaches symmetry since the 

squared deviation values are always greater than zero. 

The results in table 15 show that the sample CI appears to be a slightly biased estimator of the 

population CI since the average of the CI estimates is always lower than the population CI. This 

bias decreases as the sample size increases. The table also shows that the standard deviation 

values decrease as the sample size increases and the standard deviation values increase as the CI 

value for the population increases. These results are illustrated in figures 17 and 18. 

CONCLUSION 

Both the PWL and AAD estimators were unbiased and exhibited similar trends with respect to 

the variability of their individual estimates. The CI performed essentially the same as AAD, 

except that it appeared to be a slightly biased estimator. Since the CI offered no benefits 

compared to AAD and since it appeared slightly biased, it was decided to eliminate the CI and to 

concentrate further study on only PWL and AAD. 
57 

Table 15. Accuracy and precision for CI estimates 

 (based on the results of 10,000 simulated lots). 

Tests per Lot, 

n

Actual CI of 

Population 

Mean of 

Estimated CI 

Values Minus 

Actual CI 

Standard 

Deviation of 

Estimated CI 

Values 

1.00 –0.09 0.39 

1.10 –0.07 0.43 

1.25 –0.10 0.44 

1.40 –0.07 0.41 

1.50 –0.08 0.49 

1.75 –0.08 0.52 

2.00 –0.07 0.54 

2.25 –0.05 0.53 

2.50 –0.05 0.56 

3 

3.00 –0.05 0.56 

1.00 –0.05 0.31 

1.10 –0.05 0.34 

1.25 –0.04 0.37 

1.40 –0.06 0.38 

1.50 –0.08 0.39 

1.75 –0.02 0.39 

2.00 –0.06 0.43 

2.25 –0.01 0.43 

2.50 –0.04 0.44 

5 

3.00 –0.03 0.44 

1.00 –0.01 0.22 

1.10 –0.03 0.24 

1.25 –0.03 0.27 

1.40 –0.03 0.27 

1.50 –0.03 0.27 

1.75 –0.02 0.29 

2.00 –0.02 0.30 

2.25 –0.01 0.31 

2.50 0.00 0.30 

10 

3.00 –0.02 0.30 
58 

-0.10

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

Actual CI

Average Difference

n= 3

n = 5

n =10

Figure 17. Plot of average difference of simulated CI values versus 

 actual CI values for sample sizes = 3, 5, and 10. 

 

0.00

0.10

0.20

0.30

0.40

0.50

0.60

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

Actual CI

Standard Deviation

n = 3

n = 5

n = 10

Figure 18. Plot of how the standard deviation of CI estimates varies 

with the population CI value. 
59 

6. QUALITY MEASURES: NORMALITY ASSUMPTION 

INTRODUCTION 

As noted in chapter 5, the PWL and AAD quality measures were selected for additional analyses, 

while the CI was eliminated from further study. The use of each of these measures is based on 

the assumption that the population that is being sampled is normally distributed. To determine 

how robust these measures are, computer simulation was used to evaluate how they performed 

when the sampled population was not normally distributed. 

The most likely departures from normality would be skewed or bimodal distribution of data. 

Computer simulation programs were developed to evaluate the performance of the PWL and 

AAD estimators under each of these situations. The following sections of this chapter present the 

results for analyses conducted using these simulation programs. 

SKEWED DISTRIBUTIONS 

Skewed distributions usually occur because of some physical boundary that comes into play for a 

particular characteristic. For example, the percent passing a sieve for gradation analysis cannot 

exceed 100 percent. Thus, if the average percent passing is near 100, say 95 percent, it is 

possible to have greater spread on the low side of the average than on the high side. Another 

barrier might be pavement thickness, which cannot be less than zero. These cases are similar in 

concept to the discussions in chapter 5 regarding why the distributions of sample AAD and CI 

values have skewed distributions. The following sections present the results of computer 

simulation analyses on the effects of skewness on the estimates for PWL and AAD. 

PWL Evaluation 

Two computer simulation programs were developed to investigate the performance of the PWL 

estimating procedures when working with a skewed distribution. One program is for use with 

one-sided specifications, while the other is for two-sided specifications. Skewness is measured 

by a skewness coefficient that equals zero for a symmetrical distribution and which increases as 

the distribution becomes more highly skewed. The skewness for a data set can be calculated from 

the following equation: 

3

( 1)( 2)

∑ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −

− − = s

x X

n n

n skewness i (6) 

A skewness value of zero indicates that the distribution of data is symmetrical. The skewness can 

be either positive or negative. Positive skewness indicates a distribution with the long tail to the 

right, while negative skewness indicates that the long tail is to the left. Table 16 illustrates 

distributions with various levels of skewness. 
60 

Table 16. Distributions with various levels of skewness. 

Skewness Coefficient Positive Negative 

0.0 

0.5 

1.0 

1.5 

2.0 

3.0 
61 

Both of the programs begin with a data set that is normally distributed. An exponent, entered by 

the user, is then used to transform these data to the skewed distribution desired. 

One-Sided Limit: Computer simulation was used to evaluate the PWL estimates for populations 

with skewness coefficients of 0, +0.5, –0.5, +1.0, –1.0, +1.5, –1.5, +2.0, –2.0, +2.5, –2.5, +3.0, 

and –3.0. Sample sizes = 3, 5, and 10 were used to estimate the PWL values for populations with 

each of these skewness coefficients. In each analysis, 10,000 samples of the appropriate size 

were generated from a population with one of the skewness coefficients. The average bias and 

variability were then determined for the 10,000 PWL estimates. The results of these simulations 

are shown in tables 17 through 19 for sample sizes = 3, 5, and 10, respectively. 

For efficiency, the simulation program calculates the bias and standard deviation only for 

positive skewness coefficients, it then reverses these numbers and their signs to obtain the values 

for negative skewness (i.e., the bias for 90 PWL with positive skewness is equal to –1 times the 

bias for 10 PWL with negative skewness). This is apparent in tables 17 through 19. 

Figures 19 through 21 graphically present the bias results from tables 17 through 19, 

respectively. These figures plot the bias versus the actual PWL value for each of the levels of 

skewness. Figures 19 through 21 present the results for samples sizes = 3, 5, and 10, 

respectively. 

The trends are obvious. The greater the skewness, the greater the bias in the PWL estimate. This 

is logical since the distributions deviate more from normality (on which the PWL calculation is 

based) as the skewness increases. The values for positive and negative skewness are mirror 

images of each other. 

Table 20 presents a portion of the bias results from tables 17 through 19. The results are sorted 

by the amount of skewness, with skewness coefficients of 1.0, 2.0, and 3.0 included. Figure 22 

presents the bias results from table 20 plotted by sample sizes = 3, 5, and 10 for equal levels of 

skewness. It should be noted that for negative skewness, the magnitude of the biases would be 

the same; however, the plots would be flipped vertically about the bias value of zero. 

Interestingly, figure 22 shows that the larger the sample size, the greater the bias of the estimate. 

This probably stems from the fact that the larger sample does a better job of estimating PWL for 

the assumed symmetrical normal distribution. This leads to a greater bias since the population is 

actually a skewed distribution rather than a symmetrical normal distribution. 
62 

Table 17. Bias in estimating PWL for various skewness coefficients 

 and one-sided limits (3 tests per lot and 10,000 simulated lots). 

PWL +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 

99 +0.04 –0.35 –1.26 –2.33 –3.57 –4.96 –6.57 

95 –0.04 –0.60 –1.27 –2.35 –3.14 –4.31 –5.43 

90 –0.01 –0.33 –1.04 –1.63 –2.11 –2.88 –3.65 

80 +0.35 +0.23 +0.25 –0.23 –0.41 –0.45 –0.53 

70 +0.64 +0.72 +0.77 +1.42 +0.81 +1.41 +1.39 

60 –0.12 +0.11 +1.21 +1.69 +2.04 +2.53 +2.82 

50 –0.35 +0.61 +1.61 +2.13 +2.95 +3.20 +3.67 

40 +0.14 +0.83 +0.81 +2.03 +2.94 +3.13 +3.76 

30 –0.10 +0.61 +1.11 +1.63 +2.07 +2.22 +2.95 

20 –0.31 +0.07 +0.54 +0.90 +1.77 +1.43 +1.87 

10 +0.11 –0.19 –0.10 –0.16 +0.33 +0.51 +0.23 

 5 +0.02 –0.21 –0.05 –0.27 –0.33 –0.17 –0.10 

 1 +0.06 –0.12 –0.19 –0.22 –0.21 –0.20 –0.22 

PWL –0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 

99 –0.06 +0.12 +0.19 +0.22 +0.21 +0.20 +0.22 

95 –0.02 +0.21 +0.05 +0.27 +0.33 +0.17 +0.10 

90 –0.11 +0.19 +0.10 +0.16 –0.33 –0.51 –0.23 

80 +0.31 –0.07 –0.54 –0.90 –1.77 –1.43 –1.87 

70 +0.10 –0.61 –1.11 –1.63 –2.07 –2.22 –2.95 

60 –0.14 –0.83 –0.81 –2.03 –2.94 –3.13 –3.76 

50 +0.35 –0.61 –1.61 –2.13 –2.95 –3.20 –3.67 

40 +0.12 –0.11 –1.21 –1.69 –2.04 –2.53 –2.82 

30 –0.64 –0.72 –0.77 –1.42 –0.81 –1.41 –1.39 

20 –0.35 –0.23 –0.25 +0.23 +0.41 +0.45 +0.53 

10 +0.01 +0.33 +1.04 +1.63 +2.11 +2.88 +3.65 

 5 +0.04 +0.60 +1.27 +2.35 +3.14 +4.31 +5.43 

 1 –0.04 +0.35 +1.26 +2.33 +3.57 +4.96 +6.57 

Note: The values shown in the table are calculated by subtracting the actual population 

 PWL value from the average of the 10,000 estimated lot PWL values. 
63 

Table 18. Bias in estimating PWL for various skewness coefficients 

 and one-sided limits (5 tests per lot and 10,000 simulated lots). 

PWL +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 

99 +0.03 –0.78 –2.27 –4.31 –6.27 –8.22 –9.96 

95 –0.20 –0.97 –2.34 –4.05 –5.32 –6.89 –8.26 

90 –0.03 –0.56 –1.65 –2.54 –3.72 –4.76 –5.68 

80 –0.20 +0.22 +0.15 +0.14 +0.01 –0.40 –0.65 

70 –0.46 +1.04 +1.79 +2.37 +2.74 +3.04 +3.14 

60 –0.19 +1.24 +2.54 +3.58 +4.90 +5.36 +5.47 

50 –0.02 +1.15 +3.12 +4.42 +5.54 +6.50 +7.17 

40 –0.08 +1.44 +3.24 +4.68 +5.42 +6.36 +7.30 

30 –0.03 +1.17 +2.13 +3.02 +4.21 +5.29 +6.13 

20 +0.39 +0.16 +0.97 +1.58 +2.26 +2.82 +3.81 

10 –0.13 –0.44 –0.47 –0.12 +0.35 +0.43 +0.80 

 5 –0.04 –0.77 –0.86 –0.70 –0.77 –0.65 –0.57 

 1 0.00 –0.30 –0.40 –0.47 –0.45 –0.46 –0.36 

PWL –0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 

99 0.00 +0.30 +0.40 +0.47 +0.45 +0.46 +0.36 

95 +0.04 +0.77 +0.86 +0.70 +0.77 +0.65 +0.57 

90 +0.13 +0.44 +0.47 +0.12 –0.35 –0.43 –0.80 

80 –0.39 –0.16 –0.97 –1.58 –2.26 –2.82 –3.81 

70 +0.03 –1.17 –2.13 –3.02 –4.21 –5.29 –6.13 

60 +0.08 –1.44 –3.24 –4.68 –5.42 –6.36 –7.30 

50 +0.02 –1.15 –3.12 –4.42 –5.54 –6.50 –7.17 

40 +0.19 –1.24 –2.54 –3.58 –4.90 –5.36 –5.47 

30 +0.46 –1.04 –1.79 –2.37 –2.74 –3.04 –3.14 

20 +0.20 –0.22 –0.15 –0.14 –0.01 +0.40 +0.65 

10 +0.03 +0.56 +1.65 +2.54 +3.72 +4.76 +5.68 

 5 +0.20 +0.97 +2.34 +4.05 +5.32 +6.89 +8.26 

 1 –0.03 +0.78 +2.27 +4.31 +6.27 +8.22 +9.96 

Note: The values shown in the table are calculated by subtracting the actual population 

 PWL value from the average of the 10,000 estimated lot PWL values. 
64 

Table 19. Bias in estimating PWL for various skewness coefficients 

 and one-sided limits (10 tests per lot and 10,000 simulated lots). 

PWL +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 

99 +0.01 –1.15 –3.21 –5.77 –8.28 –10.61 –12.63 

95 +0.01 –1.35 –3.24 –5.24 –7.32 –9.02 –10.47 

90 +0.08 –0.76 –2.14 –3.55 –5.08 –6.39 –7.64 

80 0.00 +0.26 +0.41 –0.05 –0.54 –0.98 –1.69 

70 –0.29 +1.66 +2.49 +3.15 +3.45 +3.46 +3.49 

60 –0.18 +2.08 +4.11 +5.47 +6.41 +7.27 +7.72 

50 +0.04 +2.42 +4.76 +6.47 +8.27 +9.27 +10.17 

40 +0.02 +2.48 +4.54 +6.33 +8.29 +9.60 +11.11 

30 –0.05 +1.43 +3.63 +5.34 +7.02 +8.08 +9.75 

20 +0.09 +0.46 +1.60 +2.99 +3.79 +5.03 +6.29 

10 +0.06 –0.53 –0.64 –0.22 +0.03 +0.20 +1.06 

 5 –0.04 –0.78 –1.18 –1.29 –1.12 –0.97 –0.72 

 1 +0.01 –0.44 –0.64 –0.64 –0.69 –0.63 –0.60 

PWL –0.0 –0.5 –1.0 –1.5 –2.0 –2.5 –3.0 

99 –0.01 +0.44 +0.64 +0.64 +0.69 +0.63 +0.60 

95 +0.04 +0.78 +1.18 +1.29 +1.12 +0.97 +0.72 

90 –0.06 +0.53 +0.64 +0.22 –0.03 –0.20 –1.06 

80 –0.09 –0.46 –1.60 –2.99 –3.79 –5.03 –6.29 

70 +0.05 –1.43 –3.63 –5.34 –7.02 –8.08 –9.75 

60 –0.02 –2.48 –4.54 –6.33 –8.29 –9.60 –11.11 

50 –0.04 –2.42 –4.76 –6.47 –8.27 –9.27 –10.17 

40 +0.18 –2.08 –4.11 –5.47 –6.41 –7.27 –7.72 

30 +0.29 –1.66 –2.49 –3.15 –3.45 –3.46 –3.49 

20 +0.00 –0.26 –0.41 +0.05 +0.54 +0.98 +1.69 

10 –0.08 +0.76 +2.14 +3.55 +5.08 +6.39 +7.64 

 5 –0.01 +1.35 +3.24 +5.24 +7.32 +9.02 +10.47 

 1 –0.01 +1.15 +3.21 +5.77 +8.28 +10.61 +12.63 

Note: The values shown in the table are calculated by subtracting the actual population 

 PWL value from the average of the 10,000 estimated lot PWL values. 
65 

Figure 19a. Plots of bias versus actual PWL for 10,000 simulated lots with 3 

 tests per lot and one-sided limits showing positive skewness.

Figure 19b. Plots of bias versus actual PWL for 10,000 simulated lots with 3 

 tests per lot and one-sided limits showing negative skewness. 

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

99 95 90 80 70 60 50 40 30 20 10 5 1

Actual PWL

Bias

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Positive Skewness

n = 3

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

99 95 90 80 70 60 50 40 30 20 10 5 1

Acutal PWL

Bias

-0.0

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

Negative Skewness

n = 3
66 

Figure 20a. Plots of bias versus actual PWL for 10,000 simulated lots with 

5 tests per lot and one-sided limits showing positive skewness. 

Figure 20b. Plots of bias versus actual PWL for 10,000 simulated lots with 

 5 tests per lot and one-sided limits showing negative skewness.

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

99 95 90 80 70 60 50 40 30 20 10 5 1

Actual PWL

Bias

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Positive Skewness

n = 5

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

99 95 90 80 70 60 50 40 30 20 10 5 1

Actual PWL

Bias

-0.0

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

Negative Skewness

n = 5
67 

Figure 21a. Plots of bias versus actual PWL for 10,000 simulated lots with 10 

 tests per lot and one-sided limits with positive skewness.

Figure 21b. Plots of bias versus actual PWL for 10,000 simulated lots with 10 

 tests per lot and one-sided limits with negative skewness. 

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

99 95 90 80 70 60 50 40 30 20 10 5 1

Actual PWL

Bias

-0.0

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

Negative Skewness

n = 10

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

99 95 90 80 70 60 50 40 30 20 10 5 1

Actual PWL

Bias

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Positive Skewness

n = 10
68 

Table 20. Bias in estimating PWL for various skewness coefficients, 

 sample sizes, and one-sided limits (10,000 simulated lots). 

Skewness = +1.0 Skewness = +2.0 Skewness = +3.0 PWL

n = 3 n = 5 n = 10 n = 3 n = 5 n = 10 n = 3 n = 5 n = 10 

99 –1.26 –2.27 –3.21 –3.57 –6.27 –8.28 –6.57 –9.96 –12.63 

95 –1.27 –2.34 –3.24 –3.14 –5.32 –7.32 –5.43 –8.26 –10.47 

90 –1.04 –1.65 –2.14 –2.11 –3.72 –5.08 –3.65 –5.68 –7.64 

80 +0.25 +0.15 +0.41 –0.41 +0.01 –0.54 –0.53 –0.65 –1.69 

70 +0.77 +1.79 +2.49 +0.81 +2.74 +3.45 +1.39 +3.14 +3.49 

60 +1.21 +2.54 +4.11 +2.04 +4.90 +6.41 +2.82 +5.47 +7.72 

50 +1.61 +3.12 +4.76 +2.95 +5.54 +8.27 +3.67 +7.17 +10.17 

40 +0.81 +3.24 +4.54 +2.94 +5.42 +8.29 +3.76 +7.30 +11.11 

30 +1.11 +2.13 +3.63 +2.07 +4.21 +7.02 +2.95 +6.13 +9.75 

20 +0.54 +0.97 +1.60 +1.77 +2.26 +3.79 +1.87 +3.81 +6.29 

10 –0.10 –0.47 –0.64 +0.33 +0.35 +0.03 +0.23 +0.80 +1.06 

 5 –0.05 –0.86 –1.18 –0.33 –0.77 –1.12 –0.10 –0.57 –0.72 

 1 –0.19 –0.40 –0.64 –0.21 –0.45 –0.69 –0.22 –0.36 –0.60 

Note: The values shown in the table are calculated by subtracting the actual population PWL value from the 

average of the 10,000 estimated lot PWL values. 
69 

Figure 22a. Plot of bias versus actual PWL for 10,000 simulated lots with 

 various tests per lot and one-sided limits with +1 skewness.

Figure 22b. Plot of bias versus actual PWL for 10,000 simulated lots with 

 various tests per lot and one-sided limits with +2 skewness. 

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

99 95 90 80 70 60 50 40 30 20 10 5 1

Actual PWL

Bias

n = 3

n = 5

n = 10

Skewness = +1.0

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

99 95 90 80 70 60 50 40 30 20 10 5 1

Actual PWL

Bias

n = 3

n = 5

n = 10

Skewness = +2.0
70 

Figure 22c. Plot of bias versus actual PWL for 10,000 simulated lots with 

 various tests per lot and one-sided limits with +3 skewness. 

Two-Sided Limits: Computer simulation was used to study a number of different cases of 

skewed populations when there were two-sided specification limits. Skewness coefficients of 0, 

0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 were considered along with sample sizes of n = 3, 5, and 10. 

Because of the relative ease of use of PD for two-sided limits (in comparison to using PWL), the 

simulations were based on PD. Since it has been shown earlier in this report that PD and PWL 

are equivalent, but complementary, measures, the discussions of two-sided limits are based on 

PD rather than PWL. 

With two-sided specification limits, there are many ways in which a given PD value can be 

divided between being outside the lower and upper limits. The program (SKEWBIAS2H) that 

was developed simulates a number of different divisions of the PD areas. The divisions can be 

noted as PDL and PDU, where PDL is the percent defective below the lower specification limit 

and PDU is the percent defective above the upper specification limit. The ratio of the division of 

PDL and PDU used are 8/0, 7/1, 6/2, 5/3, 4/4, 3/5, 2/6, 1/7, and 0/8. The diagrams presented in 

figure 23 illustrate the PD divisions used. 

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

99 95 90 80 70 60 50 40 30 20 10 5 1

Actual PWL

Bias

n = 3

n = 5

n = 10

Skewness = +3.0
71 

Figure 23a. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the 

 PWL estimate for two-sided specification limits (skewness coefficient = +1.0, split=8/0).

Figure 23b. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the 

 PWL estimate for two-sided specification limits (skewness coefficient = +1.0, split=7/1).

Figure 23c. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the 

PWL estimate for two-sided specification limits (skewness coefficient = +1.0, split=6/2). 

Skewness = 1.0 PD = 50 

Split = 8/0 PD L = 50 

PD U = 0 

Skewness = 1.0, PD = 50 

Split = 8/0 PD L = 50 

PDU = 0 

Skewness = 1.0, PD = 50 

Split = 6/2 PD L = 37.5 

PDU = 12.5 

Skewness = 1.0, PD = 50 

Split = 7/1 PD L = 43.75 

PDU = 6.25 
72 

Figure 23d. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the 

 PWL estimate for two-sided specification limits (skewness coefficient = +1.0, split=5/3). 

Figure 23e. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the 

 PWL estimate for two-sided specification limits (skewness coefficient = +1.0, split=4/4). 

Figure 23f. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the 

 PWL estimate for two-sided specification limits (skewness coefficient = +1.0, split=3/5). 

Skewness = 1.0, PD = 50 

Split = 5/3 PD L = 31.25 

PDU = 18.75 

Skewness = 1.0, PD = 50

Split = 4/4

PD L = 25 PDU = 25

Skewness = 1.0, PD = 50 

Split = 3/5 PD L = 18.75 

PDU = 31.25 
73 

Figure 23g. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the 

PWL estimate for two-sided specification limits (skewness coefficient =+1.0, split = 2/6). 

Figure 23h. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the 

PWL estimate for two-sided specification limits (skewness coefficient = +1.0, split = 1/7). 

Figure 23i. Illustration of the divisions that SKEWBIAS2H uses to calculate bias in the PWL 

estimate for two-sided specification limits (skewness coefficient = +1.0, split = 0/8). 

Skewness = 1.0, PD = 50 

Split = 1/7 PD L = 6.25 

PDU = 43.75 

Skewness = 1.0, PD = 50 

Split = 0/8 PD L = 0 

PDU = 50

Skewness = 1.0, PD = 50 

Split = 2/6 PD L = 12.5 

PDU = 37.5 
74 

Tables 21 through 23 show simulation results that illustrate how the bias in estimating PD and, 

hence, PWL varies with the PDL/PDU division and sample size. In each analysis, 10,000 samples 

of the appropriate size were generated from a population with one of the skewness coefficients. 

The average bias and variability were then determined for the 10,000 PD estimates. 

Figures 24 through 26 present plots of the results from tables 21 through 23, which indicate that 

the amount of bias in the PD estimate depends on the amount of skewness, the actual PD of the 

sampled population, the division of PDL/PDU, and the sample size used. Figures 27 through 29, 

which also plot the results from tables 21 through 23, highlight the effect of sample size on bias 

in the PD estimate. 

As in the case of the one-sided specification limit shown earlier in this chapter, the larger the 

sample size, the greater the bias of the estimate. This probably stems from the fact that the larger 

sample does a better job of estimating PD for the assumed symmetrical normal distribution. This 

leads to a greater bias since the population is actually a skewed distribution rather than a 

symmetrical normal distribution. 

To illustrate how this estimating bias arises for skewed distributions, figure 30 shows the plots of 

two populations—one that is normally distributed and one that has a skewness coefficient = 

+1.0. These two populations have the same mean, 5.0, and the same standard deviation, 1.0. The 

solid vertical line in the figure represents the mean for the two distributions. For the normal 

distribution, the median, or 50th percentile, occurs at the same value as the mean (i.e., 5.0). 

However, for the skewed distribution, the median is at a smaller value of 4.85. 

Assume that the single lower specification limit is 5.0, meaning that the normal distribution has 

50 PD (or 50 PWL). However, since the skewed distribution has a median of 4.85, it will have 

greater than 50 PD (or less than 50 PWL). Since the quality index is used to estimate PD (or 

PWL) and since this requires an assumption that the population is normally distributed, when the 

quality index determines an estimated PD of 50 for the normal distribution, the skewed 

distribution will have greater than 50 PD. Thus, for a population with 50 PD and for a single 

lower specification limit, if the skewness is away from the limit, the quality index will, on 

average, underestimate the PD value of the skewed distribution. If the skewness were in the 

direction of the lower specification limit, then the quality index would, on average, overestimate 

the PD value. This is exactly what is shown in figures 26 and 29. Appendix F shows additional 

simple illustrations of how biased estimates can be obtained. 
75 

Table 21. Results of simulations with PD = 10, 10,000 simulated lots, 

 sample sizes = 3, 5, and 10, and two-sided limits. 

Skewness 

PDL/PDU

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

Sample Size = 3

8/0 –0.22 +0.05 +0.73 +1.83 +2.14 +3.20 +3.80 

7/1 +0.27 +0.14 +0.63 +1.34 +2.13 +3.24 +3.87 

6/2 –0.14 +0.01 +1.03 +1.25 +2.61 +3.33 +4.09 

5/3 –0.05 +0.12 +0.77 +1.89 +2.67 +3.87 +4.74 

4/4 –0.07 +0.15 +1.07 +1.64 +3.58 +4.25 +5.43 

3/5 +0.02 +0.14 +0.86 +2.17 +2.97 +4.52 +5.54 

2/6 –0.15 –0.03 +1.13 +2.03 +3.58 +4.79 +6.59 

1/7 –0.01 +0.43 +0.89 +1.94 +3.68 +5.31 +6.51 

0/8 –0.13 –0.06 –0.13 –0.05 +0.40 +0.11 +0.31 

Sample Size = 5

8/0 +0.07 +0.71 +1.56 +2.70 +3.63 +4.58 +5.64 

7/1 –0.02 +0.45 +1.44 +2.32 +3.43 +4.81 +5.78 

6/2 –0.29 +0.20 +1.41 +2.46 +3.81 +4.19 +6.39 

5/3 –0.23 +0.21 +1.48 +2.68 +4.47 +5.74 +6.82 

4/4 –0.02 +0.36 +1.76 +3.27 +4.66 +6.20 +7.77 

3/5 –0.36 +0.27 +1.80 +3.43 +5.47 +7.14 +8.28 

2/6 +0.13 +0.50 +1.79 +3.79 +5.61 +7.75 +9.76 

1/7 –0.01 +0.59 +1.98 +3.47 +6.19 +8.46 +10.03 

0/8 +0.11 –0.26 –0.36 –0.18 +0.45 +0.45 +0.64 

Sample Size = 10

8/0 +0.05 +0.92 +2.22 +3.57 +5.07 +6.27 +7.40 

7/1 +0.06 +0.56 +1.76 +3.26 +4.94 +6.26 +7.58 

6/2 +0.11 +0.42 +1.66 +3.31 +5.03 +6.65 +8.25 

5/3 –0.16 +0.46 +1.87 +3.64 +5.69 +7.52 +8.90 

4/4 +0.07 +0.50 +2.10 +3.97 +6.13 +8.03 +9.77 

3/5 +0.07 +0.60 +2.24 +4.40 +6.53 +8.79 +10.93 

2/6 –0.08 +0.68 +2.37 +4.96 +7.27 +9.77 +11.91 

1/7 +0.14 +0.53 +2.46 +5.32 +7.88 +10.77 +12.77 

0/8 +0.02 –0.61 –0.69 –0.21 –0.10 +0.43 +1.18 
76 

Table 22. Results of simulations with PD = 30, 10,000 simulated lots, 

 sample sizes = 3, 5, and 10, and two-sided limits. 

Skewness 

PDL/PDU

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

Sample Size = 3

8/0 +0.11 –0.49 –0.86 –1.34 –1.46 –1.33 –1.36 

7/1 –0.08 –0.64 –0.61 –1.01 –1.01 –1.13 –1.37 

6/2 +0.18 –0.36 –0.68 –0.48 –0.53 –0.01 +0.12 

5/3 –0.28 +0.13 +0.14 +0.40 +0.60 +1.01 +1.94 

4/4 +0.34 –0.37 +0.81 +1.37 +1.73 +2.30 +3.67 

3/5 –0.04 +0.63 +1.07 +2.15 +2.73 +3.75 +4.46 

2/6 –0.40 +0.64 +1.86 +3.05 +4.18 +5.50 +6.67 

1/7 –0.47 +0.59 +2.12 +3.66 +5.46 +7.58 +8.19 

0/8 +0.03 +0.27 +0.78 +1.19 +1.84 +2.10 +2.67 

Sample Size = 5

8/0 –0.07 –0.84 –1.96 –2.26 –2.69 –2.94 –3.08 

7/1 +0.35 –1.36 –2.02 –2.46 –2.52 –2.61 –2.32 

6/2 +0.13 –0.75 –1.31 –1.27 –1.06 –0.56 –0.27 

5/3 +0.20 –0.50 –0.66 –0.01 +1.04 +1.48 +1.80 

4/4 +0.04 +0.18 +0.88 +2.04 +2.75 +4.08 +5.25 

3/5 –0.02 +0.61 +1.97 +3.61 +5.10 +6.63 +7.79 

2/6 –0.26 +1.17 +3.19 +5.18 +7.25 +9.28 +11.22 

1/7 +0.02 +1.87 +3.99 +7.05 +9.18 +12.10 +14.07 

0/8 –0.03 +1.18 +1.81 +3.69 +4.31 +5.27 +5.99 

Sample Size = 10

8/0 +0.03 –1.44 –2.40 –3.14 –3.56 –3.63 –3.53 

7/1 +0.12 –2.01 –2.97 –3.34 –3.36 –3.04 –2.80 

6/2 +0.07 –1.28 –2.21 –1.83 –1.24 –0.57 +0.45 

5/3 –0.10 –0.75 –0.43 +0.36 +1.36 +2.59 +3.83 

4/4 –0.01 +0.33 +1.28 +2.57 +4.55 +6.19 +8.34 

3/5 +0.06 +1.11 +3.08 +5.29 +7.73 +10.08 +12.51 

2/6 –0.23 +2.13 +5.13 +7.98 +10.88 +13.38 +15.99 

1/7 –0.01 +2.59 +6.44 +10.01 +13.97 +16.84 +19.92 

0/8 +0.16 +1.56 +3.35 +5.40 +6.80 +8.17 +9.70 
77 

Table 23. Results of simulations with PD = 50, 10,000 simulated lots, 

 sample sizes = 3, 5, and 10, and two-sided limits. 

Skewness 

PDL/PDU

0.0 0.5 1.0 1.5 2.0 2.5 3.0 

Sample Size = 3

8/0 –0.08 –0.82 –1.37 –2.29 –3.12 –3.38 –3.40 

7/1 –0.39 –0.94 –1.77 –2.11 –2.39 –3.13 –3.11 

6/2 +0.41 –0.67 –1.42 –1.64 –1.59 –2.45 –2.33 

5/3 +0.05 –0.48 –0.40 –0.77 –0.48 –0.60 +0.08 

4/4 +0.13 –0.15 +0.59 +0.55 +1.00 +1.47 +1.96 

3/5 +0.25 +0.44 +0.97 +1.97 +2.56 +2.97 +4.16 

2/6 –0.04 +1.12 +1.69 +2.91 +4.14 +5.32 +5.71 

1/7 –0.12 +1.38 +2.64 +3.86 +5.23 +6.80 +8.22 

0/8 +0.05 +0.87 +1.41 +1.99 +2.92 +3.03 +4.11 

Sample Size = 5

8/0 –0.15 –1.44 –3.06 –4.67 –5.64 –6.45 –7.55 

7/1 –0.01 –2.15 –3.56 –4.90 –5.80 –6.24 –6.57 

6/2 +0.06 –1.39 –2.70 –3.46 –3.79 –3.97 –3.97 

5/3 +0.25 –1.16 –1.07 –1.11 –0.86 –0.60 –0.34 

4/4 –0.19 –0.04 +0.48 +1.17 +2.28 +2.91 +3.74 

3/5 +0.03 +0.89 +2.22 +3.53 +5.00 +6.51 +7.51 

2/6 –0.02 +1.73 +3.88 +6.27 +8.03 +9.95 +11.50 

1/7 0.00 +2.41 +5.30 +8.06 +10.93 +12.89 +15.06 

0/8 +0.07 +1.65 +3.17 +4.57 +5.42 +6.41 +7.43 

Sample Size = 10

8/0 –0.07 –2.26 –4.66 –6.69 –8.06 –9.33 –9.98 

7/1 –0.17 –3.22 –5.56 –6.96 –8.20 –8.83 –9.08 

6/2 +0.15 –2.22 –3.78 –4.69 –4.71 –4.90 –4.67 

5/3 +0.26 –1.01 –1.56 –1.31 –0.48 +0.41 +0.94 

4/4 +0.17 +0.04 +1.15 +2.38 +4.09 +5.65 +6.74 

3/5 –0.01 +1.41 +3.70 +5.56 +8.17 +10.36 +12.33 

2/6 0.00 +2.62 +5.93 +8.88 +11.84 +14.59 +17.13 

1/7 –0.05 +3.48 +7.56 +11.71 +15.36 +18.38 +20.71 

0/8 –0.08 +2.34 +4.93 +6.61 +8.03 +9.35 +10.39 
78 

Figure 24a. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots 

 with PD = 10, sample = 3, and two-sided limits.

Figure 24b. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots 

 with PD = 10, sample = 5 and two-sided limits. 

-2

0

2

4

6

8

10

12

14

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Positive Skewness

n = 3

-2

0

2

4

6

8

10

12

14

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Positive Skewness

n = 5
79 

Figure 24c. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots 

 with PD = 10, sample = 10, and two-sided limits. 

-2

0

2

4

6

8

10

12

14

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

0.0 0.5 1.0

1.5 2.0 2.5

3.0

Positive Skewness

n = 10
80 

Figure 25a. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots 

 with PD = 30, sample = 3, and two-sided limits.

Figure 25b. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots 

 with PD = 30, sample = 5, and two-sided limits. 

-4

-2

0

2

4

6

8

10

12

14

16

18

20

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Positive Skewness

n = 3

-4

-2

0

2

4

6

8

10

12

14

16

18

20

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Positive Skewness

n = 5
81 

Figure 25c. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots 

 with PD = 30, sample = 10, and two-sided limits. 

-4

-2

0

2

4

6

8

10

12

14

16

18

20

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Positive Skewness

n = 10
82 

Figure 26a. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots 

 with PD = 50, sample = 3 and two-sided limits.

Figure 26b. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots 

 with PD = 50, sample = 5, and two-sided limits. 

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

22

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Positive Skewness

n = 5

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

22

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Positive Skewness

n = 3
83 

Figure 26c. Plot of bias versus PDL/PDU divisions for 10,000 simulated lots 

 with PD = 50, sample = 10, and two-sided limits.

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

22

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Positive Skewness

n = 10
84 

Figure 27a. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots 

with PD = 10, skewness = 1, and two-sided limits. 

Figure 27b. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots 

 with PD = 10, skewness = 2, and two-sided limits. 

-2

0

2

4

6

8

10

12

14

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

n = 3

n = 5

n = 10

Skewness = +1.0

PD = 10

-2

0

2

4

6

8

10

12

14

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

n = 3

n = 5

n = 10

Skewness = +2.0

PD = 10
85 

Figure 28a. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots 

with PD = 30, skewness = 1, and two-sided limits.

Figure 28b. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots 

 with PD = 30, skewness = 2, and two-sided limits.

-4

-2

0

2

4

6

8

10

12

14

16

18

20

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

n = 3

n = 5

n = 10

Skewness = +1.0

PD = 30

-4

-2

0

2

4

6

8

10

12

14

16

18

20

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

n = 3

n = 5

n = 10

Skewness = +2.0

PD = 30
86 

Figure 29a. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots 

 with PD = 50, skewness = 1, and two-sided limits. 

Figure 29b. Plot of bias versus SKEWBIAS2H divisions for 10,000 simulated lots 

 with PD = 50, skewness = 2, and two-sided limits.

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

22

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

n = 3

n = 5

n = 10

Skewness = +1.0

PD = 50

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

22

8/0 7/1 6/2 5/3 4/4 3/5 2/6 1/7 0/8

PDL / PDU

Bias

n = 3

n = 5

n = 10

Skewness = +2.0

PD = 50
87 

Figure 30. Comparison of a normal population with a population 

 with skewness coefficient = +1.0. 

Variability of PWL/PD Estimates: Simulation was also used to evaluate the variability of PD 

estimates and to show how sample size affects this variability. Figure 31 shows an example of an 

output screen for a normal population (i.e., skewness coefficient = 0.00) that actually has a PD = 

10 and sample size = 5. The results of the nine divisions for PDL/PDU are clearly indicated. 

The BIAS/SE results indicate the average bias and the standard error of the bias, respectively. 

The bias values are relatively closely distributed about zero, indicating that the method for 

estimating PD (and, hence, PWL) is not biased for a symmetrical distribution. This is to be 

expected since the PD estimating method assumes a symmetrical normal distribution. 

The standard error values for sample size n = 5 are all 0.11. The standard error is determined by 

calculating the standard deviation of the 10,000 lot sample means and then dividing this standard 

deviation by the square root of 10,000 (i.e., the number of sample averages). The standard error 

is a measure of the variability of the estimate for the population mean. The standard error is not a 

measure of the variability of the individual lot sample means. However, it can be related to this 

individual sample mean variability. 

Table 8 shows the standard deviation values calculated from 1000 sample means. If the standard 

deviation values in table 8 were taken as estimates for the variability of the individual sample 

means, then these values would estimate the standard error values in figure 31 by dividing the 

table 8 values by the square root of 10,000, which is 100. The value in table 8 for a population 

with 90 PWL, which corresponds to 10 PD in figure 31, is 11.42. If you divide this number by 

100, the result is 0.1142, which is consistent with the standard error of 0.11 shown in figure 31. 

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5

Normal

Skew 1.0

 Normal Skewed

Mean 5.00 5.00

Median 5.00 4.85 
88 

Figure 32 shows portions of the output from three simulations, each with a population of PD = 

10 and skewness coefficient = 0.00. In the figure, the divisions for 8/0, 7/1, and 6/2 are shown 

for sample sizes = 3, 5, and 10. These histograms represent the individual bias values for the 

10,000 lots that were simulated. 

One thing is immediately apparent—the bias plots are not symmetrical. This is because of the 

natural boundaries of 0 PD and 100 PD. It is not possible to have less than 0 PD, so it is not 

possible to have a bias estimate of less than –10 PD. It is possible, however, to overestimate PD 

by as much as +90 PD. This is reflected in the plots that are skewed to the right, while having no 

bias values of less than –10. The second point to notice in the plots is that the variability of the 

bias results is related to sample size. This is shown in the standard error values and in the spread 

of the histogram plots. The larger the sample size, the smaller the standard error and the less 

spread there is in the bias histogram. 

As already shown in tables and figures above, the bias values vary with the amount of skewness, 

the sample size, the PD or PWL of the population, and the division of PD material outside the 

upper and lower specification limits. Figure 32 shows that the shape and the spread of the 

distribution of the sample means vary with the sample size. These distributions also vary in 

shape considerably with the population PD or PWL value. Appendix G contains a number of 

sample output screens that illustrate the shape and spread of the sample means for a variety of 

sample sizes and population PD values. 

While appendix G contains populations with skewness coefficients up to 3.00, in practice, it is 

unlikely that highway materials will have skewness values much greater than 1.00. Figure 33 

shows an example of an output screen for a population with a skewness coefficient = 1.00 that 

has a PD = 30 and sample size = 5. Figure 34 shows portions of the output screens for the same 

population, but with sample sizes = 3, 5, and 10. 
89 

Figure 31. Sample program output screen for a population with PD = 10, 

 skewness coefficient = 0.00, and sample size = 5. 
90 

Figure 32a. Portions of program output screen for PD = 10, skewness coefficient = 0.00, 

 sample=3. 

Figure 32b. Portion of program output screens for PD = 10, skewness coefficient = 0.00, 

 sample = 5.

Figure 32c. Portion of program output screens for PD = 10, skewness coefficient = 0.00, 

 sample = 10.
91 

Figure 33. Sample program output screen for a population with PD = 30, 

 skewness coefficient = 1.00, and sample size = 5.
92 

Figure 34a. Portion of program output screens for PD = 30, skewness coefficient = 1.00, 

 and sample = 3. 

Figure 34b. Portion of program output screens for PD = 30, skewness coefficient = 1.00, 

 and sample = 5. 

Figure 34c. Portion of program output screens for PD = 30, skewness coefficient = 1.00, 

 and sample = 10.
93 

AAD Evaluation 

A computer simulation program was developed to investigate the performance of the AAD 

estimating procedures when working with a skewed distribution. The program can generate a 

population with any mean and standard deviation, and with skewness coefficients for 

distributions that are symmetrical (0.0), negatively skewed (–3.0, –2.5, –2.0, –1.5, –1.0, –0.5), or 

positively skewed (+3.0, +2.5, +2.0, +1.5, +1.0, +0.5). The program determines the actual AAD 

for the population with the input values for mean, standard deviation, skewness coefficient, and 

offset of the mean from the target value. The program then generates 10,000 samples from the 

input distribution and determines the estimated AAD for each sample. The sample size can be 3 

to 30 values per sample. The program then determines the mean, standard deviation, standard 

error, minimum value, and maximum value for the 10,000 AAD values. It also prints a histogram 

for the generated AAD values. 

Figure 35 shows the effect of a skewed population on the actual population AAD for populations 

that are centered on the target value and that have skewness coefficients ranging from 0.0 to +3.0 

in increments of 0.5. The actual AAD values would be the same for negative skewness 

coefficients of the same magnitude as long as the population means were centered on the target. 

Since the population means are centered on the target values, the population AAD values 

actually decrease as the populations become more skewed. 

Figure 36 shows the effect of sample sizes = 3, 5, and 10 on the sampling distributions for the 

AAD estimates for populations that are centered on the target value and that have skewness 

coefficients of 0.0, +0.5, and +1.0. As would be expected, the spread of the AAD estimates 

decreases as the sample size increases. This is reflected in the reduced spread in the histogram 

plots and in the smaller standard deviation values as the sample size increases. It also appears 

that the mean of the sampling distribution approximates the actual population AAD value quite 

well regardless of the level of skewness in the population. 

Figure 37 shows three populations with the same standard deviation, but with mean offsets from 

the target = 0.0, 1.0, and 2.0. These mean offsets are measured in standard deviation units. As 

would be expected, the actual population AAD increases as the mean moves away from the 

target value. The mean of the estimated AAD values is quite close to the actual values regardless 

of the mean offset. However, the standard deviation (i.e., the spread) of the AAD sample values 

increases as the mean offset increases. 

Figure 38 shows three populations that have extremely different distributions, but which all have 

essentially equal AAD values. Although the means of the sampling distributions are about the 

same, the skewed distributions have greater spread in the AAD values for individual lots. It is not 

certain, but it seems unlikely that each of these populations would perform identically in service, 

even though they have the same population AAD values. 
94 

Figure 35a. Example of AAD distribution and actual values for populations centered on 

the target and with skewness coefficient of 0.

Figure 35b. Examples of AAD distribution and actual values for populations centered on 

the target and with skewness coefficients between .5 and 1.5.

Figure 35c. Examples of AAD distribution and actual values for populations centered on 

the target and with skewness coefficients between 2.0 and 3.0.
95 

Figure 36a. Comparison of the shapes and spread of estimated AAD values for populations 

centered on the target and with various skewness coefficients. 

Figure 36b. Comparison of the shapes and spread of estimated AAD values for populations 

centered on the target and with various skewness coefficients, sample size=3. 

Figure 36c. Comparison of the shapes and spread of estimated AAD values for populations 

centered on the target and with various skewness coefficients, sample size = 5. 
96 

Figure 36d. Comparison of the shapes and spread of estimated AAD values for populations 

centered on the target and with various skewness coefficients, sample size = 10. 
97 

Figure 37. Comparison of the shapes and spread of estimated AAD values for 

 normal populations centered on and offset from the target. 
98 

Figure 38. Example of populations that are very dissimilar in shape, 

but have approximately the same AAD. 
99 

There are obviously an infinite number of combinations of population mean offsets from the 

target and skewness coefficients. It is not possible to evaluate all of these combinations. 

However, it is possible to hold the others constant and change only one variable to see what 

effect this might have. 

For sample sizes = 3, 5, and 10, table 24 lists the actual AAD values, the mean of the AAD 

sample values (i.e., the bias), and the standard deviation values for a number of populations that 

are all centered on the target, but have varying skewness levels. Figure 39 plots the results from 

table 24. It is apparent that the means of the AAD sample estimates are generally close to the 

true AAD values. In other words, the AAD estimating process appears to be unbiased. However, 

the standard deviations of the AAD sample estimates increase as the skewness level increases, 

while they decrease with increasing sample size. 

For a sample size = 5, table 25 lists the actual AAD values, the mean of the AAD sample values 

(i.e., the bias), and the standard deviation values for a number of normal populations (i.e., 

skewness coefficient = 0.0) with centers that vary with respect to the amount of their offset from 

the target. Figure 40 plots the results from table 25. It is apparent that the means of the AAD 

sample estimates are generally close to the true AAD values. In other words, the AAD estimating 

process appears to be unbiased. However, the standard deviations of the AAD sample estimates 

increase as the mean offset from the target increases. 

For a sample size = 5, table 26 lists the actual AAD values, the mean of the AAD sample values 

(i.e., the bias), and the standard deviation values for a number of normal populations that are all 

centered on the target, but with varying population standard deviation values. Figure 41 plots the 

results from table 26. It is apparent that the means of the AAD sample estimates are generally 

close to the true AAD values. In other words, the AAD estimating process appears to be 

unbiased. However, the standard deviations of the AAD sample estimates increase as the 

population standard deviation increases. 
100 

Table 24. Bias and spread of the AAD sample estimates for populations centered 

on the target, but with various levels of skewness. 

n Offset Skewness Std. 

Dev. AAD Bias Std. Dev. of AAD Values 

0.00 0.7977 0.0042 0.3502 

0.50 0.7934 0.0048 0.3524 

1.00 0.7776 0.0081 0.3659 

1.50 0.7544 0.0013 0.3783 

2.00 0.7273 –0.0067 0.3997 

2.50 0.6993 0.0061 0.4202 

3 0.00 

3.00 

1.00 

0.6718 –0.0047 0.4158 

0.00 0.7977 0.0004 0.2693 

0.50 0.7934 0.0007 0.2709 

1.00 0.7776 –0.0008 0.2811 

1.50 0.7544 –0.0026 0.2908 

2.00 0.7273 –0.0012 0.3064 

2.50 0.6993 –0.0041 0.3136 

5 0.00 

3.00 

1.00 

0.6718 –0.0023 0.3316 

0.00 0.7977 –0.0009 0.1899 

0.50 0.7934 –0.0022 0.1919 

1.00 0.7776 0.0018 0.1975 

1.50 0.7544 0.0009 0.2069 

2.00 0.7273 0.0036 0.2191 

2.50 0.6993 –0.0012 0.2233 

10 0.00 

3.00 

1.00 

0.6718 0.0009 0.2334 

n = sample size, tests per lot 

Offset = offset of population mean from the target value, in units of standard 

 deviation 

Skewness = skewness coefficient for the population 

Std. Dev. = standard deviation for the population 

AAD = actual AAD value for the population 

Bias = average of the estimated AAD values minus the actual AAD value 

Std. Dev. of 

AAD Values = standard deviation of the estimated AAD values 
101 

Figure 39a. Bias of the AAD sample estimates for populations centered on the target, but 

with various levels of skewness. 

Figure 39b. Spread of the AAD sample estimates for populations centered on the 

target, but with various levels of skewness. 

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Skewness Coefficient

Bias

n = 3

n = 5

n = 10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Skewness Coefficient

Std Dev of Estimated AADs

n = 3

n = 5

n = 10
102 

Table 25. Bias and spread of the AAD sample estimates for normal populations 

 with various offsets from the target and n = 5. 

n Offset Skewness Std. 

Dev. AAD Bias Std. Dev. of AAD Values 

0.00 0.7977 –0.0014 0.2682 

0.25 0.8225 –0.0014 0.2808 

0.50 0.8955 0.0012 0.3000 

0.75 1.0123 0.0003 0.3258 

1.00 1.1666 0.0020 0.3590 

1.25 1.3511 –0.0030 0.3866 

1.50 1.5585 0.0037 0.4027 

1.75 1.7822 0.0007 0.4227 

5 

2.00 

0.00 1.00 

2.0168 0.0024 0.4280 

n = sample size, tests per lot 

Offset = offset of population mean from the target value, in units of standard 

 deviation 

Skewness = skewness coefficient for the population 

Std. Dev. = standard deviation for the population 

AAD = actual AAD value for the population 

Bias = average of the estimated AAD values minus the actual AAD value 

Std. Dev. of 

AAD Values = standard deviation of the estimated AAD values 
103 

Figure 40a. Bias of the AAD samples estimates for normal populations with various offsets 

from the target and n=5. 

Figure 40b. Spread of the Estimated AAD sample estimates for normal 

populations with various offsets from the target and n = 5.

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

0.005

0.00 0.50 1.00 1.50 2.00

Mean Offset from Target

Bias

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.00 0.50 1.00 1.50 2.00

Mean Offset from Target

Std Dev of Estimated AADs
104 

Table 26. Bias and spread of the AAD sample estimates for normal populations 

 with various standard deviation values and n = 5. 

n Offset Skewness Std. 

Dev. AAD Bias Std. Dev. of AAD Values 

1.00 0.7977 0.0014 0.2683 

1.25 0.9971 0.0010 0.3328 

1.50 1.1965 –0.0011 0.4037 

1.75 1.3960 0.0015 0.4784 

2.00 1.5954 –0.0037 0.5358 

2.25 1.7948 –0.0007 0.6050 

2.50 1.9942 0.0055 0.6773 

2.75 2.1936 0.0001 0.7341 

5 0.00 0.00 

3.00 2.3931 0.0012 0.8164 

n = sample size, tests per lot 

Offset = offset of population mean from the target value, in units of standard 

 deviation 

Skewness = skewness coefficient for the population 

Std. Dev. = standard deviation for the population 

AAD = actual AAD value for the population 

Bias = average of the estimated AAD values minus the actual AAD value 

Std. Dev. of 

AAD Values = standard deviation of the estimated AAD values 
105 

 

 Figure 41a. Bias of the AAD sample estimates for normal populations centered on 

 the target, with various standard deviation values, and n=5.

 Figure 41b. Standard Deviation of the AAD sample estimates for normal 

 populations centered on the target, with various standard deviation 

 values, and n = 5. 

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

1.00 1.50 2.00 2.50 3.00

Population Standard Deviation

Bias

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.00 1.50 2.00 2.50 3.00

Population Standard Deviation

Std Dev of Estimated AADs
106 

Comparison of PWL and AAD 

Both PWL and AAD seemed to perform reasonably well when used with skewed distributions. 

They both provided bias estimates that were close to the actual values. They both had decreasing 

variability of the estimated values as the sample size increased. They both had increasing 

variability of the estimated values as the departures from normality became more pronounced. 

However, it is difficult to compare the performance of the two since the magnitude of the 

numbers that are calculated varies so greatly. The PWL (or PD) values can range from 0 to 100, 

whereas AAD values will typically range from 0 to a value of less than 10. Also, if the PWL and 

AAD results were to be converted to payment factors by some equation, the multiplier for the 

AAD equation is likely to be much larger than the one for the PWL equation. 

While no definitive conclusion can be drawn, both methods will probably provide satisfactory 

results for the amount of skewness that is likely to be found in highway materials applications. 

BIMODAL DISTRIBUTIONS 

It is possible that a process that is believed to be normally distributed could change, thereby 

resulting in a bimodal distribution. This could happen, for instance, if the asphalt content was 

changed at a mixing plant and the data from before and after the change were considered in the 

same lot calculations. Another way in which a bimodal distribution could appear is if there were 

two different plants providing materials for the same project. If the plants had different process 

means, then the combined output from the two plants might result in a bimodal distribution. 

PWL Evaluation 

A computer simulation program was developed to investigate the performance of the PWL 

estimating procedures when working with a bimodal distribution. The program forms a bimodal 

distribution in the same manner in which it would happen in actual practice—by combining two 

different normal populations. The output screen for the program provides a plot of the 

distribution of each of the individual populations and a plot for the combined distribution. 

When two populations are combined, they can differ in three ways: 

• Different means, but the same standard deviations. 

• Same means, but different standard deviations. 

• Both different means and different standard deviations. 

The program can handle any of these three situations. Each of these three scenarios is considered 

in the following sections: 

Different Means: A sample output screen from the program is shown in figure 42. In this 

example, two identical populations are combined. Each population, shown as Distribution #1 and 

Distribution #2 in figure 42, has a mean of 10.00 and a standard deviation of 1.00. Each 

population also has a PD of 9.99. The top two plots in figure 42 represent these populations. The 

combined population has a mean = 10.00, standard deviation = 1.00, and PD = 9.99. Figure 42 
107 

also shows the lower specification limit, Limit1, to be 8.72 and the upper specification limit, 

Limit2, to be 999.00. These limits are input by the user. Inputting a very large upper limit that 

will never be reached makes this an example for a specification item that has only a lower limit. 

By trial and error, input limits can be easily developed to represent any value for the population 

PD (or PWL) value. The bottom plot in figure 42 represents the combined population. 

The right side of the screen shows the results of the simulation of, in this example, 1000 

replications of a sample size = 5. The average of these 1000 sample PD values is shown as 10.16, 

with a standard deviation of 11.46. A histogram of the differences between the individual sample 

PD values and the true population PD value is also shown. In the bottom right corner of the 

screen, Summary of Results shows that the PD bias was 0.16 and that we are 95-percent 

confident that the true value falls within the range of 0.16 ±0.71. 

It is to be expected that if two identical distributions are combined, the result will be the same 

distribution. What will happen then if distributions with different means, but the same standard 

deviations, are combined? Figure 43 shows, for a sample size = 5, the output screens for cases 

where the offset between the population means varies from 1 to 5 standard deviations. Note that 

for 1 and 2 standard deviation offsets, the combined populations still appear to be approximately 

normal. Not until the mean offset reaches 3 does the bimodal shape begin to become apparent. It 

is unlikely that distributions that are so different would be combined, and if they were combined, 

problems would probably arise that would make the error obvious. This means that for most 

practical cases, if two distributions with the same standard deviation were combined, the 

combined distribution would not differ enough from normal to adversely affect the PD or PWL 

estimates. 

Table 27 presents the results of simulation analyses in which two distributions with equal 

standard deviations and a variety of mean offsets were combined. This table represents the case 

of a single specification limit. Table 28 provides similar results for simulation analyses of twosided specifications. The tables indicate that the average bias values are reasonably small, 

particularly for mean offsets of up to 2 or 3 standard deviations. 

Different Standard Deviations: The combining of two normal distributions with different 

standard deviations, but with equal means, presents no different situation with respect to 

estimating PWL than does a single normal distribution. This is because whenever two normal 

distributions with equal means are combined, the resulting distribution will always be another 

normal distribution, regardless of the values of their standard deviations. This is illustrated in 

figure 44, which shows example output screens in which several pairs of normal distributions 

with differing standard deviations are combined. The resulting combined distribution always has 

the same mean as the two initial distributions and the only difference is the standard deviation for 

the combined distribution. Therefore, no additional discussion is necessary for the case of 

combining normal distributions with equal means. 
108 

Figure 42. Sample output screen from the simulation program 

 for bimodal distributions. 
109 

Figure 43a. Program output screens for sample size = 5 and mean offsets = 1.

 Figure 43b. Program output screens for sample size = 5 and mean offsets = 2.
110 

Figure 43c. Program output screens for sample size = 5 and mean offsets = 3.

 Figure 43d. Program output screens for sample size = 5 and mean offsets = 4.
111 

 Figure 43e. Program output screens for sample size = 5 and mean offsets = 5. 
112 

Table 27. Bias results for combining two populations with equal σ

with one-sided limits. 

n Offset Actual PD Average 

PD 

Average 

Bias 95% CI Different? 

5 0.0 9.99 10.16 +0.16 ±0.71 N 

5 1.0 9.99 10.18 +0.19 ±0.68 N 

5 2.0 9.99 9.37 –0.62 ±0.62 N 

5 3.0 10.00 9.90 –0.09 ±0.64 N 

5 4.0 9.99 10.80 +0.80 ±0.62 S 

5 5.0 9.99 11.82 +1.82 ±0.60 S 

3 0.0 9.99 10.00 +0.01 ±0.97 N 

3 1.0 9.99 10.32 +0.33 ±0.94 N 

3 2.0 9.99 9.25 –0.74 ±0.89 N 

3 3.0 10.00 9.74 –0.26 ±0.88 N 

3 4.0 9.99 10.57 +0.58 ±0.92 N 

3 5.0 9.99 11.85 +1.86 ±0.89 S 

5 0.0 30.02 29.62 –0.39 ±1.05 N 

5 1.0 30.01 29.25 –0.75 ±1.06 N 

5 2.0 30.01 27.27 –1.74 ±1.09 S 

5 3.0 30.01 26.40 –3.61 ±0.99 S 

5 4.0 30.00 24.67 –5.33 ±0.99 S 

5 5.0 30.01 23.17 –6.84 ±0.94 S 

3 0.0 30.02 29.91 –0.10 ±1.54 N 

3 1.0 30.01 29.81 –0.20 ±1.52 N 

3 2.0 30.01 30.38 +0.37 ±1.45 N 

3 3.0 30.01 27.39 –2.62 ±1.44 S 

3 4.0 30.00 26.63 –3.37 ±1.36 S 

3 5.0 30.01 25.47 –4.54 ±1.34 S 

n = sample size 

Offset = offset of the two population means in terms of the average σ of the 

 populations 

Actual PD = actual percent defective (PD) of the combined population 

Average PD = average PD of 1000 simulated samples from the combined population 

Average Bias = average simulated PD minus the actual PD 

95% CI = interval within which 95 percent of the average bias values should fall 

Different? = whether or not the average bias is significantly different from zero at the 

 0.05 level (N = not significantly different, S = significantly different) 
113 

Table 28. Bias results for combining two populations with equal σ

 with two-sided limits equidistant from the mean. 

n Offset Actual PD Average 

PD 

Average 

Bias 95% CI Different? 

5 0.0 10.00 9.38 –0.61 ±0.66 N 

5 1.0 10.00 10.21 +0.21 ±0.66 N 

5 2.0 9.99 10.47 +0.48 ±0.66 N 

5 3.0 9.99 12.46 +2.47 ±0.63 S 

5 4.0 9.99 14.51 +4.52 ±0.59 S 

5 5.0 9.99 16.40 +6.41 ±0.57 S 

3 0.0 10.00 10.70 +0.71 ±0.98 N 

3 1.0 10.00 10.33 +0.34 ±0.94 N 

3 2.0 9.99 10.98 +0.99 ±0.96 S 

3 3.0 9.99 11.30 +1.31 ±0.91 S 

3 4.0 9.99 13.06 +3.07 ±0.90 S 

3 5.0 9.99 11.30 +1.31 ±0.91 S 

5 0.0 30.02 29.83 –0.19 ±1.03 N 

5 1.0 30.02 30.05 +0.03 ±0.99 N 

5 2.0 30.02 28.77 –1.24 ±0.93 S 

5 3.0 30.00 27.23 –2.77 ±0.81 S 

5 4.0 30.02 28.90 –1.12 ±0.70 S 

5 5.0 30.02 29.03 –0.99 ±0.59 S 

3 0.0 30.02 30.14 +0.12 ±1.54 N 

3 1.0 30.02 28.80 –1.21 ±1.52 N 

3 2.0 30.02 28.33 –1.69 ±1.45 S 

3 3.0 30.00 29.89 –0.11 ±1.44 N 

3 4.0 30.02 29.61 –0.42 ±1.36 N 

3 5.0 30.02 30.71 +0.70 ±1.34 N 

n = sample size 

Offset = offset of the two population means in terms of the average σ of the 

 populations 

Actual PD = actual percent defective (PD) of the combined population 

Average PD = average PD of 1000 simulated samples from the combined population 

Average Bias = average simulated PD minus the actual PD 

95% CI = interval within which 95 percent of the average bias values should fall 

Different? = whether or not the average bias is significantly different from zero at the 

 0.05 level (N = not significantly different, S = significantly different) 
114 

 Figure 44a. Illustration 1 of program output screens when combining distributions 

 with equal means for sample size = 5. 

 Figure 44b. Illustration 2 of program output screens when combining distributions 

 with equal means for sample size = 5. 
115 

 Figure 44c. Illustration 3 of program output screens when combining distributions 

 with equal means for sample size = 5. 

 Figure 44d. Illustration 4 of program output screens when combining distributions 

 with equal means for sample size = 5. 
116 

Different Means and Different Standard Deviations: When populations with different means 

and different standard deviations are combined, the shape of the combined distribution depends 

on the amount of difference between them. This is illustrated in the combined distribution shapes 

in table 29. The original distributions in this figure are different in both mean and standard 

deviation. The ratio of the standard deviations of the two populations is 2.0. The offset of the two 

means is stated in terms of the weighted standard deviation. This is simply the arithmetic average 

of the standard deviations for the two distributions that are combined. 

For small offsets, the combined distribution is simply a unimodal skewed distribution. It is not 

until the weighted offset reaches 2.0 that a slight bulge appears in the long tail of the skewed 

distribution. This bulge becomes more distinct and the combined distribution approaches a true 

bimodal shape as the weighted offset increases to above 3.0. 

To investigate the performance of the PWL estimator when two different distributions are 

combined, the simulation program was used to generate 10,000 samples of size n = 5 from 

bimodal distributions with 10 PD (or 90 PWL). The PD estimate for each sample was then 

calculated and compared with the known actual PD of 10 (PWL of 90). The bias was then 

calculated as the difference between the sample PD and the actual PD. The average bias and 

variability for the 10,000 samples were then determined. 

In the analyses, three different cases were considered: 

• One-sided lower specification limit. 

• Two-sided specification limit with the limits equidistant about the population mean. 

• Two-sided specification limit with 5 percent of the population outside of each limit. 

For each of the three specification limit cases, three scenarios were considered for the bimodal 

distributions. When combining the distributions, three different standard deviation relationships 

were used. First, equal standard deviations were used for the two distributions. This generated 

bimodal distributions that were symmetrical. These were used as controls against which to 

compare the other two cases. Two different standard deviation relationships were used to 

generate skewed bimodal distributions. In these two cases, the ratios of the standard deviations 

for the two combined populations were 1.5 and 2.0. 

The t-statistic for a significance level of 0.05 and the 95-percent confidence interval for the 

average PD were calculated and used to determine whether the average bias was statistically 

significantly different from zero. The results of these analyses are presented in table 30. The bias 

in the PD (and, therefore, the PWL) estimates is more pronounced as the weighted offset 

between the two combined populations increases. 

However, for the smaller weighted mean offsets, up to about 1.0 to 2.0, depending on the 

standard deviation ratio, the bias in estimating PD is reasonably small and sometimes not 

significantly different from zero. 
117 

Table 29. Shapes of combined distributions when the combined distributions 

have different means and standard deviations.

Ratio of Standard 

Deviations 

Mean Offset, in Weighted 

Standard Deviation Units 

Shape of Combined 

Distribution 

2.0 0.00 

2.0 0.67 

2.0 1.33 

2.0 2.00 

2.0 2.67 

2.0 3.33 
118 

Table 30. Bias in estimating PD when two populations are combined 

 (sample size = 5, 10,000 simulated lots). 

Std. Dev. 

Ratio 

Weighted 

Mean Offset Actual PD Bias* 95% CI 

One-Sided Specification Limits 

0.00 9.99 0.11 0.22 

1.00 9.99 –0.07 0.22 

2.00 9.99 –0.07 0.21 

3.00 10.00 0.24 0.20 

4.00 9.99 1.02 0.19 

1.0 

5.00 9.99 1.80 0.19 

0.00 10.00 0.13 0.23 

0.80 10.00 0.56 0.22 

1.60 10.00 1.08 0.20 

2.40 10.00 1.37 0.20 

3.20 10.00 1.93 0.19 

1.5 

4.00 10.00 2.53 0.02 

0.00 10.00 0.98 0.24 

0.67 10.00 1.50 0.23 

1.33 9.99 2.14 0.21 

2.00 10.00 2.37 0.20 

2.67 9.99 2.72 0.19 

2.0 

3.33 10.00 3.13 0.19 

Two-Sided Specification Limits—Equidistant From the Mean 

0.00 10.00 0.05 0.22 

1.00 10.00 –0.02 0.21 

2.00 9.99 0.56 0.20 

3.00 9.99 2.27 0.20 

4.00 9.99 4.61 0.19 

1.0 

5.00 9.99 6.95 0.18 

0.00 9.99 –0.11 0.22 

0.80 10.00 0.07 0.22 

1.60 10.00 0.57 0.22 

2.40 10.00 1.42 0.21 

3.20 10.00 2.94 0.20 

1.5 

4.00 10.00 4.70 0.19 

0.00 10.00 –0.37 0.24 

0.67 10.00 –0.25 0.24 

1.33 10.00 0.54 0.23 

2.00 10.00 1.16 0.23 

2.67 10.00 2.26 0.22 

2.0 

3.33 10.00 3.44 0.22 

*Bias values that are NOT significantly different from zero are bolded. 
119 

Table 30. Bias in estimating PD when two populations are combined 

 (sample size = 5, 10,000 simulated lots) (continued). 

Std. Dev. 

Ratio 

Weighted 

Mean Offset Actual PD Bias* 95% CI 

Two-Sided Specification Limits—5 PD in Each Tail 

0.00 10.00 0.05 0.22 

1.00 10.00 –0.02 0.21 

2.00 9.99 0.56 0.20 

3.00 9.99 2.27 0.20 

4.00 9.99 4.61 0.19 

1.0 

5.00 9.99 6.95 0.18 

0.00 9.99 0.16 0.23 

0.80 10.00 0.30 0.22 

1.60 10.00 0.75 0.21 

2.40 10.00 1.79 0.20 

3.20 10.00 3.23 0.20 

1.5 

4.00 10.01 5.04 0.19 

0.00 10.00 –0.32 0.24 

0.67 10.00 0.41 0.23 

1.33 10.00 1.14 0.22 

2.00 10.00 2.16 0.21 

2.67 10.00 2.97 0.20 

2.0 

3.33 10.01 4.48 0.20 

*Bias values that are NOT significantly different from zero are bolded. 

AAD Evaluation 

No AAD evaluation was conducted for bimodal distributions. The results from the PD/PWL 

evaluation presented above indicated that for the types of mean and standard deviation 

differences between populations that are likely to be combined in typical materials and 

construction operations, the result would be either approximately normal or unimodal skewed 

distributions. Since the previous AAD analyses presented above in this chapter had already 

evaluated the AAD bias with respect to normal and skewed distributions, it was decided that no 

additional AAD evaluation was necessary. 

CONCLUSION 

Both the PWL (and PD) estimators and the AAD estimators provided estimates with relatively 

little bias for the departures from normality that are likely to be encountered. Neither was 

obviously better or worse than the other. Based on the non-normality analysis, there is no reason 

to eliminate either of these quality measures, nor is there a compelling reason to select one over 

the other. It was decided to continue investigating both PWL and AAD as possible quality 

measures for use when developing payment relationships.

121 

7. VERIFICATION PROCEDURES 

INTRODUCTION 

As part of the acceptance procedures and requirements, one question that must be answered is 

“Who is going to perform the acceptance tests?” The agency may either decide to do the 

acceptance testing, assign the testing to the contractor, have a combination of agency and 

contractor acceptance testing, or require a third party to do the testing. 

The decision as to who does the testing usually emanates from the agency’s personnel 

assessment, particularly in the days of agency downsizing. Many agencies are requiring the 

contractor to do the acceptance testing. This is at least partially because of agency staff 

reductions. What has often evolved is that the contractor is required to perform both QC and 

acceptance testing. If the contractor is assigned the acceptance function, the contractor’s 

acceptance tests must be verified by the agency. The agency’s verification sampling and testing 

function has the same underlying function as the agency’s acceptance sampling and testing—to 

verify the quality of the product. Statistically sound verification procedures must be developed 

that require a separate verification program. There are several forms of verification procedures 

and some forms are more efficient than others. To avoid conflict, it is in the best interests of both 

parties to make the verification process as effective and efficient as possible. 

The sources of variability are important when deciding what type of verification procedures to 

use. This decision depends on what the agency wants to verify. Independent samples (i.e., those 

obtained without respect to each other) contain up to four sources of variability: material, 

process, sampling, and testing. Split samples contain variability only in the testing method. Thus, 

if the agency wishes to verify only that the contractor’s testing methods are correct, then the use 

of split samples is best. This is referred to as test method verification. If the agency wishes to 

verify the contractor’s overall production, sampling, and testing processes, then the use of 

independent samples is required. This is referred to as process verification. Each of these types 

of verification is evaluated in the following sections. 

HYPOTHESIS TESTING AND LEVELS OF SIGNIFICANCE 

Before discussing the various procedures that can be used for test method verification or process 

verification, two concepts must be understood: hypothesis testing and level of significance. When 

it is necessary to test whether or not it is reasonable to accept an assumption about a set of data, 

statistical tests (called hypothesis tests) are conducted. Strictly speaking, a statistical test neither 

proves nor disproves a hypothesis. What it does is prescribe a formal manner in which evidence 

is to be examined to make a decision regarding whether or not the hypothesis is correct. 

To perform a hypothesis test, it is first necessary to define an assumed set of conditions known as 

the null hypothesis (H0). Additionally, an alternative hypothesis (Ha) is, as the name implies, an 

alternative set of conditions that will be assumed to exist if the null hypothesis is rejected. The 

statistical procedure consists of assuming that the null hypothesis is true and then examining the 

data to see if there is sufficient evidence that it should be rejected. The H0 cannot actually be 
122 

proved, only disproved. If the null hypothesis cannot be disproved (or, to be statistically correct, 

rejected), it should be stated that we fail to reject, rather than prove or accept, the hypothesis. In 

practice, some people use accept rather than fail to reject, although this is not exactly statistically 

correct. 

Verification testing is simply hypothesis testing. For test method or process verification 

purposes, the null hypothesis would be that the contractor’s tests and the agency’s tests have 

equal means, while the alternate hypothesis would be that the means are not equal. 

Hypothesis tests are conducted at a selected level of significance, α, where α is the probability of 

incorrectly rejecting the H0 when it is actually true. The value of α is typically selected as 0.10, 

0.05, or 0.01. For example, if α = 0.01 and the null hypothesis is rejected, then there is only 1 

chance in 100 that H0 is true and was rejected in error. 

The performance of hypothesis tests, or verification tests, can be evaluated by using OC curves. 

OC curves plot either the probability of not detecting a difference (i.e., accepting the null 

hypothesis that the populations are equal) or the probability of detecting a difference (i.e., 

rejecting the null hypothesis that the populations are equal) versus the actual difference between 

the two populations being compared. Curves that plot the probability of detecting a difference 

are sometimes call power curves because they plot the power of the statistical test procedure to 

detect a given difference. 

Just as there is a risk of incorrectly rejecting the H0 when it is actually true, which is called the 

type I (or α) error, there is also a risk of failing to reject the H0 when it is actually false. This is 

called the type II (or β) error. The power is the probability of rejecting the H0 when it is actually 

false and it is equal to 1 – β. Both α and β are important and are used with the OC curves when 

determining the appropriate sample size to be used. 

TEST METHOD VERIFICATION 

The procedures for verifying the testing procedures should be based on split samples so that the 

testing method is the only source of variability present. The two procedures used most often for 

test method verification are: (1) comparing the difference between the split-sample results to a 

maximum allowable difference, and (2) the use of the t-test for paired measurements (i.e., the 

paired t-test). In this report, these are referred to as the maximum allowable difference and the 

paired t-test, respectively, and each is discussed below. 

Maximum Allowable Difference 

This is the simplest procedure that can be used for verification, although it is the least powerful. 

In this method, usually a single sample is split into two portions, with one portion tested by the 

contractor and the other portion tested by the agency. The difference between the two test results 

is then compared to a maximum allowable difference. Because the procedure uses only two test 

results, it cannot detect real differences unless the results are far apart. 
123 

The value selected for the maximum allowable difference is usually selected in the same manner 

as the D2S limits contained in many American Association of State Highway and Transportation 

Officials (AASHTO) and American Society for Testing and Materials (ASTM) test procedures. 

The D2S limit indicates the maximum acceptable difference between two results obtained on test 

portions of the same material (and thus applies only to split samples) and is provided for singleand multi-laboratory situations. It represents the difference between two individual test results 

that has approximately a 5-percent chance of being exceeded if the tests are actually from the 

same population. 

Stated in general statistical terminology, the maximum allowable difference is set at two times 

the standard deviation of the distribution of the differences that would be obtained if the two test 

populations (the contractor’s and the agency’s) were actually equal. In other words, if the two 

populations are truly the same, there is approximately a 0.05 chance that this verification method 

will find them to be not equal. Therefore, the level of significance is 0.05 (5 percent). 

OC Curves: OC curves were developed to evaluate the performance of the maximum allowable 

difference method for test method verification. In this method, a test is performed on a single 

split sample to compare the agency’s and the contractor’s test results. If we assume that both of 

these split test results are from normally distributed subpopulations, then we can calculate the 

variance of the difference and use it to calculate two standard deviation limits (approximately 95 

percent) for the sample difference quantity. 

Suppose that the agency’s subpopulation has a variance 2 σ A and the contractor’s subpopulation 

has a variance 2 σ C . Since the variance of the difference in two independent random variables is 

the sum of the variances, the variance of the difference in an agency’s observation and a 

contractor’s observation is 2 2 σ A +σ C . The maximum allowable difference is based on the test 

standard deviation, which may be provided in the form of D2S limits. Let us call this test 

standard deviationσ test . Under an assumption that 2 2 2 σ A = σ C = σ test , this variance of a difference 

becomes 2 2σ test . 

The maximum allowable difference limits are set as two times the standard deviation of the test 

differences (i.e., approximately 95-percent limits). This, therefore, sets the limits at 2 2 2 ± σ test , 

which is σ test ± 2 2 (or ±2.8284σ test ). Without loss of generality, we can assume 1 σ test = , along 

with an assumption of a mean difference of 0, and use the standard normal distribution with a 

region between –2.8284 and +2.8284 as the acceptance region for the difference in an agency’s 

test result and a contractor’s test result. With these two limits fixed, we can calculate the power 

of this decisionmaking process relative to various true differences in the underlying 

subpopulation means and/or various ratios of the true underlying subpopulation standard 

deviations. 

These power values can conveniently be displayed as a three-dimensional surface. If we vary the 

mean difference along the first axis and the standard deviation ratio along a second axis, we can 

show power on the vertical axis. The agency’s subpopulation, the contractor’s subpopulation, or 

both, could have standard deviations that are smaller, about the same, or larger than the supplied 
124 

σ test value. To develop OC curves, these situations were represented in terms of the minimum 

standard deviation between the contractor’s population and the agency’s population as follows: 

• Minimum standard deviation equals the test standard deviation (σ test ). 

• Minimum standard deviation equals half the test standard deviation. 

• Minimum standard deviation equals twice the test standard deviation. 

Figures 45 through 47 show the OC curves for each of the above cases. The power values are 

shown where the ratio of the larger of the agency’s or the contractor’s standard deviation to the 

smaller of the agency’s or contractor’s standard deviation is varied over the values 0, 1, 2, 3, 4, 

and 5. The mean difference given along the horizontal axis (values of 0, 1, 2, and 3) represents 

the difference in the agency’s and contractor’s subpopulation means expressed as multiples 

ofσ test . 

In figure 45, which shows the case when the minimum standard deviation equals the test 

standard deviation (σ test ), even when the ratio of the contractor’s and agency’s standard 

deviations is 5 and the difference between the contractor’s and the agency’s means is three times 

the value forσ test , there is less than a 70-percent chance of detecting the difference based on the 

results from a single split sample. As would be expected, the power values decrease when the 

minimum standard deviation is half of σ test (figure 46) and increase when the minimum standard 

deviation is twice σ test (figure 47). 

As is the case with any method based on a sample size = 1, the D2S method does not have much 

power to detect the differences between the contractor’s and the agency’s populations. The 

appeal of the maximum allowable difference method lies in its simplicity, rather than in its 

power. 

Average Run Length: The maximum allowable difference method was also evaluated based on 

the average run length. The average run length is the average number of lots that it takes to 

identify a difference between dissimilar populations. As such, the shorter the average run length, 

the better. 

Various actual differences between the contractor’s and the agency’s population means and 

standard deviations were considered in the analysis. In the results that are presented, i refers to 

the difference (in units of the agency’s population standard deviation) between the agency’s and 

the contractor’s population means. Also, j refers to the ratio of the contractor’s population 

standard deviation to the agency’s population standard deviation. In the analyses, i values of 0, 1, 

2, and 3 were used, while the j values used were 0.5, 1.0, 1.5, and 2.0. Some examples of these i

and j values are illustrated in figure 48. 
125 

Figure 45. OC surface for the maximum allowable difference test method 

 verification method (assuming the smaller σ = σtest).

Figure 46. OC surface for the maximum allowable difference test method 

 verification method (assuming the smaller σ = 0.5σtest).

Prob. of Detecting

a Difference, %

0

1

2

3

4

5

1

2 3

0

20

40

60

80

Mean Difference, in σtest Units

Std. Dev.

Ratio

Prob. of Detecting

a Difference, %

0

1

2

3

4

5

1

2 3

0

20

40

60

80

Mean Difference, in σtest Units

Std. Dev.

Ratio

Prob. of Detecting

a Difference, %

0

1

2

3

4

5

1

2 3

0

20

40

60

80

Mean Difference, in σtest Units

Std. Dev.

Ratio

Prob. of Detecting

a Difference, %

0

1

2

3

4

5

1

2 3

0

20

40

60

80

Mean Difference, in σtest Units

Std. Dev.

Ratio
126 

Figure 47. OC surface for the maximum allowable difference test method 

 verification method (assuming the smaller σ = 2σtest).

0

1

2

3

4

5

1

2 3

0

20

40

60

80

Mean Difference, in σtest Units

Std. Dev.

Ratio

Prob. of Detecting

a Difference, %

0

1

2

3

4

5

1

2 3

0

20

40

60

80

Mean Difference, in σtest Units

Std. Dev.

Ratio

Prob. of Detecting

a Difference, %
127 

Figure 48a. Example 1 of some of the cases considered in the average run length analysis 

 for the maximum allowable difference method. 
128 

Figure 48b. Example 2 of some of the cases considered in the average run length analysis 

 for the maximum allowable difference method.
129 

Figure 48c. Example 3 of some of the cases considered in the average run length analysis 

 for the maximum allowable difference method. 
130 

The results of the analyses are presented in table 31 and figure 49. These values are based on 

5000 simulated projects. As shown in the table, when i = 0 and j = 1.0 (meaning that the 

contractor’s and the agency’s populations are the same), the average run length is approximately 

21.5 project lots. This is consistent with what would be expected. Since the limits are set at 2 

standard deviations and since there is only 0.0455 chance of a value outside of 2 standard 

deviations, there is only 1 chance in 22 of declaring the populations to be different for this 

situation. It should also be noted in the table that the standard deviation values are nearly as large 

as the average run lengths. This means that for any individual simulated project, the run length 

could have varied greatly from the average. Indeed, for this case, the individual run lengths 

varied from 1 to more than 200. 

Table 31 clearly shows that as the difference between the population means (i) increases, the 

average run length decreases since it is easier to detect a difference between the two populations. 

This is also true for the ratio of the population standard deviations (j). 

Table 31. Average run length results for the single split-sample method 

 (5000 simulated lots).

Mean Difference, Run Length 

units of agency’s σ

Contractor’s σ 

÷

Agency’s σ Average Std. Dev. 

0.5 85.57 85.44 

1.0 21.55 20.88 

1.5 8.43 8.04 0 

2.0 4.83 4.19 

0.5 19.16 19.11 

1.0 9.86 9.14 

1.5 5.83 5.25 1 

2.0 4.07 3.53 

0.5 4.38 3.82 

1.0 3.58 3.03 

1.5 3.10 2.56 2 

2.0 2.67 2.09 

0.5 1.77 1.14 

1.0 1.85 1.27 

1.5 1.88 1.29 3 

2.0 1.88 1.30 
131 

Paired t-Test 

Since the maximum allowable difference is not a very powerful test, another procedure that uses 

multiple test results to conduct a more powerful hypothesis test can be used. For the case in 

which it is desirable to compare more than one pair of split-sample test results, the t-test for 

paired measurements (i.e., the paired t-test) can be used. This test uses the differences between 

pairs of tests and determines whether the average difference is statistically different from zero. 

Thus, it is the difference within the pairs, not between the pairs, that is being tested. The tstatistic for the paired t-test is: 

n

s

X

t

d

d = (7) 

where: Xd = average of the differences between the split-sample test results 

d s = standard deviation of the differences between the split-sample test results 

 n = number of split samples 

The calculated t-value is then compared to the critical value (tcrit) obtained from a table of 

t-values at a level of α/2 and n – 1 degrees of freedom. Computer programs, such as Microsoft®

Excel, contain statistical test procedures for the paired t-test. This makes the implementation 

process straightforward. 

OC Curves: OC curves can be consulted to evaluate the performance of the paired t-test in 

identifying the differences between population means. OC curves are useful in answering the 

question, “How many pairs of test results should be used?” This form of the OC curve, for a 

given level of α, plots on the vertical axis the probability of either not detecting (β) or detecting 

(1 − β) a difference between two populations. The standardized difference between the two 

population means is plotted on the horizontal axis. 

For a paired t-test, the standardized difference (d) is measured as: 

d

c a d

σ

μ − μ = (8) 

where: ⏐μc − μa⏐ = true absolute difference between the mean (μc) of the contractor’s 

 test result population (which is unknown) and the mean (μa) of the 

 agency’s test result population (which is unknown)

 σd = standard deviation of the true population of signed differences 

 between the paired tests (which is unknown) 

The OC curves are developed for a given level of significance (α). OC curves for α values of 

0.05 and 0.01 are shown in figures 49 and 50, respectively. It is evident from the OC curves that 
132 

for any probability of not detecting a difference (β (value on the vertical axis)), the required n

will increase as the difference (d) decreases (value on the horizontal axis). In some cases, the 

desired β or difference may require prohibitively large sample sizes. In that case, a compromise 

must be made between the discriminating power desired, the cost of the amount of testing 

required, and the risk of claiming a difference when none exists. 

To use this OC curve, the true standard deviation of the signed differences (σd) is assumed to be 

known (or approximated based on past data or published literature). After experience is gained 

with the process, σd can be more accurately defined and a better idea of the required number of 

tests can be determined. 

As an example of how to use the OC curves, assume that the number of pairs of split-sample 

tests for verification of some test method is desired. The probability of not detecting a difference 

(β) is chosen as 10 percent or 0.10. (Some OC curves, which are often called power curves, use 

1 − β (known as the power of the test) on the vertical axis; however, the only difference is the 

scale change (in this case, 1 − β) being 90 percent or 0.90.) Assume that the absolute difference 

between μc and μa should not be greater than 20 units, that the standard deviation of the 

differences is 20 units, and that α is selected as 0.05. This produces a d value of 20 ÷ 20 = 1.0. 

Reading this value on the horizontal axis and a β of 0.20 on the vertical axis shows that about 10 

paired split-sample tests are necessary for the comparison. 
133 

Figure 49. OC curves for a two-sided t-test (α = 0.05) (Natrella, M.G., “Experimental 

 Statistics,” National Bureau of Standards Handbook 91, 1963).

Probability of NOT Detecting a Difference, 

β

Standardized Difference, d
134 

Figure 50. OC curves for a two-sided t-test (α = 0.01) (Natrella, M.G., “Experimental 

 Statistics,” National Bureau of Standards Handbook 91, 1963).

Probability of NOT Detecting a Difference, 

β

Standardized Difference, d
135 

PROCESS VERIFICATION 

Procedures to verify the overall process should be based on independent samples so that all of 

the components of variability (i.e., process, materials, sampling, and testing) are present. Two 

procedures for comparing independently obtained samples appear in the AASHTO 

Implementation Manual for Quality Assurance.

(2) These two methods appear in the AASHTO 

manual in appendix G, which is based on the comparison of a single agency test with 5 to 10 

contractor tests, and in appendix H, which is based on the use of the F-test and t-test to compare 

a number of agency tests with a number of contractor tests. These methods are referred to as the 

AASHTO appendix G method and the AASHTO appendix H method, respectively. Each of 

these methods is discussed and analyzed in the following sections. 

AASHTO Appendix G Method 

In this method, a single agency test result must fall within an interval that is defined from the 

average and range of 5 to 10 contractor test results. The allowable interval within which the 

agency’s test must fall is X ± CR , where X and R are the mean and range, respectively, of the 

contractor’s tests, and C is a factor that varies with the number of contractor tests. The factor C is 

the product of a factor to estimate the sample standard deviation from the sample range and the 

t-value for the 99th percentile of the t-distribution. This is not a particularly efficient approach, 

although this statement can be made for any method that is based on the use of a single agency 

test. Table 32 indicates the allowable interval based on the number of contractor tests. 

Table 32. Allowable intervals for the AASHTO appendix G method. 

Number of Contractor Tests Allowable Interval 

10 X ± 0.91R

9 X ± 0.97R

8 X ±1.05R

7 X ±1.17R

6 X ±1.33R

5 X ±1.61R

OC Curves: Computer simulation was used to develop OC curves (plotted as power curves) that 

indicate the probability of detecting a difference between test populations with various 

differences in means and in the ratios of their standard deviations. The differences between the 

means of the contractor’s and the agency’s populations (∆ = (μContr – μAgency)/σAgency), stated in 

units of the agency’s standard deviation, were varied from 0 to 3.0. Various ratios of the 

contractor’s standard deviation to the agency’s standard deviation (σContr/σAgency) were varied 

from 0.50 to 3.00. 
136 

Since there are two parameters that varied, OC surfaces were plotted, with each surface 

representing a different number of contractor tests (5 to 10) that were compared to a single 

agency test. These OC surfaces are shown in figure 51. As shown in the plots, the power of this 

procedure is quite low, even when a large number of contractor tests are used and when there are 

large differences in the means and standard deviations for the contractor’s and the agency’s 

populations. For example, for five contractor tests, even when the contractor’s standard deviation 

is three times that of the agency and the contractor’s mean is three of the agency’s standard 

deviations from the agency’s mean, there is less than a 50-percent chance of detecting a 

difference. Even if the number of contractor tests is 10, the probability of detecting a difference 

is still less than 60 percent. 

Average Run Length: The method in appendix G was also evaluated based on the average run 

length. Various actual differences between the contractor’s and the agency’s population means 

and standard deviations were considered in the analysis. In the results that are presented, i refers 

to the difference (stated in units of the agency’s population standard deviation) between the 

agency’s and the contractor’s population means. Also, j refers to the ratio of the contractor’s 

population standard deviation to the agency’s population standard deviation. In the analyses, i

values of 0, 1, 2, and 3 were used, while j values of 0.5, 1.0, 1.5, and 2.0 were used. 

The results of the simulation analyses, for the case of five contractor tests and one agency test 

per lot, are presented in table 33. The use of 5 and 10 contractor tests represents the upper and 

lower bounds, respectively, for the results since these are the fewest and most tests for the 

procedure. As shown in table 33, the run lengths can be quite large, particularly when the 

contractor’s population standard deviation is larger than that of the agency. The values in the 

table are based on 5000 simulated projects. 

Also note that the use of 10 tests gives a better performance than that of 5 tests when the 

contractor’s standard deviation is equal to or less than that of the agency (ratios of 1.0 and 0.5). 

However, the opposite is true when the contractor’s standard deviation is greater than that of the 

agency (ratios of 1.5 and 2.0). This is contrary to the desire to use a larger sample to identify the 

differences between the contractor’s and the agency’s populations. 
137 

Figure 51a. OC Surfaces (also called power surfaces) for the appendix G method for 

 5 contractor tests compared to a single agency test. 

Figure 51b. OC surfaces (also called power surfaces) for the appendix G method for 

 6 contractor tests compared to a single agency test.

Prob. of Detecting

a Difference, %

0

1

2

3

0.50

1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σ A Units

Std. Dev. Ratio, σ C / σ A

5 Contractor Tests, 1 Agency Test

Prob. of Detecting

a Difference, %

0

1

2

3

0.50

1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σ A Units

Std. Dev. Ratio, σ C / σ A

5 Contractor Tests, 1 Agency Test

Prob. of Detecting

a Difference, %

0

1

2

3

0.50

1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σA Units

Std. Dev. Ratio, σ C / σ A

6 Contractor Tests, 1 Agency Test

Prob. of Detecting

a Difference, %

0

1

2

3

0.50

1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σA Units

Std. Dev. Ratio, σ C / σ A

6 Contractor Tests, 1 Agency Test
138 

 Figure 51c. OC surfaces (also called power surfaces) for the appendix G method for 7 

 contractor tests compared to a single agency test. 

 Figure 51d. OC surfaces (also called power surfaces) for the appendix G method for 8 

 contractor tests compared to a single agency test. 

Prob. of Detecting

a Difference, %

0

1

2

3

0.50

1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σA Units

Std. Dev. Ratio, σ C / σ A

7 Contractor Tests, 1 Agency Test Prob. of Detecting

a Difference, %

0

1

2

3

0.50

1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σA Units

Std. Dev. Ratio, σ C / σ A

7 Contractor Tests, 1 Agency Test

Prob. of Detecting

a Difference, %

0

1

2

3

0.50 1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σA Units

Std. Dev. Ratio, σ C / σ A

8 Contractor Tests, 1 Agency Test Prob. of Detecting

a Difference, %

0

1

2

3

0.50 1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σA Units

Std. Dev. Ratio, σ C / σ A

8 Contractor Tests, 1 Agency Test
139 

 Figure 51e. OC surfaces (also called power surfaces) for the appendix G method for 

 9 contractor tests compared to a single agency test. 

 

Figure 51f. OC surfaces (also called power surfaces) for the appendix G method 

for 10 contractor tests compared to a single agency test.

Prob. of Detecting

a Difference, %

0

1

2

3

0.50

1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σA Units

Std. Dev. Ratio, σ C / σ A

9 Contractor Tests, 1 Agency Test

Prob. of Detecting

a Difference, %

0

1

2

3

0.50

1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σA Units

Std. Dev. Ratio, σ C / σ A

9 Contractor Tests, 1 Agency Test

Prob. of Detecting

a Difference, %

0

1

2

3

0.50 1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σA Units

Std. Dev. Ratio, σ C / σ A

10 Contractor Tests, 1 Agency Test Prob. of Detecting

a Difference, %

0

1

2

3

0.50 1.33

2.17 3.00

0

20

40

60

80

Mean Diff., 

σA Units

Std. Dev. Ratio, σ C / σ A

10 Contractor Tests, 1 Agency Test
140 

Table 33. Average run length results for the appendix G method 

 (5000 simulated lots). 

Mean Difference, Run Length 

units of agency’s σ

Contractor’s σ 

÷

Agency’s σ Average Std. Dev. 

5 Contractor Tests and 1 Agency Test 

0.5 7.92 7.57 

1.0 43.30 42.68 

1.5 124.19 126.40 0 

2.0 234.45 234.56 

0.5 4.04 3.51 

1.0 18.04 17.78 

1.5 54.78 53.93 1 

2.0 114.63 114.98 

0.5 1.82 1.24 

1.0 6.21 5.69 

1.5 17.61 17.23 2 

2.0 39.30 38.33 

0.5 1.22 0.51 

1.0 2.88 2.34 

1.5 7.23 6.80 3 

2.0 16.23 15.74 

10 Contractor Tests and 1 Agency Test 

0.5 5.15 4.70 

1.0 40.50 39.90 

1.5 230.83 226.93 0 

2.0 887.62 882.77 

0.5 2.74 2.18 

1.0 12.76 12.04 

1.5 62.33 61.14 1 

2.0 229.00 227.47 

0.5 1.39 0.73 

1.0 3.76 3.32 

1.5 13.30 12.61 2 

2.0 46.17 46.19 

0.5 1.07 0.28 

1.0 1.75 1.20 

1.5 4.46 3.94 3 

2.0 12.77 12.15 
141 

AASHTO Appendix H Method 

This procedure involves two hypothesis tests where the null hypothesis for each test is that the 

contractor’s tests and the agency’s tests are from the same population. In other words, the null 

hypotheses are that the variability of the two data sets is equal for the F-test and that the means 

of the two data sets are equal for the t-test. 

The procedures for the F-test and the t-test are more complicated and involved than that for the 

appendix G method discussed above. The F-test and the t-test approach also requires more 

agency test results before a comparison can be made. However, the use of the F-test and the 

t-test is much more statistically sound and has more power to detect actual differences than the 

appendix G method, which relies on a single agency test for the comparison. Any comparison 

method that is based on a single test result will not be very effective in detecting differences 

between data sets. 

When comparing two data sets that are assumed to be normally distributed, it is important to 

compare both the means and the variances. A different test is used for each of these comparisons. 

The F-test provides a method for comparing the variances (standard deviations squared) of the 

two sets of data. The differences in the means are assessed by the t-test. To simplify the use of 

these tests, they are available as built-in functions in computer spreadsheet programs such as 

Microsoft® Excel. For this reason, the procedures involved are not discussed in this report. The 

procedures are fully discussed in the QA manual that was prepared as part of this project.(1) 

A question that needs to be answered is: What power do these statistical tests have, when used 

with small to moderate sample sizes, to declare that various differences in the means and 

variances are statistically significant? This question is addressed separately for the F-test and the 

t-test with the development of the OC curves in the following sections. 

F-Test for Variances (Equal Sample Sizes): Suppose that we have two sets of measurements 

that are assumed to come from normally distributed populations and we wish to conduct a test to 

see if they come from populations that have the same variances (i.e.,σ σ x y

2 2 = ). Furthermore, 

suppose that we select a level of significance of α = 0.05, meaning that we are allowing up to a 

5-percent chance of incorrectly deciding that the variances are different when they are really the 

same. If we assume that these two samples are nx x , x ,Kx 1 2 and ny y , y ,Ky 1 2 , we can calculate 

the sample variances 2

x s and 2

y s , and construct: 

2 2 / x y F = s s (9) 

and accept 2 2 : Ho σ x = σ y for the values of F in the interval [ , ] F1−α 2,nx−1,ny−1 Fα 2,nx−1,ny−1 . 

For this two-sided or two-tailed test, figure 52 shows the probability that we have accepted the 

two samples as coming from populations with the same variability. This probability is usually 

referred to as β and the power of the test is usually referred to as 1 – β. Notice that the horizontal 

axis is the quantity λ, where λ = σx/σy, the true standard deviation ratio. Thus, for λ = 1, where 

the hypothesis of equal variance should certainly be accepted, it is accepted with a probability of 
142 

0.95, reduced from 1.00 only by the magnitude of our type I error risk (α). One significant 

limiting factor for the use of figure 52 is the restriction that nx = ny = n. This limitation is 

addressed in subsequent sections of the report. 

Example: Suppose that we have nx = 6 contractor tests and ny = 6 agency tests, conduct an α = 

0.05 level test and accept (or fail to reject) that these two sets of tests represent populations with 

equal variances. What power did our test have to discern whether the populations from which 

these two sets of tests came were really rather different in variability? Suppose that the true 

population standard deviation of the contractor’s tests (σx) was twice as large as that of the 

agency’s tests (σy), giving λ = 2. If we enter figure 52 with λ = 2 and nx = ny = 6, we find that 

β ≈ 0.74 or that the power (1 – β) is 0.26. This tells us that with samples of nx = 6 and ny = 6, we 

only have a 26-percent chance of detecting a standard deviation ratio of 2 (and, correspondingly, 

a fourfold difference in variance) as being different. 

Suppose that we are not comfortable with the power of 0.26, so subsequently we increase the 

number of tests used. Then suppose that we now have nx = 20 and ny = 20. If we again consider 

λ = 2, we can determine from figure 52 that the power of detecting these sets of tests as coming 

from populations with unequal variances to be more than 0.80 (approximately 82 to 83 percent). 

If we proceed to conduct our F-test with these two samples and conclude that the underlying 

variances are equal, we will certainly feel much more comfortable with our conclusions. 

Figure 53 gives the appropriate OC curves to be used if we choose to conduct an α = 0.01 level 

test. Again, we see that for equal variances 2 σ x and 2 σ y (i.e., λ = 1), that β = 0.99, reduced from 

1.00 only by the size of α. 

F-Test for Variances (Unequal Sample Sizes): Up to now, the discussions and OC curves have 

been limited to equal sample sizes. Routines were developed for this project to calculate the 

power for this test for any combination of sample sizes nx and ny. There are obviously an infinite 

number of possible combinations for nx and ny. Thus, it is not possible to present OC curves for 

every possibility. However, three sets of tables were developed to provide a subset of power 

calculations using some sample sizes that are of potential interest for comparing the contractor’s 

and the agency’s samples. These power calculations are presented in table form since there are 

too many variables to be presented in a single chart, and the data can be presented in a more 

compact form in tables than in a long series of charts. Table 34 gives power values for all 

combinations of sample sizes of 3 to 10, with the ratio of the two subpopulation standard 

deviations = 1, 2, 3, 4, and 5. Table 35 gives power values for the same sample sizes, but with 

the standard deviation ratios = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0. Table 36 gives power values for all 

combinations for sample sizes = 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, and 100, with the 

standard deviation ratio = 1, 2, or 3. 
143 

Figure 52. OC curves for the two-sided F-test for level of significance 

 α = 0.05 (Bowker, A.H., and G.J. Lieberman, Engineering Statistics). 

Figure 53. OC curves for the two-sided F-test for level of significance 

 α = 0.01 (Bowker, A.H., and G.J. Lieberman, Engineering Statistics). 
144 

Table 34. F-test power values for n = 3-10 and s-ratio λ = 1-5. 

λ nx ny Power λ nx ny Power λ nx ny Power 

3 0.05000 3 0.05000 3 0.24820 

4 0.05000 4 0.05000 4 0.27854 

5 0.05000 5 0.05000 5 0.30055 

6 0.05000 6 0.05000 6 0.31744 

7 0.05000 7 0.05000 7 0.33086 

8 0.05000 8 0.05000 8 0.34179 

9 0.05000 9 0.05000 9 0.35087 

3 

10 0.05000 

9 

10 0.05000 

7 

10 0.35853 

3 0.05000 3 0.05000 3 0.26768 

4 0.05000 4 0.05000 4 0.30567 

5 0.05000 5 0.05000 5 0.33401 

6 0.05000 6 0.05000 6 0.35619 

7 0.05000 7 0.05000 7 0.37410 

8 0.05000 8 0.05000 8 0.38888 

9 0.05000 9 0.05000 9 0.40129 

4 

10 0.05000 

1 

10 

10 0.05000 

8 

10 0.41187 

3 0.05000 3 0.09939 3 0.28308 

4 0.05000 4 0.09753 4 0.32758 

5 0.05000 5 0.09663 5 0.36144 

6 0.05000 6 0.09620 6 0.38837 

7 0.05000 7 0.09600 7 0.41036 

8 0.05000 8 0.09590 8 0.42869 

9 0.05000 9 0.09586 9 0.44421 

5 

10 0.05000 

3 

10 0.09585 

9 

10 0.45752 

3 0.05000 3 0.14835 3 0.29549 

4 0.05000 4 0.15169 4 0.34549 

5 0.05000 5 0.15385 5 0.38414 

6 0.05000 6 0.15544 6 0.41521 

7 0.05000 7 0.15668 7 0.44081 

8 0.05000 8 0.15767 8 0.46230 

9 0.05000 9 0.15848 9 0.48060 

6 

10 0.05000 

4 

10 0.15915 

2 

10 

10 0.49639 

3 0.05000 3 0.19036 3 0.19034 

4 0.05000 4 0.20240 4 0.19354 

5 0.05000 5 0.21041 5 0.19556 

6 0.05000 6 0.21622 6 0.19696 

7 0.05000 7 0.22064 7 0.19798 

8 0.05000 8 0.22413 8 0.19875 

9 0.05000 9 0.22694 9 0.19934 

7 

10 0.05000 

5 

10 0.22926 

3 

10 0.19981 

3 0.05000 3 0.22309 3 0.31171 

4 0.05000 4 0.24464 4 0.33525 

5 0.05000 5 0.25968 5 0.35007 

6 0.05000 6 0.27093 6 0.36030 

7 0.05000 7 0.27968 7 0.36777 

8 0.05000 8 0.28669 8 0.37347 

9 0.05000 9 0.29243 9 0.37795 

1 

8 

10 0.05000 

2 

6 

10 0.29722 

3 

4 

10 0.38157 
145 

Table 34. F-test power values for n = 3-10 and s-ratio λ = 1-5 (continued). 

λ nx ny Power λ nx ny Power λ nx ny Power 

3 0.39758 3 0.29251 3 0.69798 

4 0.44454 4 0.30367 4 0.79871 

5 0.47603 5 0.31010 5 0.85988 

6 0.49872 6 0.31427 6 0.89907 

7 0.51588 7 0.31717 7 0.92520 

8 0.52931 8 0.31930 8 0.94321 

9 0.54011 9 0.32093 9 0.95598 

5 

10 0.54899 

3 

10 0.32222 

9 

10 0.96525 

3 0.45403 3 0.46558 3 0.71073 

4 0.51906 4 0.51179 4 0.81311 

5 0.56396 5 0.54104 5 0.87423 

6 0.59696 6 0.56126 6 0.91256 

7 0.62225 7 0.57608 7 0.93751 

8 0.64225 8 0.58742 8 0.95427 

9 0.65846 9 0.59637 9 0.96583 

6 

10 0.67186 

4 

10 0.60363 

4 

10 

10 0.97399 

3 0.49230 3 0.56455 3 0.39165 

4 0.57007 4 0.63665 4 0.41270 

5 0.62436 5 0.68356 5 0.42481 

6 0.66443 6 0.71649 6 0.43266 

7 0.69516 7 0.74084 7 0.43815 

8 0.71943 8 0.75955 8 0.44219 

9 0.73906 9 0.77437 9 0.44530 

7 

10 0.75523 

5 

10 0.78638 

3 

10 0.44776 

3 0.51945 3 0.62143 3 0.58713 

4 0.60623 4 0.70759 4 0.64932 

5 0.66693 5 0.76314 5 0.68814 

6 0.71159 6 0.80150 6 0.71467 

7 0.74565 7 0.82932 7 0.73394 

8 0.77236 8 0.85027 8 0.74858 

9 0.79378 9 0.86652 9 0.76007 

8 

10 0.81129 

6 

10 0.87943 

4 

10 0.76932 

3 0.53955 3 0.65697 3 0.68068 

4 0.63285 4 0.75074 4 0.76196 

5 0.69797 5 0.81002 5 0.81171 

6 0.74560 6 0.84993 6 0.84479 

7 0.78161 7 0.87808 7 0.86811 

8 0.80958 8 0.89866 8 0.88527 

9 0.83177 9 0.91416 9 0.89836 

9 

10 0.84970 

7 

10 0.92613 

5 

10 0.90860 

3 0.55494 3 0.68090 3 0.72975 

4 0.65311 4 0.77901 4 0.81790 

5 0.72136 5 0.83976 5 0.86956 

6 0.77092 6 0.87961 6 0.90223 

7 0.80803 7 0.90692 7 0.92409 

8 0.83654 8 0.92628 8 0.93936 

9 0.85890 9 0.94042 9 0.95041 

3 

10 

10 0.87675 

4 

8 

10 0.95100 

5 

6 

10 0.95864 
146 

Table 34. F-test power values for n = 3-10 and s-ratio λ = 1-5 (continued). 

λ nx ny Power 

3 0.75893 

4 0.84940 

5 0.90024 

6 0.93086 

7 0.95030 

8 0.96318 

9 0.97201 

7 

10 0.97824 

3 0.77800 

4 0.86909 

5 0.91845 

6 0.94695 

7 0.96423 

8 0.97513 

9 0.98225 

8 

10 0.98704 

3 0.79133 

4 0.88238 

5 0.93024 

6 0.95690 

7 0.97244 

8 0.98184 

9 0.98772 

9 

10 0.99150 

3 0.80115 

4 0.89188 

5 0.93838 

6 0.96351 

7 0.97767 

8 0.98594 

9 0.99092 

5 

10 

10 0.99400 
147 

Table 35. F-test power values for n = 3-10 and s-ratio λ = 0-1. 

λ nx ny Power λ nx ny Power λ nx ny Power 

3 1.00000 3 1.00000 3 0.43815 

4 1.00000 4 1.00000 4 0.73394 

5 1.00000 5 1.00000 5 0.86811 

6 1.00000 6 1.00000 6 0.92409 

7 1.00000 7 1.00000 7 0.95030 

8 1.00000 8 1.00000 8 0.96423 

9 1.00000 9 1.00000 9 0.97244 

3 

10 1.00000 

9 

10 1.00000 

7 

10 0.97767 

3 1.00000 3 1.00000 3 0.44219 

4 1.00000 4 1.00000 4 0.74858 

5 1.00000 5 1.00000 5 0.88527 

6 1.00000 6 1.00000 6 0.93936 

7 1.00000 7 1.00000 7 0.96318 

8 1.00000 8 1.00000 8 0.97513 

9 1.00000 9 1.00000 9 0.98184 

4 

10 1.00000 

0.0 

10 

10 1.00000 

8 

10 0.98594 

3 1.00000 3 0.39165 3 0.44530 

4 1.00000 4 0.58713 4 0.76007 

5 1.00000 5 0.68068 5 0.89836 

6 1.00000 6 0.72975 6 0.95041 

7 1.00000 7 0.75893 7 0.97201 

8 1.00000 8 0.77800 8 0.98225 

9 1.00000 9 0.79133 9 0.98772 

5 

10 1.00000 

3 

10 0.80115 

9 

10 0.99092 

3 1.00000 3 0.41270 3 0.44776 

4 1.00000 4 0.64932 4 0.76932 

5 1.00000 5 0.76196 5 0.90860 

6 1.00000 6 0.81790 6 0.95864 

7 1.00000 7 0.84940 7 0.97824 

8 1.00000 8 0.86909 8 0.98704 

9 1.00000 9 0.88238 9 0.99150 

6 

10 1.00000 

4 

10 0.89188 

0.2 

10 

10 0.99400 

3 1.00000 3 0.42481 3 0.14221 

4 1.00000 4 0.68814 4 0.22806 

5 1.00000 5 0.81171 5 0.29564 

6 1.00000 6 0.86956 6 0.34398 

7 1.00000 7 0.90024 7 0.37868 

8 1.00000 8 0.91845 8 0.40429 

9 1.00000 9 0.93024 9 0.42380 

7 

10 1.00000 

5 

10 0.93838 

3 

10 0.43906 

3 1.00000 3 0.43266 3 0.14250 

4 1.00000 4 0.71467 4 0.24034 

5 1.00000 5 0.84479 5 0.32488 

6 1.00000 6 0.90223 6 0.38884 

7 1.00000 7 0.93086 7 0.43614 

8 1.00000 8 0.94695 8 0.47159 

9 1.00000 9 0.95690 9 0.49879 

0.0 

8 

10 1.00000 

0.2 

6 

10 0.96351 

0.4 

4 

10 0.52015 
148 

Table 35. F-test power values for n = 3-10 and s-ratio λ = 0-1 (continued). 

λ nx ny Power λ nx ny Power λ nx ny Power 

3 0.14291 3 0.07564 3 0.06891 

4 0.24808 4 0.10273 4 0.10161 

5 0.34448 5 0.12665 5 0.13711 

6 0.42028 6 0.14614 6 0.17223 

7 0.47749 7 0.16173 7 0.20526 

8 0.52079 8 0.17425 8 0.23545 

9 0.55411 9 0.18444 9 0.26265 

5 

10 0.58029 

3 

10 0.19283 

9 

10 0.28698 

3 0.14332 3 0.07283 3 0.06870 

4 0.25345 4 0.10212 4 0.10168 

5 0.35863 5 0.13003 5 0.13786 

6 0.44371 6 0.15430 6 0.17409 

7 0.50889 7 0.17470 7 0.20854 

8 0.55851 8 0.19170 8 0.24035 

9 0.59674 9 0.20593 9 0.26925 

6 

10 0.62671 

4 

10 0.21791 

0.6 

10 

10 0.29529 

3 0.14369 3 0.07120 3 0.05467 

4 0.25739 4 0.10174 4 0.06163 

5 0.36934 5 0.13222 5 0.06758 

6 0.46187 6 0.15988 6 0.07248 

7 0.53357 7 0.18396 7 0.07649 

8 0.58837 8 0.20461 8 0.07980 

9 0.63057 9 0.22225 9 0.08255 

7 

10 0.66355 

5 

10 0.23736 

3 

10 0.08487 

3 0.14399 3 0.07022 3 0.05202 

4 0.26041 4 0.10157 4 0.05929 

5 0.37772 5 0.13386 5 0.06587 

6 0.47638 6 0.16407 6 0.07156 

7 0.55351 7 0.19107 7 0.07642 

8 0.61261 8 0.21472 8 0.08057 

9 0.65804 9 0.23528 9 0.08412 

8 

10 0.69341 

6 

10 0.25314 

4 

10 0.08719 

3 0.14424 3 0.06960 3 0.05017 

4 0.26278 4 0.10153 4 0.05755 

5 0.38447 5 0.13516 5 0.06448 

6 0.48825 6 0.16736 6 0.07067 

7 0.56996 7 0.19675 7 0.07612 

8 0.63266 8 0.22292 8 0.08090 

9 0.68076 9 0.24600 9 0.08508 

9 

10 0.71805 

7 

10 0.26628 

5 

10 0.08875 

3 0.14445 3 0.06919 3 0.04883 

4 0.26470 4 0.10155 4 0.05626 

5 0.39001 5 0.13622 5 0.06340 

6 0.49813 6 0.17003 6 0.06995 

7 0.58375 7 0.20139 7 0.07584 

8 0.64952 8 0.22972 8 0.08109 

9 0.69984 9 0.25499 9 0.08577 

0.4 

10 

10 0.73868 

0.6 

8 

10 0.27741 

0.8 

6 

10 0.08994 
149 

Table 35. F-test power values for n = 3-10 and s-ratio λ = 0-1 (continued). 

λ nx ny Power λ nx ny Power 

3 0.04785 3 0.05000 

4 0.05529 4 0.05000 

5 0.06258 5 0.05000 

6 0.06938 6 0.05000 

7 0.07560 7 0.05000 

8 0.08124 8 0.05000 

9 0.08633 9 0.05000 

7 

10 0.09092 

5 

10 0.05000 

3 0.04709 3 0.05000 

4 0.05453 4 0.05000 

5 0.06193 5 0.05000 

6 0.06893 6 0.05000 

7 0.07541 7 0.05000 

8 0.08136 8 0.05000 

9 0.08680 9 0.05000 

8 

10 0.09175 

6 

10 0.05000 

3 0.04650 3 0.05000 

4 0.05393 4 0.05000 

5 0.06141 5 0.05000 

6 0.06856 6 0.05000 

7 0.07527 7 0.05000 

8 0.08148 8 0.05000 

9 0.08721 9 0.05000 

9 

10 0.09248 

7 

10 0.05000 

3 0.04603 3 0.05000 

4 0.05345 4 0.05000 

5 0.06099 5 0.05000 

6 0.06827 6 0.05000 

7 0.07516 7 0.05000 

8 0.08159 8 0.05000 

9 0.08757 9 0.05000 

0.8 

10 

10 0.09312 

8 

10 0.05000 

3 0.05000 3 0.05000 

4 0.05000 4 0.05000 

5 0.05000 5 0.05000 

6 0.05000 6 0.05000 

7 0.05000 7 0.05000 

8 0.05000 8 0.05000 

9 0.05000 9 0.05000 

3 

10 0.05000 

9 

10 0.05000 

3 0.05000 3 0.05000 

4 0.05000 4 0.05000 

5 0.05000 5 0.05000 

6 0.05000 6 0.05000 

7 0.05000 7 0.05000 

8 0.05000 8 0.05000 

9 0.05000 9 0.05000 

1.0 

4 

10 0.05000 

1.0 

10 

10 0.05000 
150 

Table 36. F-test power values for n = 5-100 and s-ratio λ = 1-3. 

λ nx ny Power λ nx ny Power λ nx ny Power 

5 0.05 5 0.05 5 0.05 

10 0.05 10 0.05 10 0.05 

15 0.05 15 0.05 15 0.05 

20 0.05 20 0.05 20 0.05 

25 0.05 25 0.05 25 0.05 

30 0.05 30 0.05 30 0.05 

40 0.05 40 0.05 40 0.05 

50 0.05 50 0.05 50 0.05 

60 0.05 60 0.05 60 0.05 

70 0.05 70 0.05 70 0.05 

80 0.05 80 0.05 80 0.05 

90 0.05 90 0.05 90 0.05 

5 

100 0.05 

20 

100 0.05 

40 

100 0.05 

5 0.05 5 0.05 5 0.05 

10 0.05 10 0.05 10 0.05 

15 0.05 15 0.05 15 0.05 

20 0.05 20 0.05 20 0.05 

25 0.05 25 0.05 25 0.05 

30 0.05 30 0.05 30 0.05 

40 0.05 40 0.05 40 0.05 

50 0.05 50 0.05 50 0.05 

60 0.05 60 0.05 60 0.05 

70 0.05 70 0.05 70 0.05 

80 0.05 80 0.05 80 0.05 

90 0.05 90 0.05 90 0.05 

10 

100 0.05 

25 

100 0.05 

50 

100 0.05 

5 0.05 5 0.05 5 0.05 

10 0.05 10 0.05 10 0.05 

15 0.05 15 0.05 15 0.05 

20 0.05 20 0.05 20 0.05 

25 0.05 25 0.05 25 0.05 

30 0.05 30 0.05 30 0.05 

40 0.05 40 0.05 40 0.05 

50 0.05 50 0.05 50 0.05 

60 0.05 60 0.05 60 0.05 

70 0.05 70 0.05 70 0.05 

80 0.05 80 0.05 80 0.05 

90 0.05 90 0.05 90 0.05 

1 

15 

100 0.05 

1 

30 

100 0.05 

1 

60 

100 0.05 
151 

Table 36. F-test power values for n = 5-100 and s-ratio λ = 1-3 (continued). 

λ nx ny Power λ nx ny Power λ nx ny Power 

5 0.05 5 0.05 5 0.45487 

10 0.05 10 0.05 10 0.62152 

15 0.05 15 0.05 15 0.70573 

20 0.05 20 0.05 20 0.75560 

25 0.05 25 0.05 25 0.78820 

30 0.05 30 0.05 30 0.81099 

40 0.05 40 0.05 40 0.84054 

50 0.05 50 0.05 50 0.85870 

60 0.05 60 0.05 60 0.87092 

70 0.05 70 0.05 70 0.87969 

80 0.05 80 0.05 80 0.88626 

90 0.05 90 0.05 90 0.89137 

70 

100 0.05 

1 100 

100 0.05 

15 

100 0.89545 

5 0.05 5 0.21041 5 0.49087 

10 0.05 10 0.22926 10 0.68548 

15 0.05 15 0.23658 15 0.78230 

20 0.05 20 0.24043 20 0.83747 

25 0.05 25 0.24281 25 0.87192 

30 0.05 30 0.24442 30 0.89495 

40 0.05 40 0.24646 40 0.92304 

50 0.05 50 0.24770 50 0.93906 

60 0.05 60 0.24853 60 0.94918 

70 0.05 70 0.24913 70 0.95606 

80 0.05 80 0.24958 80 0.96099 

90 0.05 90 0.24993 90 0.96468 

80 

100 0.05 

5 

100 0.25022 

20 

100 0.96753 

5 0.05 5 0.38414 5 0.51241 

10 0.05 10 0.49639 10 0.72299 

15 0.05 15 0.55109 15 0.82516 

20 0.05 20 0.58353 20 0.88085 

25 0.05 25 0.60501 25 0.91389 

30 0.05 30 0.62027 30 0.93485 

40 0.05 40 0.64053 40 0.95864 

50 0.05 50 0.65336 50 0.97099 

60 0.05 60 0.66221 60 0.97817 

70 0.05 70 0.66869 70 0.98272 

80 0.05 80 0.67363 80 0.98578 

90 0.05 90 0.67753 90 0.98795 

1 

90 

100 0.05 

2 

10 

100 0.68068 

2 

25 

100 0.98955 
152 

Table 36. F-test power values for n = 5-100 and s-ratio λ = 1-3 (continued). 

λ nx ny Power λ nx ny Power λ nx ny Power 

5 0.52669 5 0.56187 5 0.57339 

10 0.74730 10 0.80456 10 0.82226 

15 0.85174 15 0.90914 15 0.92497 

20 0.90637 20 0.95588 20 0.96762 

25 0.93725 25 0.97764 25 0.98564 

30 0.95585 30 0.98820 30 0.99345 

40 0.97551 40 0.99632 40 0.99851 

50 0.98476 50 0.99869 50 0.99962 

60 0.98968 60 0.99948 60 0.99989 

70 0.99256 70 0.99977 70 0.99997 

80 0.99436 80 0.99989 80 0.99999 

90 0.99556 90 0.99995 90 1.00000 

30 

100 0.99639 

60 

100 0.99997 

90 

100 1.00000 

5 0.54439 5 0.56683 5 0.57568 

10 0.77664 10 0.81224 10 0.82571 

15 0.88220 15 0.91614 15 0.92793 

20 0.93379 20 0.96120 20 0.96968 

25 0.96067 25 0.98137 25 0.98696 

30 0.97548 30 0.99073 30 0.99425 

40 0.98924 40 0.99745 40 0.99879 

50 0.99462 50 0.99921 50 0.99972 

60 0.99702 60 0.99972 60 0.99993 

70 0.99821 70 0.99989 70 0.99998 

80 0.99886 80 0.99996 80 0.99999 

90 0.99923 90 0.99998 90 1.00000 

40 

100 0.99945 

70 

100 0.99999 

2 

100 

100 1.00000 

5 0.55491 5 0.57053 5 0.47603 

10 0.79358 10 0.81791 10 0.54899 

15 0.89881 15 0.92118 15 0.57700 

20 0.94770 20 0.96490 20 0.59187 

25 0.97160 25 0.98387 25 0.60108 

30 0.98387 30 0.99235 30 0.60736 

40 0.99414 40 0.99810 40 0.61537 

50 0.99757 50 0.99947 50 0.62026 

60 0.99888 60 0.99984 60 0.62355 

70 0.99943 70 0.99994 70 0.62593 

80 0.99969 80 0.99998 80 0.62772 

90 0.99982 90 0.99999 90 0.62911 

2 

50 

100 0.99989 

2 

80 

100 1.00000 

3 5 

100 0.63024 
153 

Table 36. F-test power values for n = 5-100 and s-ratio λ = 1-3 (continued).

λ nx ny Power λ nx ny Power λ nx ny Power 

5 0.72136 5 0.82417 5 0.84999 

10 0.87675 10 0.96743 10 0.98107 

15 0.92836 15 0.99254 15 0.99738 

20 0.95158 20 0.99797 20 0.99960 

25 0.96404 25 0.99936 25 0.99993 

30 0.97154 30 0.99977 30 0.99999 

40 0.97985 40 0.99996 40 1.00000 

50 0.98420 50 0.99999 50 1.00000 

60 0.98681 60 1.00000 60 1.00000 

70 0.98853 70 1.00000 70 1.00000 

80 0.98973 80 1.00000 80 1.00000 

90 0.99062 90 1.00000 90 1.00000 

10 

100 0.99130 

25 

100 1.00000 

50 

100 1.00000 

5 0.78336 5 0.83321 5 0.85393 

10 0.93786 10 0.97267 10 0.98279 

15 0.97640 15 0.99463 15 0.99783 

20 0.98918 20 0.99877 20 0.99971 

25 0.99431 25 0.99968 25 0.99996 

30 0.99669 30 0.99990 30 0.99999 

40 0.99860 40 0.99999 40 1.00000 

50 0.99928 50 1.00000 50 1.00000 

60 0.99957 60 1.00000 60 1.00000 

70 0.99972 70 1.00000 70 1.00000 

80 0.99980 80 1.00000 80 1.00000 

90 0.99985 90 1.00000 90 1.00000 

15 

100 0.99988 

30 

100 1.00000 

60 

100 1.00000 

5 0.80975 5 0.84390 5 0.85668 

10 0.95808 10 0.97822 10 0.98394 

15 0.98816 15 0.99654 15 0.99812 

20 0.99597 20 0.99938 20 0.99976 

25 0.99841 25 0.99987 25 0.99997 

30 0.99930 30 0.99997 30 1.00000 

40 0.99982 40 1.00000 40 1.00000 

50 0.99994 50 1.00000 50 1.00000 

60 0.99998 60 1.00000 60 1.00000 

70 0.99999 70 1.00000 70 1.00000 

80 0.99999 80 1.00000 80 1.00000 

90 1.00000 90 1.00000 90 1.00000 

3 

20 

100 1.00000 

3 

40 

100 1.00000 

3 

70 

100 1.00000 
154 

Table 36. F-test power values for n = 5-100 and s-ratio λ = 1-3 (continued). 

λ nx ny Power 

5 0.85871 

10 0.98476 

15 0.99831 

20 0.99980 

25 0.99998 

30 1.00000 

40 1.00000 

50 1.00000 

60 1.00000 

70 1.00000 

80 1.00000 

90 1.00000 

80 

100 1.00000 

5 0.86026 

10 0.98537 

15 0.99844 

20 0.99983 

25 0.99998 

30 1.00000 

40 1.00000 

50 1.00000 

60 1.00000 

70 1.00000 

80 1.00000 

90 1.00000 

90 

100 1.00000 

5 0.86150 

10 0.98584 

15 0.99855 

20 0.99985 

25 0.99998 

30 1.00000 

40 1.00000 

50 1.00000 

60 1.00000 

70 1.00000 

80 1.00000 

90 1.00000 

3 

100 

100 1.00000 
155 

From these tables, it is obvious that the limiting factor in how well the F-test will be able to 

identify differences will be the number of agency verification tests. The power of the F-test is 

limited not by the larger of the sample sizes, but by the smaller of the sample sizes. For example, 

in table 34, when nx = 3 and ny = 10, the power is only about 20 percent, even when there is a 

threefold difference in the true standard deviations (i.e., λ = 3). The limiting aspect of the smaller 

sample size is also noticeable in table 36 for larger sample sizes. For example, for λ = 2 and for 

ny = 100, the power when nx = 5 is only about 25 percent. The power increases to 68 percent for 

nx = 10, 90 percent for nx = 15, and 97 percent for nx = 20. Since the agency will have fewer 

verification tests than the number of contractor tests, the agency’s verification sampling and 

testing rate will determine the power to identify variability differences when they exist. 

t-Test for Means: As with the appendix G method, the performance of the t-test for means can 

be evaluated with OC curves or by considering the average run length. 

OC Curves: Suppose that we have two sets of measurements that are assumed to be from 

normally distributed populations and that we wish to conduct a two-sided or two-tailed test to see 

if these populations have equal means (i.e., μ x = μ y). Suppose that we assume that these two 

samples are from populations with unknown, but equal, variances. If these two samples are x1, x2, 

..., xnx, with sample mean X and sample variance 2

x s , and y1, y2, ..., yny, with sample mean Y

and sample variance 2

y s , we can calculate: 

x y x y

x x y y

n n n n

s n s n

X Y

t

1 1

2

( 1) ( 1) 2 2

× +

+ −

− + −

− = (10) 

and accept H0: μ x = μ y for values of t in the interval[ t , t ] − α/2,nx +ny −2 α/2,nx +ny −2 . 

For this test, figure 49 or 50, depending on the α value, shows the probability that we have 

accepted the two samples as coming from populations with the same means. The horizontal axis 

scale is: 

σ

μ x μ y

d − = (11)

where: σ = σ x = σ y = true common population standard deviation 

We can access the OC curves in figure 49 or 50 with a value for d of d* and a value for n of n′

where: 

n′ = nx + ny −1 (12) 

and 
156 

x y

x y

n n

n n

n

d d

+

×

×

′ * (13) =

Example: Suppose that we have nx = 8 contractor tests and ny = 8 agency tests, conduct an α = 

0.05 level test and accept that these two sets of tests represent populations with equal means. 

What power did our test really have to discern if the populations from which these two sets of 

tests came had different means? Suppose that we consider a difference in these population means 

of 2 or more standard deviations as a noteworthy difference that we would like to detect with 

high probability. This would indicate that we are interested in d = 2. Calculating 

′ = + −1 = 8 + 8 −1 = 15 n nx ny (14) 

and 

1.0328

8 8

8 8

15

2 * = +× = ×

+

×

×

′ =

x y

x y

n n

n n

n

d d (15) 

we find from figure 50 that β ≈ 0.05, so that our power of detecting a mean difference of 2 or 

more standard deviations would be approximately 95 percent. 

Now suppose that we consider an application where we still have a total of 16 tests, but with nx = 

12 contractor tests and ny = 4 agency tests. Suppose that we are again interested in the t-test 

performance in detecting a means difference of 2 standard deviations. Again, calculating 

 15 n′ = nx + ny −1 = 8 + 8 −1 = (16) 

but now 

0.8944

12 4

12 4

15

2 * = +× = ×

+

×

×

′ =

x y

x y

n n

n n

n

d d (17) 

we find from figure 50 that β ≈ 0.12, indicating that our power of detecting a mean difference of 

2 or more standard deviations would be approximately 88 percent. 

Figure 51 gives the appropriate OC curves for use in conducting an α = 0.01 level test on the 

means. This figure is accessed in the same manner as described above for figure 50. 

Average Run Length: The effectiveness of the t-test procedure was evaluated by determining 

the average run length in terms of project lots. The evaluation was performed by simulating 1000 

projects and determining, on average, how many lots it took to determine that there was a 

difference between the contractor’s and the agency’s population means. 
157 

The results of the simulation analyses, for the case of five contractor tests and one agency test 

per lot, are presented in table 37. The results are shown only for the case where five contractor 

tests and one agency test are performed on each project lot. Similar results were obtained for 

cases where fewer and more contractor tests were conducted per lot. As shown in table 37, when 

there is no difference between the population means, the run lengths are quite large (as they 

should be). The values with asterisks are biased on the low side, because to speed up the 

simulation time, the maximum run lengths were limited to 100. Therefore, the actual average run 

length would be greater than those shown in the table since the maximum cutoff value was 

reached in more than half of the 1000 projects simulated for each i and j combination. 

The average run lengths become relatively small as the actual difference between the 

contractor’s and the agency’s population means increases. This is obviously what is desired. 

Table 37. Average run length results for the appendix H method (5 contractor 

 tests and 1 agency test per lot) for 1000 simulated lots.

Mean Difference, Run Length 

units of agency’s σ

Contractor’s σ 

÷

Agency’s σ Average Std. Dev. 

0.5 55.47* 46.01* 

1.0 70.15* 41.91* 

1.5 77.78* 36.95* 0 

2.0 75.72* 38.56* 

0.5 4.83 4.05 

1.0 5.75 4.28 

1.5 8.63 5.70 

1 

2.0 9.83 5.94 

0.5 2.60 1.18 

1.0 2.64 1.02 

1.5 3.51 1.52 

2 

2.0 4.40 2.03 

0.5 2.35 0.73 

1.0 2.10 0.37 

1.5 2.36 0.66 3 

2.0 2.88 1.03 

* These values are lower than the actual values. To reduce the simulation processing time, the maximum 

number of lots was limited to 100. For these cases, more than half of the projects were truncated at 100 lots. 
158 

CONCLUSIONS AND RECOMMENDATIONS 

Based on the analyses that were conducted and were summarized in this chapter, the following 

recommendations were made: 

Recommendation for Test Method Verification 

The comparison of a single split sample by using the maximum allowable limits (such as the 

D2S limits) is simple and can be done for each split sample that is obtained. However, since it is 

based on comparing only single data values, it is not very powerful for identifying differences 

where they exist. It is recommended that each individual split sample be compared using the 

maximum allowable limits, but that the paired t-test also be used on the accumulated splitsample results to allow for a comparison with more discerning power. If either of these 

comparisons indicates a difference, then an investigation to identify the cause of the difference 

should be initiated. 

Recommendation for Process Verification 

Since they are both based on five contractor tests and one agency test per lot, the results in tables 

33 and 37 can be used to compare the appendix H and appendix G methods. The average run 

lengths for the appendix H method (t-test) were better than those for the appendix G method 

(single agency test compared to five contractor tests). Compared to the appendix G method, the 

appendix H method had longer average run lengths where there was no difference in the means 

and shorter lengths where there was a difference in the means. This is what is desirable in the 

verification procedure. The appendix H method is recommended for use in verifying the 

contractor’s test results when the agency obtains independent samples for evaluating the total 

process. 

From the OC curves that were developed, it is apparent that the number of agency verification 

tests will be the deciding factor when determining the validity of the contractor’s overall process. 

When using the OC curves in figure 50 or 51, the lower the value of d*, the lower the power of 

the test for a given number of test results. The value for d* will decrease as the agency’s portion 

of the total number of tests declines (this is shown in equation 13). If, in the expression under the 

square root sign, the total number of tests (nx + ny) is fixed, then the value of d* will decrease as 

the value of either nx or ny goes down. 

An example will illustrate this point. Suppose that the total of nx + ny is fixed at 16, then the 

maximum value under the square root sign will be when nx = ny = 8. This is true because the 

denominator is fixed at 16 and 8 × 8 = 64 is larger than any other combination of numbers that 

total 16. As one of the values gets smaller (and the other gets correspondingly larger), the 

product of the two numbers will decrease, thereby decreasing d* and reducing the power of the 

test. 

The amount of verification sampling and testing is a subjective decision for each individual 

agency. However, with the OC (or power) curves and tables in this chapter, an agency can 

determine the risks that are associated with any frequency of verification testing and can make an 

informed decision regarding this testing frequency. 
159 

When using the appendix H method, first, an F-test is used to determine whether or not the 

variances (and, hence, standard deviations) are different for the two populations. The result of 

the F-test determines how the subsequent t-test is conducted to compare the averages of the 

contractor’s and the agency’s test results. Given some of the low powers associated with small 

sample sizes in tables 34 through 36, it could be argued that an agency will rarely be able to 

conclude from the F-test that a difference in variances exists. Given this fact, it may be 

reasonable to just assume that the populations have equal variances and run the t-test for equal 

variances and ignore the F-test altogether. This argument has some merit. However, with the 

ease of conducting the F-test and the t-test by computer, once the test results are input, there is 

essentially no additional effort associated with conducting the F-test before the t-test. 
 
 161 

8. QUALITY MEASURES AND PAYMENT 

CANDIDATE QUALITY MEASURES 

In chapter 5, candidate quality measures were investigated with respect to their accuracy and 

precision when estimating the true population parameters. The candidate quality measures that 

were analyzed were PWL (or its complement, PD), AAD, and CI. 

From the simulation analyses summarized in chapter 5, both the sample PWL and the sample 

AAD were unbiased estimators of the corresponding population values. They both showed 

decreasing variability in the estimated values as the sample size increased. For PWL, the 

variability values peaked at a population PWL of 50 and then decreased as the population PWL 

approached either 100 or 0. The AAD variability values were lowest when the population mean 

was centered at the target and they increased as the offset from the target of the population mean 

increased, leveling off as the offset approached 3 standard deviations. 

The simulation analyses for CI showed similar variability relationships as were found for AAD. 

However, the CI appeared to be a slightly biased estimator of the population CI since all of the 

simulated average CI sample values were negative. 

Since there appears to be no appreciable benefit to using CI rather than AAD (i.e., they have 

similar variability traits, but AAD is unbiased while the CI is slightly biased), the CI was 

eliminated from further analyses. This chapter presents the results of the further analyses that 

were conducted on PWL and AAD. 

ANALYSES PERFORMED 

The quality measure that is selected needs to be capable of being used to determine the payment 

factor. Therefore, it must be easy to develop an equation that relates the quality measure to the 

anticipated performance of the pavement. For the analyses to be conducted at this point, it was 

sufficient to accept the fact that the quality of the material in the population would increase as 

either its PWL value increased or its AAD value decreased. The relationship of payment to 

projected performance is discussed in a later chapter. 

For each quality measure, it was necessary to determine whether or not it could provide an 

unbiased estimate for the payment factor for a given population. The amount of variability in the 

payment factor estimates for individual lots also required study. These results were both 

generated by computer simulation studies. 

Since it is rare that only a single quality characteristic is used to determine the payment factor for 

a lot, it was necessary to consider how the quality measures performed when multiple acceptance 

characteristics were used and to consider methods for combining these individual estimated 

quality measures into a combined (or composite) payment factor. While it was not within the 

scope of the original project, it was decided that, if multiple quality characteristics were to be 

investigated, it would be necessary to consider potential correlations among these characteristics. 
 162 

Finally, it was necessary to investigate the risks associated with the quality measures and how 

they would be used for acceptance decisions. The concept of risks for acceptance is similar to 

that discussed in chapter 7 for verification testing. If pass and fail are the only options, the 

acceptance decision is simply a form of hypothesis testing and the concept is fairly 

straightforward. When acceptance at payments other than 100 percent is possible (either 

incentives or disincentives), the evaluation of the risks becomes much more complicated and not 

nearly as clear-cut as for pass or fail decisions. The analysis of the payment risks associated with 

PWL and AAD quality measures was conducted for single quality characteristics and for two 

quality characteristics that were either independent or correlated. 

PAYMENT FOR A SINGLE QUALITY CHARACTERISTIC 

In chapter 5, it was determined that both PWL and AAD were unbiased estimators for the true 

population values. As such, they should also provide unbiased payment factor estimates when 

used in a payment equation. Computer simulation studies were conducted to verify that this was 

true. 

PWL Payment Factors 

Computer simulation was used to generate expected payment (EP) curves for a payment equation 

that based payment on the estimated PWL value. Since PWL is an unbiased estimator of the 

actual PWL value, the EP curve should follow the payment equation exactly. This will be true 

provided that an appropriate incentive provision is included in the specification. This is discussed 

at length in a later section on evaluating risks. 

For the purposes of the simulation, any payment equation could have been used. For the 

simulation studies, the payment equation from the AASHTO Quality Assurance Guide 

Specification was used.(3) This payment equation is: 

 Pay = 55 + (0.5 × PWL) (18) 

This equation was used with sample sizes = 3, 5, and 10. Figures 54 and 55 show the results of 

the analyses. The plots in figure 54 show the EP curves and the limits within which 95 percent of 

the estimated payment values fell. As expected, the EP (i.e., the average payment in the long run) 

follows the same line as the payment equation (equation 18). However, there is considerable 

variability in the payment values for individual lots. This variability (or spread about the EP line) 

decreases as the sample size increases from 3 to 10. This was also expected since it follows 

exactly what was found when PWL values were simulated and since it is the PWL values that 

determine the payment factor. 

To further illustrate the distribution of individual lot payment factors, figure 55 presents 

histograms of both the individual PWL and the payment factors for populations with an actual 

PWL = 90 and 50, for sample sizes = 3, 5, and 10. 
 163 

 Figure 54c. EP curves for PWL payment schedule with sample size = 10. 

 

 Figure 54a. EP curves for PWL payment schedule with sample size = 3. 

 Figure 

 

 Figure 54b. EP curves for PWL payment schedule with sample size = 5. 

 

n = 3

Pay = 55 + (0.5 × PWL) n = 5

Pay = 55 + (0.5 ×PWL)

Pay = 55 + (0.5 x PWL) n = 10 
 164 

Actual PWL = 50, n = 3 

Figure 55b. Distribution of actual PWL = 50, sample size = 3, and the resulting 

payment factors. 

al 

PWL = 90 and resulting payment factors. 

Figure 55a. Distribution of actual PWL = 90, sample size = 3, and resulting 

payment factors. 

Actual PWL = 50, n = 3
 165 

Figure 55d. Distribution of actual PWL = 50, sample size = 5, and the 

 resulting payment factors. 

Actual PWL = 90, n = 5

Figure 55c. Distribution of actual PWL = 90, sample size = 5, 

and the resulting payment factors. 

Actual PWL = 50, n = 5
 166 

Figure 55f. Distribution of actual PWL = 50, sample size = 10, and the 

 resulting payment factors. 

Actual PWL = 90, n = 10

Figure 55e. Distribution of actual PWL = 90, sample size = 10, and 

the resulting payment factors. 

Actual PWL = 50, n = 10
 167 

AAD Payment Factors 

Computer simulation was also used to generate EP curves for a payment equation that based 

payment on the estimated AAD value. Since AAD is an unbiased estimator of the actual AAD 

value, the EP curve should follow the payment equation exactly. For the purposes of the 

simulation, any payment equation could have been used. 

Since it is known that the expected value for the distribution of absolute deviations for a normal 

distribution is 0.798σ, a logical equation for relating AAD to payment is: 

 Pay = 105 – 24.75(AAD – 0.798), and Pay ≤ 105% (19) 

This equation provides a maximum payment of 105 percent for an AAD = 0.798σ (i.e., the value 

for a normal distribution centered on the target). This payment equation was used in the 

comparison analysis. It should be noted that this equation was not intended to provide exactly the 

same EP curve as the PWL payment equation. 

Figure 56 shows an EP curve for a sample size = 5, using equation 19 to determine the payment 

factor. It is shown in the figure that the EP curve follows a straight line. This is true because the 

payment equation is a straight line and AAD is an unbiased estimator. Also, the same trends of 

increasing variability about the EP line with a smaller sample size and decreased variability with 

a larger sample size would hold true for AAD. 

Table 38 shows the EP values from the plot in figure 56 and the payment factor that should have 

been received for a population with the AAD value shown. The final column shows the 

difference between the EP and the correct payment. For lower AAD populations (i.e., those with 

higher quality), the EPs are lower than called for in the payment equation (equation 19). As the 

AAD value increases, the EP values come in line with the correct values. This issue may be 

related to the selection of the payment equation rather than the AAD estimation process. This 

issue is discussed in the section on risks. 

Matching PWL and AAD Payment Equations 

Although at first PWL and AAD seem very different, they are both based on the assumption that 

the population being sampled is normal. All normal distributions can be related to the standard 

normal distribution (i.e., μ = 0.0 and σ = 1.0) by transforming the measurement axis to standard 

deviation units by the relationship: 

σ

− μ = X Z (20) 

where: Z = number of standard deviation units from the population mean 
 168 

Figure 56. EP curve with the 90th and 10th payment percentiles for the 

 AAD payment schedule. 

Table 38. Simulated EP factors and correct payment factors based on AAD. 

Actual AAD Expected Payment Correct Payment Difference 

0.798 102.24 105.00 –2.76 

0.912 100.40 102.18 –1.78 

1.018 98.21 99.55 –1.34 

1.128 95.96 96.83 –0.87 

1.238 93.60 94.11 –0.51 

1.349 91.06 91.36 –0.30 

1.461 88.93 88.59 +0.34 

1.574 85.89 85.79 +0.10 

1.688 82.90 82.97 –0.07 

Expected Pay Curve (Pay = 105 – 24.75[AAD – 0.798]), n = 5

65

70

75

80

85

90

95

100

105

0.75 1 1.25 1.5 1.75

Population AAD

Expected Payment, %

Pay - Avg

Pay - 90%

Pay - 10%
 169 

A similar approach can be taken for AAD. If AAD is measured in standard deviation units, then 

AAD can be related to the number of standard deviation units that the population mean is offset 

from the target value. This works for any normal distribution as long as AAD is measured in 

standard deviation units. In this way, AAD computer simulation results can be applied to any 

normal distribution in much the same manner that the standard normal curve can be used to 

calculate areas under any normal distribution. Figure 57 illustrates how this concept works. ZTarg

represents the distance that the population mean (μ) is offset from the target value (T) when 

measured in units of standard deviation (σ). 

Figure 57. Illustration of measuring AAD in standard deviation units (ZTarg) 

 from the mean. 

μ

σ

T μ

σ

T

σ

μ T

Targ

− Z =
 170 

If we also measure the specification limits to be used in calculating PWL in terms of standard 

deviation units from the target value, then once the PWL specification limits are set, we can 

develop a relationship between any given PWL value and its equivalent AAD value for the same 

normal population. 

In the simulation studies that were conducted to compare PWL results with AAD results for the 

same normal population, the PWL specification limits were set at ±1.645σ from the target value, 

which corresponds to 90 PWL for a population centered on the target. Since it is now possible to 

determine AAD and PWL values for the same offset from the target (ZTarg), an empirical 

relationship can be developed that for any normal population can convert any given PWL value 

to its corresponding AAD value, and vice versa. Equations 21 and 22 show this approximate 

empirical relationship in which all the terms have already been defined: 

1.198 Targ 1.613( ) AAD Targ 0.798 Z Z e

− = + (21) 

0.639 3.12(AAD 0.798)

Targ AAD 0.798 − − Z = − e (22) 

First, it should be noted that it is known that AAD = 0.7979σ if the population mean is centered 

at the target value. Therefore, since the AAD value for a population is approximately 0.798 times 

the population standard deviation, the only way that the AAD value could be zero is if the 

standard deviation was zero. Also, the smallest population AAD value is 0.798 times the 

population standard deviation. Therefore, if the population AAD value is measured in standard 

deviation units, then 0.798 is the smallest population AAD that is possible, thus, AAD ≥ 0.798 is 

a requirement. 

Equations 21 and 22 allow the corresponding PWL value to be obtained for a given AAD value, 

and vice versa. For example, for a population with an AAD of 1.00, the offset of the population 

mean from the target value (ZTarg), measured in standard deviation units, can be calculated as 

0.7403 by using equation 22. Since the specification limits are assumed to be set at ±1.645σ, 

PWL can be determined from tables of areas under the standard normal curve using figure 58. 

Tables 39 through 41 show the relationships between ZTarg, measured in standard deviation units, 

and the AAD and PWL values for a normal population, provided that the PWL specification 

limits are two-sided and are set at ±1.645σ. The two-side specification limits are a requirement 

since AAD is always two-sided because it is measured as the deviation from a target value. With 

these tables, it is easy to identify the appropriate PWL and AAD values for a given normal 

population, provided that the specification limits are set at ±1.645σ. 
 171 

Figure 58. Example illustrating the PWL specification limits and the offset 

 in σ units between the population mean and the target. 

μ = 0

σ = 1.0

T

0.7403

L = T – 1.645 U = T + 1.645

μ = 0

σ = 1.0

T

0.7403

L = T – 1.645 U = T + 1.645
 172 

Table 39. Relationship between ZTarg, AAD, and PWL for a normal population when the 

 PWL specification limits are set at 1.645σ.

ZTarg PWL (Lower) PWL (Upper) PWL (Total) AAD 

0.00 95.00 95.00 90.00 0.798 

0.10 95.95 93.88 89.83 0.820 

0.20 96.75 92.58 89.33 0.831 

0.30 97.41 91.07 88.48 0.845 

0.40 97.96 89.34 87.30 0.866 

0.50 98.40 87.39 85.79 0.895 

0.60 98.76 85.20 83.96 0.933 

0.70 99.05 82.77 81.82 0.979 

0.80 99.28 80.09 79.37 1.032 

0.90 99.45 77.19 76.64 1.093 

1.00 99.59 74.05 73.65 1.159 

1.10 99.70 70.71 70.41 1.231 

1.20 99.78 67.18 66.96 1.307 

1.30 99.84 63.50 63.33 1.388 

1.40 99.88 59.68 59.56 1.471 

1.50 99.92 55.76 55.68 1.558 

1.60 99.94 51.79 51.74 1.647 

1.70 99.96 47.81 47.77 1.738 

1.80 99.97 43.84 43.81 1.831 

1.90 99.98 39.94 39.92 1.925 

2.00 99.99 36.13 36.12 2.020 

2.10 99.99 32.46 32.45 2.116 

2.20 99.99 28.94 28.94 2.213 

2.30 100.00 25.62 25.62 2.310 

2.40 100.00 22.51 22.51 2.408 

2.50 100.00 19.63 19.63 2.506 

2.60 100.00 16.98 16.98 2.605 

2.70 100.00 14.57 14.57 2.704 

2.80 100.00 12.40 12.40 2.803 

2.90 100.00 10.47 10.47 2.902 

3.00 100.00 8.77 8.77 3.002 

3.10 100.00 7.28 7.28 3.102 

3.20 100.00 6.00 6.00 3.201 

3.30 100.00 4.90 4.90 3.301 

3.40 100.00 3.96 3.96 3.401 

3.50 100.00 3.18 3.18 3.501 
 173 

Table 40. Relationship between AAD, ZTarg, and PWL for a normal population when the 

 PWL specification limits are set at 1.645σ.

AAD ZTarg PWL (Lower) PWL (Upper) PWL (Total) 

0.798 0.0000 95.00 95.00 90.00 

0.800 0.0476 95.47 94.49 89.96 

0.850 0.3521 97.71 90.20 87.91 

0.900 0.5137 98.46 87.10 85.56 

0.950 0.6371 98.88 84.33 83.20 

1.000 0.7403 99.15 81.72 80.86 

1.100 0.9131 99.47 76.79 76.26 

1.200 1.0603 99.66 72.06 71.72 

1.300 1.1929 99.77 67.44 67.21 

1.400 1.3164 99.85 62.88 62.72 

1.500 1.4337 99.90 58.37 58.26 

1.600 1.5469 99.93 53.91 53.84 

1.700 1.6570 99.95 49.52 49.47 

1.800 1.7649 99.97 45.23 45.20 

1.900 1.8711 99.98 41.05 41.03 

2.000 1.9761 99.99 37.03 37.01 

2.250 2.2348 99.99 27.77 27.76 

2.500 2.4900 100.00 19.90 19.90 

2.750 2.7433 100.00 13.60 13.60 

3.000 2.9954 100.00 8.84 8.84 

3.250 3.2468 100.00 5.46 5.46 

3.500 3.4978 100.00 3.20 3.20 

3.750 3.7484 100.00 1.77 1.77 

4.000 3.9989 100.00 0.93 0.93 

4.250 4.2492 100.00 0.46 0.46 

4.500 4.4994 100.00 0.22 0.22 
 174 

Table 41. Relationship between PWL, AAD, and ZTarg for a normal population when the 

 PWL specification limits are set at 1.645σ.

PWL (Total) AAD ZTarg PWL (Lower) PWL (Upper) 

90.00 0.7980 0.0000 95.00 95.00 

85.00 0.9119 0.5456 98.58 86.42 

80.00 1.0186 0.7753 99.22 80.78 

75.00 1.1277 0.9558 99.54 75.46 

70.00 1.2380 1.1121 99.71 70.30 

65.00 1.3492 1.2546 99.81 65.19 

60.00 1.4610 1.3886 99.88 60.12 

55.00 1.5737 1.5175 99.92 55.07 

50.00 1.6878 1.6437 99.95 50.05 

45.00 1.8047 1.7699 99.97 45.03 

40.00 1.9253 1.8978 99.98 40.02 

35.00 2.0518 2.0301 99.99 35.01 

30.00 2.1863 2.1693 99.99 30.00 

25.00 2.3326 2.3194 100.00 25.00 

20.00 2.4967 2.4867 100.00 20.00 

15.00 2.6889 2.6816 100.00 15.00 

10.00 2.9314 2.9264 100.00 10.00 

 5.00 3.2930 3.2900 100.00 5.00 

 1.00 3.9740 3.9728 100.00 1.00 

Using the information from tables 39 through 41, a computer simulation program was developed 

that generated the EP and the standard deviation of the individual lot payment factors for any 

value of either PWL or AAD and the corresponding payment equation. The program output also 

identifies both the PWL and AAD values for the population. This program makes it simple to 

compare PWL and AAD payment plans provided that equivalent payment equations are 

developed for AAD and PWL. Tables 39 through 41 allow the development of such equivalent 

AAD and PWL payment schedules. 

As an example, suppose an agency was considering the use of PWL with the payment factor 

determined by equation 18. However, the agency would also like to evaluate a payment equation 

based on AAD that would provide for a given normal population the same EP as the PWL 

equation. This is not difficult to do and, indeed, was done in this project to compare PWL and 

AAD payment procedures for the same population. 

First, equation 18 was used to determine the payment factor for a number of actual PWL values 

ranging from 90 to 10. Table 41 was then used to identify the AAD value that corresponded to 

each of the PWL values. This then allowed the payment factors to be associated with their 

respective AAD values. The results of this process are shown in table 42. 
 175 

Table 42. AAD values equivalent to the corresponding PWL payment factors. 

Payment Factor Corresponding PWL 

Value Equivalent AAD Value 

100 90 0.798 

95 80 1.019 

90 70 1.238 

85 60 1.461 

80 50 1.688 

75 40 1.925 

70 30 2.186 

65 20 2.497 

60 10 2.931 

The AAD program that was developed allows for the use of a compound payment equation that 

is made up of two straight lines with different equations. To illustrate this and to help match the 

EP curve of the PWL payment equation, it was decide to use two equations that met at AAD = 

1.925. The program allows for an equation in the form: 

 Pay = A – B(AAD – 0.798) (23) 

It is easy to solve for the A and B coefficients using the values in table 42. Doing so resulted 

in the following AAD payment plan to match the PWL payment equation: 

 Pay = 100 – 22.18(AAD – 0.798) when AAD ≤ 1.925 (24) 

 Pay = 91.80 – 14.91(AAD – 0.798) when AAD > 1.925 (25) 

One should note that equations 24and 25 each provide a payment factor of 75 when AAD = 

1.925, thus providing the desired continuity in the two payment equations. 

Computer simulation was used to compare the performance of a PWL payment plan based on 

equation 18 with that of an AAD payment plan based on equations 24 and 25. The results of 

these simulations are shown in table 43 and figures 59 and 60. The plans have very similar EPs, 

with the AAD plan being slightly higher in the mean offset range of about 1.5 to 2.5. However, 

table 39 shows that this represents PWL values of about 55 and below. The PWL standard 

deviations increase to a peak at a mean offset of about 1.6 to 1.7, and are higher than those for 

AAD for the middle portion of the plot. The PWL peak corresponds to a PWL of 50, which was 

shown previously to be the point of maximum variability for PWL estimates. 
 176 

Table 43. Results of the simulation of matched PWL and AAD 

 payment equations for sample size = 5. 

PWL Results AAD Results 

Mean 

Offset PWL Expected 

Payment 

Standard 

Deviation AAD Expected Payment 

Standard 

Deviation 

0.000 90 100.26 5.40 0.798 99.76 5.05 

0.775 80 95.05 7.22 1.019 94.96 7.03 

1.112 70 89.74 8.61 1.238 89.99 7.76 

1.389 60 85.41 9.26 1.461 85.03 8.39 

1.644 50 80.15 9.52 1.688 80.71 8.21 

1.898 40 74.48 9.41 1.925 76.55 8.00 

2.169 30 69.76 8.91 2.186 71.78 7.35 

2.487 20 65.02 7.79 2.497 66.57 6.99 

2.926 10 60.11 5.51 2.931 60.14 6.65 
 177 

Figure 59. EP curves for matched PWL and AAD payment equations for sample size = 5. 

Figure 60. Standard deviations for individual payment factors for matched PWL and 

AAD payment equations for sample size = 5. 

50

60

70

80

90

100

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mean Offset from Target 

Expected Payment, %

PWL

AAD

0

2

4

6

8

10

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mean Offset from Target 

Payment St Dev, %

PWL

AAD
 178 

While there were some minor differences, the simulation study showed that when a single 

quality characteristic is used, it is possible to develop separate, but matched, PWL and AAD 

payment equations that provide similar EP curves and have reasonably close variability 

associated with the individual lot payment factors. 

PAYMENT FOR MULTIPLE QUALITY CHARACTERISTICS 

Since it is rare that only a single quality characteristic is used to determine the payment factor for 

a lot, it was also necessary to consider how the quality measures performed when multiple 

acceptance characteristics were used and to consider methods for combining these individual 

estimated quality measures into a combined, or composite, payment factor. While it was not 

within the scope of the original project, it was decided that, if multiple quality characteristics 

were to be investigated, it would also be necessary to consider potential correlations among these 

characteristics. 

Simulating Correlated Variables 

First it was necessary to develop computer routines for simulating correlated normal variables. 

These simulation routines were developed to simulate up to four correlated variables. The 

routines were then tested to verify that they were capable of generating samples with the 

intended level of correlation among the various variables. 

Correlation is related to a pair of variables and the amount of correlation between the two 

variables is measured in terms of the correlation coefficient. Correlation coefficients can vary 

from +1.0 (as one variable increases, the other also increases with perfect correlation) to –1.0 (as 

one variable increases, the other decreases with perfect correlation). A correlation coefficient of 

0.0 indicates that there is no correlation between the results of the two variables. 

Figure 61 shows the output screens from the simulations of 5000 pairs of correlated normal 

variables, each with different selected correlation coefficients. The histograms on the left side of 

the plots show the two normal populations from which the correlated values were generated. The 

scatter plots on the right side of the figures show the data that were simulated. Figure 61(a) 

shows a situation where the two variables are not correlated. The resulting plot shows no pattern 

in the results when one variable is plotted against the corresponding second variable. In this 

simulation, the desired correlation coefficient was 0.000, while the correlation coefficient for the 

5000 simulated pairs of values was –0.012, which is quite close to the desired value. 

Figure 61(b) shows a similar plot for a desired correlation coefficient of +1.000. In this case, 

there is perfect correlation between the two variables. The increasing straight line at a 45-degree 

angle that is obtained when the two variables are plotted against one another shows this 

relationship. The correlation of the simulated value was also +1.000. A plot for a correlation 

coefficient of –1.000 would also be a straight line, but would decrease at a 45-degree angle. The 

remaining plots show comparisons of various levels of positive and negative correlation, and also 

show that the simulation routine does a good job of simulating the desired correlation. 
 179 

Figure 61a. Illustration 1 of two normal variables with various values for the correlation 

 coefficient.

Figure 61b. Illustration 2 of two normal variables with various values for the correlation 

 coefficient. 
 180 

 Figure 61c. Illustration 3 of two normal variables with various values for the 

 correlation coefficient.

Figure 61d. Illustration 4 of two normal variables with various values for the 

 correlation coefficient. 
 181 

Figure 61e. Illustration 5 of two normal variables with various values for the 

 correlation coefficient. 

Figure 61f. Illustration 6 of two normal variables with various values for the 

 correlation coefficient. 
 182 

 Figure 61g. Illustration 7 of two normal variables with various values for the 

 correlation coefficient.

 Figure 61h. Illustration 8 of two normal variables with various values for the 

 correlation coefficient. 
 183 

The simulation routine that was developed is capable of simulating up to four correlated normal 

variables. When more than two correlated variables are simulated, it is not possible to clearly 

show the correlation relationships as done in figure 61 for the two-variable case. However, tables 

44 and 45 show the results of simulations for three and four correlated variables, respectively. 

While the results cannot easily be shown graphically, the tables show that the simulated data 

have correlation coefficients that compare closely with the desired values. 

Table 44. Demonstration of the simulation of three correlated normal variables with 

selected values for correlation coefficients. 

Correlated Variables Desired Correlation 

Coefficient 

Simulated Correlation 

Coefficient 

Example 1 

Variable 1—Variable 2 +0.500 +0.497 

Variable 1—Variable 3 +0.250 +0.259 

Variable 2—Variable 3 +0.500 +0.510 

Example 2 

Variable 1—Variable 2 +0.500 +0.500 

Variable 1—Variable 3 –0.250 –0.262 

Variable 2—Variable 3 –0.500 –0.509 

Example 3 

Variable 1—Variable 2 –0.500 –0.500 

Variable 1—Variable 3 –0.500 –0.496 

Variable 2—Variable 3 –0.500 –0.504 
 184 

Table 45. Demonstration of the simulation of four correlated normal variables with 

 selected values for correlation coefficients. 

Correlated Variables Desired Correlation 

Coefficient 

Simulated Correlation 

Coefficient 

Example 1 

Variable 1—Variable 2 +0.500 +0.497 

Variable 1—Variable 3 +0.250 +0.269 

Variable 1—Variable 4 +0.500 +0.506 

Variable 2—Variable 3 +0.500 +0.500 

Variable 2—Variable 4 +0.250 +0.252 

Variable 1—Variable 4 +0.500 +0.510 

Example 2 

Variable 1—Variable 2 –0.250 –0.247 

Variable 1—Variable 3 +0.500 +0.494 

Variable 1—Variable 4 –0.250 –0.248 

Variable 2—Variable 3 +0.500 +0.513 

Variable 2—Variable 4 +0.250 +0.263 

Variable 1—Variable 4 –0.500 –0.478 

Correlated Quality Characteristics

Earlier in this chapter, the results from two simulation studies—one using PWL and equation 18 

to determine payment and the other using AAD and equation 19 to determine payment—were 

compared. This comparison was expanded to address the case where two individual variables 

were considered when determining payment. While it is possible, and even likely, to have more 

than two variables involved in the acceptance and payment decision, it is not easy to picture the 

results when more than two variables are considered. Therefore, only the two-variable case was 

considered in this project. The results from two variables should be equally applicable to cases 

with three or four variables. 

Two Populations of Equal Quality: The routines discussed above were used to compare the EP 

values and the amount of variability in the individual payment factors for both PWL and AAD 

when two correlated variables were involved. In the analyses for both PWL and AAD, the 

combined payment factor was taken as the weighted average of the two individual payment 

factors. This general payment relationship is shown in equation 26. 
 185 

 Pay = (W1 × PF1) + (W2 × PF2) (26) 

where: W1 = weighting for variable 1 

W2 = weighting for variable 2 

PF1 = individual payment factor for variable 1 

PF2 = individual payment factor for variable 2 

A sample size = 5 was used in the analyses. Three different correlation coefficients between the 

two variables were considered: 0.0, –0.5, and +0.5. In each analysis, the two individual variables 

had the same actual population: 70 PWL or 1.238 AAD. Various weightings (1.00/0.00, 

0.90/0.10, 0.75/0.25, 0.50/0.50, 0.25/0.75, 0.10/0.90, and 0.00/100.00) were used to determine 

the combined payment factor. 

Since, in each case, the two individual populations were of equal quality (i.e., 70 PWL or 1.238 

AAD), the same EP should be obtained regardless of the weightings used. For the PWL case, the 

EP for each individual variable should be 55 + (0.5 × 70) = 90. For the AAD case, the EP for 

each individual variable should be 105 – [24.75 × (1.238 – 0.798)] = 94.1. 

The results from the analyses are presented in table 46 and figures 62 and 63 for PWL, and in 

table 47 and figures 64 and 65 for AAD. The results for the 1.00/0.00 and 0.00/1.00 weightings 

are equivalent to the one quality characteristic using variable 1 or variable 2, respectively. 

From table 46 and figure 62, it is shown that the EP values do not change as either the correlation 

coefficients or the variable weights change. The EP values are all around the value of 90.0 

calculated from the payment equation. Therefore, PWL provides an unbiased estimate for the 

combined payment factor regardless of the weights selected or the amount of correlation between 

the variables. 

However, the same is not true with respect to the variability of the individual simulated 

combined payment factors, as indicated by the standard deviation values in table 46 and figure 

63. There is a noticeable difference among the standard deviation values for the three different 

correlation coefficients, with the difference being greatest at the 0.50/0.50 weighting of the two 

variables. Also, within the values for each correlation coefficient, there is a similar trend of 

decreasing variability as the 0.50/0.50 weighting is approached. 

Although they are both related to the same mathematical relationship for the variance of the sum 

of two random variables, two different phenomena are at work in these two trends. It can be 

shown mathematically that if a and b are constants and X and Y are random variables, then: 

aX bY X Y XY σ a σ b σ 2abσ 2 2 2 2 2+ = + + (27) 
 186 

Table 46. Results of PWL simulation analyses for two correlated 

 normal variables. 

Correlation Coefficient 

Weightings, 0.00 +0.50 –0.50 

Var1/Var2 Expected 

Pay

Std. 

Dev.

Expected 

Pay Std. Dev. Expected Pay

Std. 

Dev.

1.00/0.00 90.62 8.75 90.02 8.75 90.05 8.52 

0.90/0.10 89.84 7.96 89.67 8.00 90.04 7.27 

0.75/0.25 90.11 6.89 90.12 7.58 90.00 5.94 

0.50/0.50 90.03 6.16 90.12 7.46 89.99 4.96 

0.25/0.75 89.72 6.82 90.02 7.66 90.14 6.30 

0.10/0.90 89.62 8.09 89.91 7.99 90.18 7.85 

0.00/1.00 90.24 8.59 89.76 8.75 89.82 8.67 

Table 47. Results of AAD simulation analyses for two correlated 

 normal variables. 

Correlation Coefficient 

Weightings, 0.00 +0.50 –0.50 

Var1/Var2 Expected 

Pay

Std. 

Dev.

Expected 

Pay Std. Dev. Expected 

Pay

Std. 

Dev.

1.00/0.00 93.92 8.44 93.60 8.53 93.61 8.51 

0.90/0.10 93.84 8.14 93.78 8.45 93.58 7.25 

0.75/0.25 93.31 6.96 93.69 7.81 93.76 6.23 

0.50/0.50 94.12 6.13 93.53 7.52 94.01 5.06 

0.25/0.75 93.84 7.14 93.84 7.62 94.30 6.03 

0.10/0.90 93.32 8.04 93.63 8.17 93.36 7.51 

0.00/1.00 93.23 8.81 93.70 8.53 93.30 8.47 
 187 

Figure 62. Expected combined weighted average payment factors for various weights 

and correlation coefficients based on PWL. 

Figure 63. Standard deviations of weighted average payment factors for various weights 

 and correlation coefficients based on PWL. 

Pi = 55 + .5PWL, PF = Wt. Avg of Pi Values

n = 5, Both Have PWL = 70

89.0

89.2

89.4

89.6

89.8

90.0

90.2

90.4

90.6

90.8

91.0

1.00-0.00 0.90-0.10 0.75-0.25 0.50-0.50 0.25-0.75 0.10-0.90 0.00-1.00

Weighting of Var1–Var2 in Pay Factor

Corr = 0

Corr. = +0.5

Corr = -0.5

Expected Pay, %

Pi = 55 + .5PWL, PF = Wt. Avg of Pi Values

n = 5, Both Have PWL = 70

89.0

89.2

89.4

89.6

89.8

90.0

90.2

90.4

90.6

90.8

91.0

1.00-0.00 0.90-0.10 0.75-0.25 0.50-0.50 0.25-0.75 0.10-0.90 0.00-1.00

Weighting of Var1–Var2 in Pay Factor

Corr = 0

Corr. = +0.5

Corr = -0.5

Expected Pay, %

n = 5, Both Have PWL = 70

Pi = 55 + .5PWL, PF = Wt. Avg of Pi Values 

4

5

6

7

8

9

1.00-0.00 0.90-0.10 0.75-0.25 0.50-0.50 0.25-0.75 0.10-0.90 0.00-1.00

Weighting of Var1–Var2 in Pay Factor

Corr = 0

Corr. = +0.5

Standard Deviation, %

 Corr = –0.5

n = 5, Both Have PWL = 70

Pi = 55 + .5PWL, PF = Wt. Avg of Pi Values 

4

5

6

7

8

9

1.00-0.00 0.90-0.10 0.75-0.25 0.50-0.50 0.25-0.75 0.10-0.90 0.00-1.00

Weighting of Var1–Var2 in Pay Factor

Corr = 0

Corr. = +0.5

Standard Deviation, %

 Corr = –0.5
 188 

Figure 64. Expected combined weighted average payment factors for various weights and 

correlation coefficients based on AAD.

Figure 65. Standard deviations of weighted average payment factors for various weights 

 and correlation coefficients based on AAD. 

Pi = 105 – 24.75(AAD – 0.798), PF = Wt. Avg of Pi Values

n = 5, Both Have AAD = 1.238 

92.8

93.0

93.2

93.4

93.6

93.8

94.0

94.2

94.4

94.6

94.8

1.00-0.00 0.90-0.10 0.75-0.25 0.50-0.50 0.25-0.75 0.10-0.90 0.00-1.00

Weighting of Var1–Var2 in Pay Factor

Corr = 0

Corr. = +0.5

 Corr = -0.5

Expected Pay, %

Pi = 105 – 24.75(AAD – 0.798), PF = Wt. Avg of Pi Values

n = 5, Both Have AAD = 1.238 

92.8

93.0

93.2

93.4

93.6

93.8

94.0

94.2

94.4

94.6

94.8

1.00-0.00 0.90-0.10 0.75-0.25 0.50-0.50 0.25-0.75 0.10-0.90 0.00-1.00

Weighting of Var1–Var2 in Pay Factor

Corr = 0

Corr. = +0.5

 Corr = -0.5

Expected Pay, %

n = 5, Both Have AAD = 1.238

Pi = 105 – 24.75(AAD – 0.798), PF = Wt. Avg of Pi Values

4

5

6

7

8

9

1.00-0.00 0.90-0.10 0.75-0.25 0.50-0.50 0.25-0.75 0.10-0.90 0.00-1.00

Weighting of Var1–Var2 in Pay Factor

Corr = 0

Corr. = +0.5

 Corr = –0.5

Standard Deviation, %

n = 5, Both Have AAD = 1.238

Pi = 105 – 24.75(AAD – 0.798), PF = Wt. Avg of Pi Values

4

5

6

7

8

9

1.00-0.00 0.90-0.10 0.75-0.25 0.50-0.50 0.25-0.75 0.10-0.90 0.00-1.00

Weighting of Var1–Var2 in Pay Factor

Corr = 0

Corr. = +0.5

 Corr = –0.5

Standard Deviation, %
 189 

In equation 27, the last term is the covariance term and it is related to the correlation coefficient. 

The constants a and b can be related to the weightings (e.g., 0.50/0.50) that are multiplied by 

each of the random variables. Although equation 27 applies to the variability of the individual 

quality characteristic results, these results are used to determine PWL, which is linearly related to 

the payment factor by the payment equation. Therefore, equation 27 can be used to explain the 

standard deviation trends shown in figure 63. 

In figure 63, for example, for a given set of weightings, such as 0.50/0.50, the standard deviation 

for +0.5 correlation is higher than for 0.0 correlation, which is higher than for –0.5 correlation. 

This follows directly from equation 27. For constant weightings, a and b in equation 27 remain 

the same for each correlation coefficient. Therefore, it is only the covariance term that changes 

the combined standard deviation value. For a correlation coefficient of 0.00, the covariance term 

in equation 27 also becomes 0.00. A positive correlation coefficient (and, therefore, a positive 

covariance term) results in a larger combined variability and, conversely, a negative correlation 

coefficient results in a smaller combined variability. Since the standard deviation also directly 

affects the AAD value, a similar standard deviation trend is shown for the AAD combined 

payment factors in table 47 and figure 65. 

Equation 27 can also be used to explain the differences in standard deviation with respect to the 

weightings that are used. These weights are represented by the a and b constants in equation 27. 

For any given covariance value, the magnitudes of a and b will determine the differences in the 

combined standard deviations. 

For example, take the case where the correlation coefficient is 0.00. In this event, the covariance 

term is eliminated from equation 27. Since the two variables have equal sample sizes, if 

measured in standard deviation units, they will also have equal variances for the purposes of 

calculating PWL and, therefore, payment. If we call this variance σ

2

, then for the case of 

a 0.50/0.50 weighting, equation 27 leads to: 

σ 0.5σ 0.5σ 0 0.5σ 0.707σ 2 2 2 = + + = = aX +bY (28) 

For example, figure 63 shows that the standard deviation for PWL, with a correlation coefficient 

of 0.00 and weighting of 1.00/0.00 or 0.00/1.00, is about 8.7. Based on the calculations above, 

the standard deviation for a 0.50/0.50 weighting should be about 0.707σ = 0.707 × 8.7 = 6.15, 

which is essentially the same value shown in table 46. Similar calculations verify the shapes 

shown in figures 63 and 65. 

Therefore, for two populations with equal PWL values, the combined payment factor, based on 

the weighted average of the two individual payment factors, has the same mean payment as for 

the individual variables, but with smaller variability. This is an unexpected result of using a 

weighted average approach for combining individual payment factors into one combined 

payment factor. 

To consider populations with additional PWL values, simulations were run for the case of 

populations with 90 PWL, 70 PWL, and 50 PWL. The simulations were run with weightings of 

0.50/0.50 and with correlation coefficients between the two variables ranging from –1 to +1. For 
 190 

both the PWL case (figure 66) and the AAD case (figure 67), the horizontal lines show that the 

EP value is not sensitive to the correlation between the individual variables, at least when the 

weighted average is used to determine the composite payment. 

However, figures 68 and 69 indicate that the variability of the payment estimates is indeed 

related to the correlation between the two individual variables. This same trend is evident for 

both PWL and AAD. This means that while the average payment stays the same, the variability 

about the average payment increases as the correlation goes from –1.0 to + 1.0 for the two lower 

quality populations (i.e., 70 PWL and 50 PWL, and 1.238 AAD and 1.688 AAD), while it 

decreases slightly as the correlation approaches 0.0 for the higher quality population (90 PWL 

and 0.798 AAD). The relationship is different for the higher quality population since the upper 

boundaries of either 100 PWL or 0.0 AAD will have a greater impact than for the lesser quality 

populations. 

Two Populations of Different Quality: All of the two-variable analyses up to now have been 

for the case where the two populations have the same quality level (i.e., the same PWL or AAD). 

Tables 47 through 51 show the results of simulation analyses for cases where the two individual 

populations have different PWL or AAD values and 50/50 weighting of the two variables when 

determining the combined payment factor. Tables 48 and 49 show that while, as expected, the 

quality levels of the individual variables affect the EP values, the EP values do not appear to be 

impacted by a moderate degree of correlation (i.e., –0.5, 0.0, and +0.5). Tables 50 and 51, 

however, show that the variability of the payment values may increase slightly with either 

positive or negative correlation between the two individual variables. This trend is evident in 

each case, except when one of the variables has the lowest quality simulated (i.e., 50 PWL or 

1.688 AAD). 

The values in tables 48 through 51 are plotted in figures 70 through 73. As shown previously, the 

payment biases using PWL are small and are centered at the correct value, regardless of the 

quality levels of the two populations. This is not the case for the payment biases using AAD, 

which are larger than for PWL and are negative for higher quality levels and positive for lower 

quality levels. For both PWL and AAD, the standard deviation values are lower for higher 

quality populations and there is little difference associated with the correlation coefficients. As 

the level of quality becomes lower, the standard deviations become larger and a difference 

develops with respect to the correlation coefficients. This trend is true for both PWL and AAD. 
 191 

Figure 66. Expected average payment factors for two populations with various 

correlation coefficients based on PWL. 

Figure 67. Expected average payment factors for two populations with various 

correlation coefficients based on AAD.

Pi = 55 + .5PWL, PF = Avg of Pi Values

n = 5, Populations Have Equal PWLs

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

Pi = 105 – 24.75(AAD – 0.798), PF = Avg of Pi Values

n = 5, Populations Have Equal PWLs

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

.798 AAD 1.238 AAD 1.688 AAD
 192 

Figure 68. Standard deviations of individual payment factors for two populations with 

 various correlation coefficients based on PWL.

Figure 69. Standard deviations of individual payment factors for two populations with 

 various correlation coefficients based on AAD.

Pi = 55 + .5PWL, PF = Avg of Pi Values

n = 5, Populations Have Equal PWLs

0

1

2

3

4

5

6

7

8

9

10

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, % Pay

90 PWL 70 PWL 50 PWL

Pi = 105 – 24.75(AAD – 0.798), PF = Avg of Pi Values

n = 5, Both Population Have Equal PWLs

0

1

2

3

4

5

6

7

8

9

10

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %

0.798 AAD 1.238 AAD 1.688 AAD
 193 

Table 48. Bias in EP for two populations with equal PWL values and 

 various correlation coefficients. 

Correlation Coefficient 

PWL–1 PWL–2 Correct 

Pay –0.50 0.00 +0.50 

90 80 97.50 –0.38 –0.30 +0.07 

90 70 95.00 –0.06 +0.22 –0.07 

90 60 92.50 +0.08 +0.16 +0.19 

90 50 90.00 0.00 –0.17 +0.02 

80 70 92.50 +0.21 +0.14 –0.08 

80 60 90.00 +0.05 +0.11 –0.04 

80 50 87.50 +0.13 –0.09 –0.13 

70 60 87.50 +0.32 +0.14 –0.20 

70 50 85.00 –0.05 –0.01 –0.04 

60 50 82.50 –0.07 +0.16 –0.06 

Table 49. Bias in EP for two populations with equal AAD values and 

 various correlation coefficients. 

Correlation Coefficient 

AAD–1 AAD–2 Correct 

Pay –0.50 0.00 +0.50 

0.798 1.019 102.25 –1.11 –1.13 –1.35 

0.798 1.238 99.55 –0.63 –0.54 –0.41 

0.798 1.461 96.80 –0.33 –0.26 –0.01 

0.798 1.688 94.00 –0.34 –0.34 –0.44 

1.019 1.238 96.80 +0.96 +0.71 +0.92 

1.019 1.461 94.05 +1.41 +1.13 +1.60 

1.019 1.688 91.25 +1.51 +1.10 +1.47 

1.238 1.461 91.35 +1.18 +1.24 +1.05 

1.238 1.688 88.55 +1.53 +1.49 +0.74 

1.461 1.688 85.80 +1.02 +1.32 +1.37 
 194 

Table 50. Standard deviations of individual payment factors for two populations with 

 equal PWL values and various correlation coefficients. 

Correlation Coefficient 

PWL–1 PWL–2 

–0.50 0.00 +0.50 

90 80 5.05 4.77 4.95 

90 70 5.41 4.99 5.23 

90 60 5.61 5.34 5.49 

90 50 5.33 5.60 5.43 

80 70 4.76 5.76 6.95 

80 60 4.99 6.15 7.06 

80 50 5.12 6.12 7.09 

70 60 5.25 6.21 7.57 

70 50 4.99 6.56 7.58 

60 50 4.94 6.44 8.39 

Table 51. Standard deviations of individual payment factors for two populations 

with equal AAD values and various correlation coefficients. 

Correlation Coefficient 

AAD–1 AAD–2 

–0.50 0.00 +0.50 

0.798 1.019 4.38 3.97 4.58 

0.798 1.238 5.39 4.88 5.39 

0.798 1.461 5.86 5.63 5.72 

0.798 1.688 6.03 6.09 6.16 

1.019 1.238 5.52 5.37 5.92 

1.019 1.461 5.96 6.05 6.54 

1.019 1.688 6.29 6.25 7.00 

1.238 1.461 5.71 6.79 7.65 

1.238 1.688 5.90 6.80 7.97 

1.461 1.688 5.94 6.88 8.44 
 195 

Figure 70. Bias for the average payment for two populations with various actual 

 PWL values. 

Figure 71. Bias for the average payment for two populations with various actual 

 AAD values. 

Pi = 55 + .5PWL, PF = Avg of Pi Values

n = 5

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

90-80 90-70 90-60 90-50 80-70 80-60 80-50 70-60 70-50 60-50

PWL of Populations 1 and 2

Payment Bias, % 

–0.5

 0.0

+0.5

Pi = 105 – 24.75(AAD – 0.798), 

PF = Avg of Pi Values, n = 5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

.798-

1.019

.798-

1.238

.798-

1.461

.798-

1.688

1.019-

1.238

1.019-

1.461

1.019-

1.688

1.238-

1.461

1.238-

1.688

1.461-

1.688

AAD of Populations 1 and 2

Payment Bias, % 

–0.5

 0.0

+0.5
 196 

Figure 72. Standard deviation for the individual average payment values for two 

 populations with various actual PWL values. 

Figure 73. Standard deviation for the individual average payment values for two 

 populations with various actual AAD values. 

Pi = 55 + .5PWL, PF = Avg of Pi Values

n = 5

3.0

4.0

5.0

6.0

7.0

8.0

9.0

90-80 90-70 90-60 90-50 80-70 80-60 80-50 70-60 70-50 60-50

PWL of Populations 1 and 2

Standard Deviation, % 

–0.5

 0.0

+0.5

Pi = 105 – 24.75(AAD – 0.798), 

PF = Avg of Pi Values, n = 5

3

4

5

6

7

8

9

.798-

1.019

.798-

1.238

.798-

1.461

.798-

1.688

1.019-

1.238

1.019-

1.461

1.019-

1.688

1.238-

1.461

1.238-

1.688

1.461-

1.688

AAD of Populations 1 and 2

Standard Deviation, %

–0.5

 0.0

+0.5
 197 

SELECTION OF THE QUALITY MEASURE 

While the analyses indicated that, in general, the AAD quality measure performed about as well 

as the PWL quality measure, there were instances where the AAD measure had greater payment 

variability and exhibited some trends regarding payment bias. These issues, in themselves, would 

not be sufficient to eliminate AAD as a potential quality measure. However, there are some other 

drawbacks to AAD that make it a less appealing quality measure than PWL. 

It is necessary to measure both center and spread when characterizing a population (lot) of 

material. Because of this, there are potential difficulties with the AAD quality measure. One 

drawback with AAD acceptance plans is that since lot variability is not really measured, a given 

lot AAD could come from a number of different populations. For example, the population could 

be centered at the target, but have a relatively large standard deviation (i.e., larger than the one 

that was assumed when developing the AAD acceptance limits or payment equation). Another 

population could have the same AAD by being centered off the target and having the same 

standard deviation that was assumed when developing the acceptance values. A third population 

could have the same AAD value by having a mean far from the target, but also by having a 

relatively small standard deviation. The fact that dissimilar populations can have the same AAD 

is shown by the distributions in figure 38 in chapter 6. 

While some of these drawbacks may also apply to PWL (or PD) acceptance plans, such as the 

fact that a given PWL can represent many different populations, there are fewer drawbacks 

because both sample mean and standard deviation are measured in the PWL method. Also, since 

the PWL method can be used with both one-sided and two-sided acceptance properties, it is more 

versatile than the AAD method, which cannot be applied to one-sided limits. 

Therefore, PWL (or its complement PD) was selected as the best quality measure for use in QA 

specifications. All remaining analyses were conducted only for the PWL (or PD) quality 

measure. 

Now that the quality measure has been selected, its use needs to be further analyzed with respect 

to the risks associated with the small sample sizes that are typically used in highway materials 

and construction applications. Risks related to PWL and PD estimates, and their related payment 

factors, are discussed in chapter 9. 
 
 199 

9. EVALUATING RISKS 

INTRODUCTION 

The concept of risk for acceptance plans, in general, and PWL, in particular, since that is our 

selected quality measure, is very similar to the concepts of accuracy and precision that are 

discussed in chapter 5 and the concepts of hypothesis testing and verification that are discussed 

in chapter 7. 

Types of Risk 

In hypothesis testing, there are two types of errors. A type I error occurs when a true null 

hypothesis is incorrectly rejected. A type II error occurs when an incorrect hypothesis is 

erroneously accepted. The probability of making a type I error is known as the α risk, while the 

probability of making a type II error is known as the β risk. When testing a null hypothesis, the α

risk is also known as the level of significance. 

The terms α and β have been applied in the highway construction industry for many years. 

However, these terms are related to a situation where the decision is either to accept or reject the 

hypothesis. In a highway construction scenario, the null hypothesis that is being tested is that the 

contractor’s product meets the specification requirements. In such a scenario, the α and β risks 

apply as they do in any other hypothesis testing situation. Oftentimes, however, the decision is 

not simply to accept or reject the material, but to accept it at an incentive payment (positive price 

adjustment, or bonus) or at a disincentive payment (negative price adjustment, or penalty). 

The Transportation Research Board (TRB) has published the following definitions for α and β

risks:(4) 

Seller’s Risk (α) (also called a type I error): The probability that an acceptance plan will 

erroneously reject acceptable quality level (AQL) material or construction with respect to a 

single acceptance quality characteristic. It is the risk that the contractor or producer takes in 

having AQL material or construction rejected. 

Buyer’s Risk (β) (also called a type II error): The probability that an acceptance plan will 

erroneously fully accept (100 percent or greater) rejectable quality level (RQL) material or 

construction with respect to a single acceptance quality characteristic. It is the risk that the 

highway agency takes in having RQL material or construction fully accepted. (The probability of 

having RQL material or construction accepted (at any pay) may be considerably greater than the 

buyer’s risk.) 

When α and β risks are applied to materials or construction, they are really only appropriate for 

the case of acceptance or rejection decisions. When materials may not only be accepted or 

rejected, but may also be accepted at an adjusted payment, then the concept of α and β risks does 

not strictly apply. In such cases, which are perhaps more common than acceptance or rejection 

decisions, then much more involved analyses than just calculating α and β risks are necessary. 
 200 

In an attempt to address the deficiencies in the use of α and β for price-adjustment acceptance 

plans, a new term (α100) has been proposed. This is defined as the probability that AQL material 

will receive less than 100-percent payment. Just as with any term that tries to define a single 

point in a payment continuum, α100 provides only a little additional information regarding the 

realm of possible payments for a given quality of material. 

OC and EP Curves 

The only single term that truly attempts to consider the totality of possible payments for a given 

quality of material is EP. However, even this term is not sufficient since it considers only the 

average payment that a contractor can expect to receive for a very large number of production 

lots of a given quality level. EP fails to consider the amount of variability in the individual 

payment values that comprise the calculation of EP. To fully evaluate the risks in a priceadjustment acceptance plan, it is necessary to determine OC curves and EP curves, and also to 

investigate the amount of variability in the individual payment values about the EP curve. 

TRB defines OC and EP curves as:(4) 

OC Curve: A graphic representation of an acceptance plan that shows the relationship between 

the actual quality of a lot and either: (1) the probability of its acceptance (for accept/reject 

acceptance plans) or (2) the probability of its acceptance at various payment levels (for 

acceptance plans that include pay-adjustment provisions). 

EP Curve: A graphic representation of an acceptance plan that shows the relationship between 

the actual quality of a lot and its EP (i.e., mathematical pay expectation, or the average pay the 

contractor can expect to receive over the long run for submitted lots of a given quality). (Both 

OC and EP curves should be used to evaluate how well an acceptance plan is theoretically 

expected to work.) 

The definition of OC curves indicates that multiple curves might be plotted for the probability of 

receiving various levels of payment for a given actual quality level. The obvious difficulty with 

this is that if, as is recommended, an equation is used to calculate the payment factor, then there 

are an infinite number of OC curves that could be developed. Certain values are obvious 

candidates (e.g., the probability of receiving full payment or, if there is a remove-and-replace 

provision for very poor quality, the probability of requiring removal and replacement). It is 

difficult to view a plot with numerous OC curves (one for each of many selected payment levels) 

and to interpret it with a great deal of significance. 

The EP curve, in essence, converts the family of OC curves into a single curve that represents the 

average payment that would be received in the long run for a given level of quality. This is the 

single curve that provides the most meaningful and useful information for a contractor or 

highway agency. However, even an EP curve is not sufficient since it does not consider the 

amount of variability in the individual payment values. This is why most of the analyses 

presented in this report consider not only bias or accuracy, but also standard deviation, to 

indicate the variability in addition to the average alone. 
 201 

CALCULATING RISKS FOR A SINGLE QUALITY CHARACTERISTIC 

Risks are traditionally determined separately for each individual quality characteristic. This 

process can be quite complicated, particularly if there are unusual provisions incorporated into 

the specification. Although not identified specifically as risk analysis at the time, much of the 

information presented in chapter 5 is directly related to the concept of risk. In this section, the 

material from chapter 5 that is related to the PWL quality measure is revisited and discussed in 

light of its application to risk analysis. 

The computer simulation routines that were used to determine figures 9 through 12 were 

developed to calculate the information necessary to plot OC and EP curves when using either 

PWL or PD as the quality measure. The routines were actually developed for PD since the 

calculations are slightly less involved. While the programs can print the results for either PD or 

PWL, the calculations are performed for PD and are then converted to PWL if necessary. 

Example 

To illustrate the process of evaluating risks, the following payment equation, which is from the 

AASHTO Quality Assurance Guide Specification and which has been used many times in other 

chapters, is used:(3)

 Pay = 55 + 0.5PWL (29) 

While this equation is not necessarily recommended, it has been used by a number of agencies. 

This equation is simple because it is a single straight line. It would be preferable to use either 

more than one straight line over different regions or to use some form of curved payment 

equation. This would allow for a shallower slope (i.e., lower payment reduction) for quality 

levels near the AQL and for a steeper slope that allows the price reductions to increase by a 

greater amount as the quality departs more from the AQL. For the purposes of illustration, 

equation 29 will work well. We also need to select values for AQL and RQL in terms of PWL. 

For this example, we will use 90 PWL as the AQL value and 60 PWL as the RQL value. 

Also, suppose that the acceptance plan calls for removal and replacement of the material if the 

calculated PWL estimate is less than 60 (i.e., less than the RQL value). Therefore, the α risk (i.e., 

the risk that the AQL material will be rejected) is the probability that an AQL population will 

have an estimated PWL value of less than 60. This can be determined by simulating a large 

number of lots from an AQL population and determining the percentage of them that have PWL 

estimated values of less than 60. 

The results of such a computer simulation of 1000 lots for a sample size = 4 are plotted in 

figure 74. The horizontal line on the plot in figure 74 indicates the α risk. It is the probability that 

AQL material will be rejected and is shown on the plot as the difference between 100-percent 

probability of acceptance and the probability of acceptance at AQL = 90 PWL. The α risk is 

about 0.025 (or 2.5 percent). This seems like a low risk, but remember that this is the risk of 

rejection. The curve in figure 74 is the probability of receiving at least some payment. While 

rejection is unlikely, there may be a significant chance of receiving less than full payment. 
 202 

Figure 74. OC curve for an acceptance plan that calls for rejection if the estimated PWL is 

 less than 60, sample size = 4. 

In fact, the probability of receiving at least 100-percent payment can be determined in the same 

way as the probability of acceptance, except that the probability in question is the probability of 

having an estimated PWL ≥ 90. This value of 90 is calculated as the PWL value that yields 100-

percent payment from equation 29. 

The probability of receiving at least 100-percent payment is shown in figure 75, along with the 

probability of acceptance at any price. While the probability of receiving at least some payment 

is quite high for AQL material, the probability of receiving 100-percent or greater payment is 

only about 0.60 (or 60 percent). Therefore, for this example, the value of α100 would be about 

1.00 – 0.60 = 0.40 (or 40 percent). That is the risk that the AQL material would receive less than 

full payment. 

From this example, it appears that α is quite low at about 2.5 percent and that α100 is quite high 

at about 40 percent. However, neither of these numbers tells the full story. For one thing, they 

apply only to one specific level of quality—AQL = 90 PWL. Secondly, they address only two 

specific points on the payment continuum—receiving greater than zero payment and greater than 

or equal to 100-percent payment. 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Probability of Acceptance

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Probability of Acceptance
 203 

Figure 75. OC curves for the probabilities of receiving at least some payment and at least 

100-percent payment, sample size = 4. 

Neither of these pieces of information is sufficient to fully evaluate the risk associated with this 

payment plan. Many other curves, such as the probability of receiving at least 90-percent 

payment or of receiving at least 104-percent payment, could be added to the plot. In reality, there 

is a continuous payment region that runs from the curve for the probability of receiving at least 

some payment to the curve for the probability of receiving some high payment. The maximum 

payment of 105 percent can only be averaged if the population has 100 PWL; it cannot be plotted 

except as a point at the upper right-hand corner. However, the curve representing, say, 104.5-

percent payment can be calculated and is plotted in figure 76, along with the curves for greater 

than zero payment and greater than or equal to 100-percent payment. 

Figure 76 shows that the probability of receiving at least 104.5-percent payment for AQL 

material is approximate 47 percent. To at least some extent, this should help to balance the fact 

that there is about a 40-percent probability of receiving less than 100-percent payment. To 

evaluate the overall long-term performance of the acceptance plan and the potential payment 

risks for the contractor, it is necessary to determine the EP curve for the plan. 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Probability of Acceptance

PF ≥ 100%

PF ≥ 0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Probability of Acceptance

PF ≥ 100%

PF ≥ 0
 204 

Figure 76. OC curves for the probabilities of receiving various payments, 

 sample size = 4. 

Figure 77 shows the EP curve for the payment schedule in equation 29, with the added provision 

that estimated PWL values of less than 60 receive no payment (i.e., they require removal and 

replacement at the contractor’s expense). Figure 77 shows that the EP in the long run for AQL 

material is less than 100 percent. It is generally accepted that the EP for AQL material should be 

100 percent. Therefore, the difference between the EP at the AQL in figure 77 and 100 percent 

can be thought of as a long-term payment risk for the contractor. 

Since it has been shown that the sample PWL is an unbiased estimator of the true PWL of the 

population, the EP curve should exactly follow the shape of the payment equation. Indeed, that is 

what happened with the simulations that were done using this same payment equation in 

chapter 5. Why, then, does the EP at the AQL not equal the value of 100 that is given by the 

equation? When the EP does not equal the value stated by the payment equation, it usually 

means that some payment barrier has interfered with the ability of the EP to average out to the 

value in the payment equation. 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Probability of Acceptance

PF ≥ 100%

PF ≥ 0

PF ≥ 104.5%

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Probability of Acceptance

PF ≥ 100%

PF ≥ 0

PF ≥ 104.5%
 205 

Figure 77. EP curve for the payment relationship Pay = 55 + 0.5PWL, 

 with an RQL provision, sample size = 4. 

For the plot in figure 77, the reason that the EP did not equal 100 at the AQL can be attributed to 

the provision requiring zero payment for estimated PWL values of less than 60. If the payment 

equation were allowed to apply throughout its total range, then the EP would have been able to 

reach 100 at the AQL. The reason that this did not happen can be shown by looking at the 

distribution of the estimated PWL values and their corresponding payment factors (shown in the 

histograms in figure 78). The histograms show that because of the variability of the small sample 

size, there were a number of times when the estimated PWL value for a lot was less than 60 even 

though the population PWL was 90. Similarly, there were many times when the estimated PWL 

was as high as 100. 

When the estimated PWL was less than 60, a payment factor of zero was used to represent the 

remove-and-replace provision. This resulted in a number of zero payments for lots that are 

shown at the far left side of the payment histogram. Had these PWL values not been assigned as 

zero, the EP value could have been 100 at the AQL. To show this, the EP curve was calculated 

again without the remove-and-replace provision. This EP curve is shown in figure 79. The EP at 

the AQL is 100 percent and the EP curve exactly follows the payment equation. 

0

10

20

30

40

50

60

70

80

90

100

110

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Expectd Payment, %

0

10

20

30

40

50

60

70

80

90

100

110

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Expectd Payment, %
 206 

 Figure 78b. Distribution of payment factors for an AQL population.

Figure 79. EP curve for the payment relationship Pay = 55 + 0.5PWL, 

 sample size = 4. 

Figure 78a. Distribution of estimated PWL values for an AQL population.

0

10

20

30

40

50

60

70

80

90

100

110

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Expectd Payment, %

0

10

20

30

40

50

60

70

80

90

100

110

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Expectd Payment, %
 207 

The most common reason that the EP curve deviates from the payment equation is when there is 

no provision to allow for incentive, or bonus, payments. This, once again, creates a boundary at 

100-percent payment. With no incentive payments, with a population at an AQL of 90 PWL, 

there will be times when the estimated PWL will be less than 90 and times when it will be 

greater than 90 (see the histogram in figure 78). When the estimated PWL is less than 90, a price 

reduction will be applied. When the estimated PWL is greater than 90, the payment will still be 

100 percent if there is no incentive payment. There will be no payments greater than 100 percent 

to balance the payments less than 100 percent to allow the average payment (i.e., the EP) to be 

100 percent. 

Effect of Sample Size 

As is shown in chapter 5, one way to reduce the variability of the individual estimated PWL 

values is to increase the size of the sample that is taken. This reduces the variability about the EP 

value and, thus, reduces the risks by increasing the likelihood that the estimated PWL values will 

be close to the true value. 

The effect of sample size is not evaluated again in this chapter. However, the reduction in 

variability is shown very clearly in figures 9 through 12. 

Distribution of Estimated PWL Values 

Figure 79 clearly shows that even the maximum 5-percent bonus provided by equation 29 was 

sufficient to offset the times when the PWL for an AQL population is underestimated. This may 

not seem reasonable in light of the fact that equation 29 allows for only a 5-percent bonus, while 

allowing penalties that can theoretically reach as much as 45 percent. The reason that this can 

happen is related to the distribution of the estimated PWL values, particularly for the small 

sample sizes that are typically used in highway materials and construction. 

The reason for this discrepancy is shown in figure 75 in the OC curve for the probability of 

receiving at least 100-percent payment. The OC curve shows that there is about a 40-percent 

chance of receiving less than 100-percent payment, while there is about a 60-percent chance of 

receiving 100-percent or greater payment. The fact that the sample PWL is an unbiased estimator 

of the population PWL does not mean that there is an equal chance of a high or low estimate. 

The fact that the average of the sampling distribution for individual PWL estimates has a mean 

equal to the true population PWL is what makes the estimator unbiased. In other words, the 

distribution of the estimates does not need to be symmetrical for the estimator to be unbiased. 

It is the skewed distribution of sample PWL estimates that allows for a smaller bonus to offset 

larger penalties. The skewness of the distribution of the estimated PWL values and, hence, the 

estimated payment values can be shown with histograms. 

Figure 80 shows histograms of the distribution of sample PWL estimates for a population with 

actual PWL = 90 and sample size = 4 for one-sided and two-sided specification limits. The 

histograms show that the distribution is skewed to the right for both one- and two-sided 

specifications, and that about 60 percent of the values lie above the actual PWL of 90. 
 208 

Figure 81 shows similar histograms for a population with an actual PWL of 50. The two-sided 

specification exhibits the same skewness to the right, but with 60 percent below the actual PWL 

of 50. However, for the one-sided specification limit, the distribution appears to be symmetrical, 

with about 50 percent on either side of the actual PWL of 50. 

The reason that the two distributions are different lies in the fact that it is the standard deviation 

that causes the estimated PWL distribution to be skewed. The only way for a one-sided 

specification to have 50 PWL is if the population mean is centered on the specification limit. 

Any population that is centered on the one specification limit will have 50 PWL, regardless of 

the value of its standard deviation. 

To illustrate why it is the standard deviation that causes the skewness in PWL estimates, it is 

necessary to look at the sampling distributions for sample means and sample standard deviations. 

It is well known that for any sample size, the distribution of the sample means from a normal 

distribution will be normally distributed with the mean equal to the population mean. This 

indicates that the sample mean should lead to a symmetrical distribution of sample PWL values. 

However, it is also known that the sample variance follows a chi-square distribution. The 

chi-square distribution is skewed to the right, but the amount of skewness decreases as the 

sample size increases. Since the sample standard deviation is the square root of the sample 

variance, the distribution of the sample standard deviations will also be skewed. This is shown in 

figure 82, which shows the distribution of 1000 sample standard deviation values for sample 

sizes = 3, 5, and 10. Note that the spread of the values becomes less and the shape of the 

distribution approaches symmetry as the sample size increases. 

Appendix H presents a more detailed discussion regarding the distributions of sample PWL 

estimates and how these distributions vary depending on the population PWL value. 
 209 

Figure 80. Distributions of sample PWL estimates for a population with 90 PWL. 

0 0 0 0 0 0 0 0 2 3 8 15

30

48

67 74 62

80 74

537

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Estimated PWLs

Actual PWL = 90, n = 4

Two–Sided Spec

90 PWL
 210 

Figure 81a. Distributions of sample PWL estimates for a population with 50 PWL 

and one-sided speculations.

Figure 81b. Distributions of sample PWL estimates for a population with 50 PWL 

and two-sided speculations.

0 1 3 4

24

85

117

101

121

113

97

89

66

37

29 29

34

11

7

32

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Estimated PWLs

Actual PWL = 50, n = 4

Two–Sided Spec

30

7

15

23

29

40

73

66

92

114

103

115

79

57

43

32

22

17

13

30

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sample PWLs

Actual PWL = 50, n = 4

One–Sided Spec
 211 

Figure 82a. Distribution of sample standard deviations for a sample size, n = 3, 

based on 1000 simulated samples. 

Figure 82b. Distribution of sample standard deviations for a sample size, n = 5, 

based on 1000 simulated samples. 

Figure 82c. Distribution of sample standard deviations for a sample size, n = 10, 

based on 1000 simulated samples. 

σ n = 3

σ n = 5

n = 10

σ
 212 

CALCULATING RISKS FOR MULTIPLE QUALITY CHARACTERISTICS 

The procedures presented above for OC and EP curves are primarily for the case of acceptance 

based on a single property. When, as will often be the case, there are multiple acceptance 

properties, it will be necessary for the agency to develop sophisticated computer simulation 

methods to complete a full analysis of the risks. These analyses will be quite involved and will 

be dependent on the quality characteristics chosen for acceptance and whether or not a 

performance model for predicting service life has been adopted by the agency. Another factor 

that will impact the analysis is whether a composite quality measure has been developed, or 

whether the individual quality measures will in some way, perhaps by adding, multiplying, or 

averaging, be combined into a composite payment factor. The topics of performance models and 

composite quality measures are addressed in chapter 11. 

To illustrate the difficulty of trying to evaluate risks when there is more than one acceptance 

property, an example will be shown based on the use of two acceptance characteristics. The 

problem is simpler if it can be assumed that the characteristics are independent and, therefore, 

have a correlation coefficient of 0. If this might not be the case, then it must be determined 

whether or not it is necessary to develop separate EP contours or surfaces for each different 

possible set of correlation coefficients. 

Calculating EP Contours 

The same simulation routines that were presented in chapter 8 for evaluating two correlated 

quality characteristics can be used to develop a form of a two-variable EP curve. In chapter 8, it 

was shown that the EPs for two correlated variables did not change as the weighting factors or 

the value of their correlation coefficients varied. However, although the EPs did not vary, the 

standard deviations of the individual combined payment factors did vary with both the 

correlation coefficient and the weightings used when combining the individual payment factors. 

Changes in the weighting factors are not likely to be a problem since they are usually part of the 

acceptance plan and, therefore, remain constant. The agency would have to decide whether or 

not it wished to evaluate the variability of the combined payment factors, or whether it was 

sufficient that the EP values matched the payment equations (i.e., that they were unbiased). If it 

is decided to only consider the EP, then a method needs to be developed to present the EP values 

in a way that can be used to assess the payment risks involved. One way to do this is by using 

contours or surfaces to represent the EP values as the actual PWL values for both acceptance 

characteristics vary. If only EP is considered, then the two variable EP contours or surfaces 

should not change with changes in the correlation coefficients. 

To illustrate how this approach might be used to develop EP curves or surfaces, 1000 samples of 

size = 5 were simulated for correlation coefficients of +0.5 and –0.5, using several different 

methods for combining the individual payment factors into a combined payment factor. The 

simulations were conducted with actual PWL values ranging from 95 to 10 for both populations. 

The results of some of these simulation analyses are presented here for illustration. 
 213 

Table 52 shows the results of simulations where the combined payments were determined as the 

average of the two individual payments. The individual payment factors were calculated using 

equation 29. The results show that there is no difference in the EP values when the correlation 

coefficient is +0.75, 0.00, or –0.75. The correct values for any cell in the table are determined by 

inserting the actual PWL values for variables 1 and 2 into equation 29. The EP values in the table 

are very close to what they should be to match the payment equation. 

To illustrate how the EP values for two variables could be shown graphically, figure 83 shows 

EP contours for the EP values shown in table 53. These EP values were determined by 

multiplying the two individual payments, which were each obtained from equation 29. The 

values were based on a correlation coefficient of +0.50. Figure 84 shows another way in which 

the EP values could be represented as a surface in a three-dimensional plot. Still another twodimensional approach for presenting the EP values for two variables is shown in figure 85. In 

this figure, the values in table 53 are plotted with one variable on the horizontal axis and the EP 

values on the vertical axis. The second variable is then plotted with a separate curve for each of a 

number of different PWL values. 

While the three-dimensional approach is theoretically a good way to visualize how the EP values 

vary with the population PWL values, the two-dimensional methods are probably a more 

practical way to present the information. None of these visualization approaches would work if 

the number of acceptance variables were three or greater. 

EFFECT OF ADDED PROVISIONS 

From the example above, it may appear that the evaluation of the risks can be complicated. The 

complexity becomes much greater when provisions in addition to a simple payment equation are 

added to the acceptance plan. In the past, a popular provision has been to state that even if the 

estimated PWL indicates a payment reduction, no price reduction will be applied if all of the 

individual test results are within the specification limits. This would not only have the effect of 

raising the EP values, it would also make it much more complicated to simulate the specification 

to establish the EP values. 

Another provision that might require analyses to fully understand the potential risks is a removeand-replace, or leave in at zero payment, provision. These are the most extreme penalties that can 

be imposed; thus, it is incumbent upon the specifying agency to fully evaluate the risks for the 

contractor before implementing such a provision. It is relatively easy to evaluate the risk that 

removal and replacement will be required when there is a single acceptance quality 

characteristic. It merely requires the same approach that was used to determine the OC curves in 

figures 74 through 76. This is illustrated with a simple example. 
 214 

Table 52. EP values using Pay = 55 + 0.5PWL for two individual payment factors 

and then averaging them, sample size = 5. 

PWL PWL Variable 2 

Variable 

1 10 30 50 60 70 80 90 95 

Correlation Coefficient = +0.75 

10 60.0 64.8 69.9 72.5 75.1 77.8 80.0 81.2 

30 64.8 70.0 75.0 77.3 80.0 82.6 85.1 86.3 

50 70.0 74.9 80.0 82.5 84.7 87.5 90.2 91.3 

60 72.5 77.7 82.2 85.0 87.7 89.9 92.5 93.5 

70 75.3 80.0 84.8 87.4 90.5 92.8 95.2 96.2 

80 77.4 82.6 87.4 90.5 92.3 95.0 97.4 98.7 

90 80.1 85.1 90.0 92.6 95.2 97.5 100.1 101.2 

95 81.2 86.3 91.5 94.0 96.2 98.9 101.2 102.5 

Correlation Coefficient = 0.00

10 59.9 64.8 70.2 72.5 74.9 77.4 79.7 81.2 

30 65.0 70.0 75.3 77.3 80.1 82.3 85.2 85.9 

50 69.9 75.0 79.8 82.6 85.1 87.3 89.9 91.4 

60 72.8 77.3 82.6 85.2 87.8 89.9 92.6 93.7 

70 74.9 80.2 85.5 87.4 90.1 92.3 95.1 96.4 

80 77.6 82.7 87.4 89.8 92.6 95.1 97.4 98.8 

90 80.0 84.8 89.6 92.6 94.9 97.7 99.7 101.2 

95 81.3 86.0 91.3 93.6 96.4 98.9 101.1 102.5 

Correlation Coefficient = –0.75

10 59.9 65.0 70.2 72.6 74.9 77.5 80.0 81.1 

30 64.9 69.7 75.0 77.6 79.9 82.4 85.0 86.0 

50 69.8 75.0 79.9 82.6 85.1 87.6 90.3 91.1 

60 72.5 77.4 82.5 84.9 87.6 89.8 92.5 94.0 

70 74.9 80.2 84.9 87.6 90.0 92.4 94.7 96.2 

80 77.5 82.6 87.5 90.2 92.8 95.0 97.6 98.8 

90 79.9 84.9 90.2 92.5 94.8 97.4 100.1 101.2 

95 81.2 86.2 91.3 94.0 96.4 98.9 101.0 102.6 
 215 

Table 53. EP values using Pay = 55 + 0.5PWL for two individual payment factors 

and then multiplying them, sample size = 5, correlation coefficient = +0.5. 

PWL PWL Variable 2 

Variable 

1 10 30 50 60 70 80 90 95 

10 35.9 42.1 48.2 50.6 54.2 56.8 60.2 61.3 

30 42.1 49.1 56.5 59.8 62.8 66.4 69.8 71.4 

50 48.9 56.7 64.7 68.6 72.7 76.6 80.6 81.6 

60 51.2 59.5 68.4 72.9 77.3 80.7 85.2 87.0 

70 54.4 63.2 72.4 77.3 81.5 85.8 90.0 92.3 

80 57.5 66.8 75.7 80.9 86.0 90.9 95.2 97.5 

90 59.9 70.1 80.5 84.9 90.6 95.1 99.7 102.0 

95 61.4 72.2 82.2 86.7 92.7 97.6 102.9 105.0 

Figure 83. EP contours for the values in table 53. 

PWL for Variable 1

PWL for Variable 2
 216 

Figure 84. EP surface for the values in table 53. 

PWL for Variable 1 PWL for Variable 2

EP
 217 

Figure 85. EP curves for the values in table 53. 

Suppose that an acceptance plan calls for a sample size = 5 and requires that the material be 

removed and replaced if the estimated PWL value is less than 60. The OC curve for the 

probability that removal and replacement will be required is determined by the probability that a 

population with any given PWL value will yield a sample PWL of less than 60. The OC curve 

(the probability of acceptance) associated with the remove-and-replace decision is shown in 

figure 86. This OC curve represents the probability that the material will NOT require removal 

and replacement. The probability that removal and replacement would be required would be 1.00 

minus the value indicated on the vertical axis. 

Since small sample sizes, usually n = 3-5, are often involved in the acceptance decision, there is 

a large amount of variability in the estimated PWL value for any given population. Table 54 

shows the probability that populations of various quality levels would have an estimated PWL of 

less than 60 for a single quality characteristic and would thus require removal and replacement. 

15

25

35

45

55

65

75

85

95

105

0 10 20 30 40 50 60 70 80 90 100

PWL for Variable 1

Expected Payment, %

10

30

50

70

90

60

80

95

15

25

35

45

55

65

75

85

95

105

0 10 20 30 40 50 60 70 80 90 100

PWL for Variable 1

Expected Payment, %

10

30

50

70

90

60

80

95
 218 

Figure 86. OC curve for an acceptance plan that calls for rejection if the estimated PWL 

 is less than 60, sample size = 5. 

Suppose that the agency had selected 90 PWL as the AQL for the acceptance plan. Table 54 

shows that when the sample size = 3 for an AQL population (i.e., 90 PWL), each single quality 

characteristic has more than a 5-percent chance of yielding a sample result that indicates that 

removal and replacement are required. For sample size = 5, this probability drops to about 

1 percent. 

To further complicate the situation, many acceptance plans call for up to four or more acceptance 

characteristics. If the same remove-and-replace provision is applied if any of the acceptance 

characteristics has an estimated PWL of less than 60, then the risks are considerably greater. 

Assume that our example acceptance plan has four acceptance characteristics and requires 

removal and replacement if any of them have estimated PWL values of less than 60. If we 

assume that the four characteristics are independent, table 54 shows the probability that at least 

one of the four characteristics will trigger the remove-and-replace provision. 

For a sample size = 3, there is about a 20-percent chance that an AQL population will require 

removal and replacement. Even for a sample size = 5, there is more than a 4-percent chance of 

the need for removal and replacement. 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60 70 80 90 100

Actual PWL

Probability of Acceptance
 219 

Table 54. Probability that populations with various quality levels would require 

 removal and replacement for the example in figure 86. 

With One Quality Characteristic With Four Independent 

Quality Characteristics Population 

PWL 

n = 3 n = 4 n = 5 n = 3 n = 4 n = 5 

100 0.000 0.000 0.000 0.000 0.000 0.000 

 95 0.014 0.003 0.001 0.055 0.012 0.004 

 90 0.054 0.027 0.011 0.199 0.104 0.043 

 85 0.121 0.068 0.044 0.403 0.245 0.165 

 80 0.199 0.146 0.106 0.588 0.468 0.361 

 75 0.285 0.240 0.197 0.739 0.666 0.584 

 70 0.366 0.339 0.302 0.838 0.809 0.763 

 65 0.471 0.449 0.423 0.922 0.908 0.889 

 60 0.563 0.555 0.549 0.964 0.961 0.959 

 55 0.631 0.651 0.670 0.981 0.985 0.988 

 50 0.710 0.735 0.758 0.993 0.995 0.997 

 45 0.790 0.822 0.854 0.998 0.999 1.000 

 40 0.841 0.887 0.906 0.999 1.000 1.000 

 35 0.894 0.927 0.944 1.000 1.000 1.000 

 30 0.938 0.961 0.971 1.000 1.000 1.000 

 25 0.958 0.979 0.988 1.000 1.000 1.000 

 20 0.981 0.993 1.000 1.000 1.000 1.000 

 15 1.000 1.000 1.000 1.000 1.000 1.000 

 10 1.000 1.000 1.000 1.000 1.000 1.000 

 5 1.000 1.000 1.000 0.000 1.000 1.000 

 0 1.000 1.000 1.000 0.055 1.000 1.000 
 220 

CLOSING COMMENTS ON RISK 

OC and EP curves describe the operation of an acceptance plan such that the risks can be 

evaluated throughout the entire quality regime. If the risks are considered to be acceptable, no 

modifications to the initial acceptance plan are necessary. However, if the risks are considered to 

be unacceptable in terms of being too high for either or both parties, a reassessment of the 

acceptance plan is necessary. 

There is no easy answer to the question “Are the risks acceptable?” The decision regarding what 

does or does not constitute an acceptable level of risk will, to a great extent, be a subjective one. 

There is, however, one factor that is not subjective. There is generally universal agreement that 

the EP should be 100 percent for quality that is at exactly the AQL. Although it should not be 

confused with the statistical risk (α), the agency may wish to consider the average payment risk 

to the contractor if the EP is less than 100 percent at the AQL, or to the agency if the EP is 

greater than 100 percent at the AQL. The EP at the RQL is another point that is often specifically 

considered. 

It must be remembered that the EP alone is not a complete measure, particularly the likelihood 

that any individual lot will receive a correct payment factor. The variability of the individual 

payment factors about the EP curve must also be considered. Ultimately, the decision regarding 

what constitutes acceptable or unacceptable risks rests with the individual agency. 
 221 

10. THE COMPOSITE PAYMENT FACTOR 

TRADITIONAL COMPOSITE PAYMENT FACTORS 

Most specifications employ multiple quality characteristics for acceptance and payment. 

Therefore, one issue that must be addressed is how to combine these multiple characteristics to 

come up with a single payment factor for a lot. In chapter 8, a quality measure (PWL) was 

selected for use in an equation to determine payment for a single quality characteristic. A number 

of different methods for determining composite payment factors are currently in use. These can 

be referred to as traditional methods for determining a composite payment factor. 

Through the years, various agencies have considered at least five different approaches for 

combining a number of payment factors for individual acceptance quality characteristics into a 

single composite payment factor. These approaches include: 

• Averaging (possibly with weighting factors) the individual payment factors. 

• Multiplying the individual payment factors. 

• Summing the individual payment adjustments. 

• Using the maximum individual payment factor. 

• Using the minimum individual payment factor. 

The approach of using the maximum individual payment factor could be thought of as allowing 

the contractor the benefit of the doubt. The approach using the minimum individual payment 

factor as the composite payment factor is based on the weak link theory (i.e., the lowest payment 

factor indicates the value for all of the quality characteristics). For the other three approaches, the 

concept is that all of the individual factors contribute to the total. However, the composite 

payment for the three approaches can be quite different depending on the value of the individual 

payment factors. All of these approaches for determining a composite payment factor were 

analyzed by computer simulation. 

ANALYSES PROCEDURES 

To simplify the simulation process and to allow for graphical presentation of the results, the 

analyses of the different methods for determining a composite payment factor were conducted 

only for the case where there are two individual, but possibly correlated, payment characteristics 

and where PWL was used for the quality measure. Only the case where the two individual 

quality characteristic populations have the same PWL value is presented here. All analyses were 

conducted using a sample size = 5, although similar trends would be expected for other sample 

sizes; however, the variability would be greater for smaller sample sizes and less for larger 

sample sizes. 

The methods considered for combining the individual payment factors were: 

• Averaging the individual payment factors ((0.5 × P1) + (0.5 × P2)). 

• Using a weighted average of the individual payment factors ((0.75 × P1) + (0.25 × P2)). 
 222 

• Multiplying the individual payment factors (P1 × P2). 

• Summing the individual payment adjustments (P1 + P2 – 100). 

• Using the maximum individual payment factor (Max(P1, P2)). 

• Using the minimum individual payment factor (Min(P1, P2)). 

Since the two simulated populations had equal PWL values, it was anticipated that the average 

and weighted average methods would have similar results for the EP and would exhibit similar 

trends for standard deviation, although the magnitudes of the standard deviation values would be 

different because of the different weighting factor multipliers used. 

Each of the above payment calculation methods was simulated for three different equal 

population PWL values (90, 70, and 50). Each of the PWL values was simulated with using 

correlation coefficients ranging from –1.0 to +1.0. The results of these analyses are presented 

below. 

ANALYSES RESULTS 

The results of the analyses are shown in figures 87 through 95. Figures 87 through 92 show the 

expected combined payment and standard deviation of the individual combined payments for 

correlation coefficients = –1.00, –0.90, –0.75, –0.50, –0.25, –0.10, 0.00, +0.10, +0.25, +0.50, 

+0.75, +0.90, and +1.00 for each of the six methods identified above. The simulation results for 

three different PWL values (90, 70, and 50) are included in each figure. 

Figures 87 through 90 show similar trends regarding EP and standard deviation values for the 

simulated combined payments using the averaging, weighted average, multiplying, and summing 

methods. These figures show that the level of correlation between the two acceptance variables 

does not affect the EP values for these four methods. These figures also show that there is a 

similar effect on the standard deviation of the payment values for both different PWL values and 

for different levels of correlation. For the 50 and 70 PWL values, there is a consistently 

increasing trend in standard deviation as the correlation coefficient goes from –1.0 to +1.0. 

However, for the 90 PWL value, the standard deviations decrease slightly as the correlation 

coefficients approach 0.0 from both the positive and negative directions. 

The reason for these different trends in the standard deviation results rests in the fact that the 

upper limit of 100 PWL (you cannot have more than 100 PWL) is reached much more often for a 

population of 90 PWL. This boundary has the effect of limiting the spread of the individual 

payment values, thereby reducing the standard deviation values. The discussions in chapter 8 that 

explained why the standard deviation was smaller for negative correlations and larger for 

positive correlations explain this similar trend in figures 87 through 90. 
 223 

Figure 87a. Simulation results of expected payment for the averaging method, combining 

two populations with equal PWL values.

Figure 87b. Simulation results of standard deviation values for the 

averaging method, combining two populations with equal PWL values.

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Avg of Pi Values, n = 5

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Avg of Pi Values, n = 5

0

1

2

3

4

5

6

7

8

9

10

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Avg of Pi Values, n = 5

0

1

2

3

4

5

6

7

8

9

10

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Avg of Pi Values, n = 5
 224 

Figure 88a. Simulation results of expected payment for the weighted average method, 

combining two populations with equal PWL values. 

Figure 88b. Simulation results of standard deviation values for the weighted average 

method, combining two populations with equal PWL values. 

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = .75P1 + .25P2, n = 5

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = .75P1 + .25P2, n = 5

0

1

2

3

4

5

6

7

8

9

10

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = .75P1 + .25P2, n = 5

0

1

2

3

4

5

6

7

8

9

10

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = .75P1 + .25P2, n = 5
 225 

Figure 89a. Simulation results of expected payment for the multiplication method, 

combing two populations with equal PWL values. 

Figure 89b. Simulation results of standard deviation values for the multiplication method, 

combining two populations with equal PWL values. 

60

65

70

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = P1 × P2, n = 5

60

65

70

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = P1 × P2, n = 5

0

2

4

6

8

10

12

14

16

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = P1 × P2, n = 5

0

2

4

6

8

10

12

14

16

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = P1 × P2, n = 5
 226 

Figure 90a. Simulation results of expected payment for the summation method, combining 

two populations with equal PWL values. 

Figure 90b. Simulation results of standard deviation values for the summation method, 

combining two populations with equal PWL values. 

55

60

65

70

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

PF = Sum of Individual Pay Adjustments, n = 5

55

60

65

70

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

PF = Sum of Individual Pay Adjustments, n = 5

0

2

4

6

8

10

12

14

16

18

20

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

PF = Sum of Individual Pay Adjustments, n = 5

0

2

4

6

8

10

12

14

16

18

20

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

PF = Sum of Individual Pay Adjustments, n = 5
 227 

91a. Simulation results of expected payment for the maximum method, combining two 

populations with equal PWL values. 

Figure 91b. Simulation results of standard deviation values for the maximum method, 

combining two populations with equal PWL values. 

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Maximum Pi, n = 5

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Maximum Pi, n = 5

0

2

4

6

8

10

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Maximum Pi, n = 5

0

2

4

6

8

10

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Maximum Pi, n = 5
 228 

Figure 92a. Simulation results of expected payment for the minimum method, combining 

two populations with equal PWL values.

Figure 92b. Simulation results of standard deviation values for the minimum method, 

combining two populations with equal PWL values. 

70

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Minimum Pi, n = 5

70

75

80

85

90

95

100

105

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Minimum Pi, n = 5

0

2

4

6

8

10

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Minimum Pi, n = 5

0

2

4

6

8

10

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

90 PWL 70 PWL 50 PWL

Pi = 55 + .5 PWL, PF = Minimum Pi, n = 5
 229 

Somewhat different trends are shown in figures 91 and 92, which show the results for the 

maximum and minimum methods, respectively, for determining the combined payment factor. 

For these methods, for populations of 70 and 50 PWL, the EP values vary with the level of 

correlation between the two acceptance variables. This is not a desirable situation. In the first 

four methods considered, the EP for each level of quality, as indicated by the PWL, was 

independent of the correlation coefficient. For these methods, the variability was affected by the 

correlation value, but the EP was centered on the correct value. 

In figure 91, not only does the variability increase as the correlation goes from –1.0 to +1.0, but 

the EP values for populations of 70 and 50 PWL decrease over this same range. In figure 92, the 

EP values for populations of 70 and 50 PWL increase as the correlation goes from –1.0 to +1.0. 

For the maximum and minimum methods, the variability increases and the increasing or 

decreasing EPs are related to one another. If the average value remains constant, but variability 

increases, then there is more spread about the average. Since it is the single largest value that 

establishes the payment for the maximum method, the increased variability means that the 

maximum values will, on average, be higher. Similarly, for the minimum method, the increased 

variability also means that the minimum values will be lower. 

Comparing the Methods 

Figures 93 through 95 show the same results as figures 87 through 92, but plotted by the actual 

PWL values rather than the payment combination method. In these figures, it is clear that the EP 

values for the averaging, weighted average, multiplying, and summing methods do not vary with 

the correlation between the acceptance variables, and that the maximum and minimum methods 

do vary. 

When the EP values vary with the correlation coefficient, it means that the average payment 

values will be different in the long run for different correlation relationships between the two 

acceptance variables. When the EP values do not vary with the correlation coefficient, but the 

standard deviation does vary, this means that even though the average payment values in the long 

run will be the same, there will be a difference in the spread or range of payment values that may 

be estimated for individual lots. 
 230 

Figure 93a. Comparison of simulation results for various methods for combining 

individual expected payment factors for two populations with PWL = 90. 

Figure 93b. Comparison of simulation results for various methods for combining 

individual standard deviation payment factors for two populations with PWL = 90. 

94

96

98

100

102

104

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

Avg Wt

Mult Sum

Max Min Pi = 55 + .5 PWL, n = 5, Equal PWLs = 90

94

96

98

100

102

104

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

Avg Wt

Mult Sum

Max Min Pi = 55 + .5 PWL, n = 5, Equal PWLs = 90

0

2

4

6

8

10

12

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

Avg Wt Mult Sum Max Min

Pi = 55 + .5 PWL, n = 5, Equal PWLs = 90

0

2

4

6

8

10

12

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

Avg Wt Mult Sum Max Min

Pi = 55 + .5 PWL, n = 5, Equal PWLs = 90
 231 

Figure 94a. Comparison of simulation results for various methods for combining 

individual expected payment factors for two populations with PWL = 70. 

Figure 94b. Comparison of simulation results for various methods for combining 

individual standard deviation payment factors for two populations with PWL = 70. 

78

80

82

84

86

88

90

92

94

96

98

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

Avg Wt Mult Sum Max Min

Pi = 55 + .5 PWL, n = 5, Equal PWLs = 70

78

80

82

84

86

88

90

92

94

96

98

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

Avg Wt Mult Sum Max Min

Pi = 55 + .5 PWL, n = 5, Equal PWLs = 70

0

2

4

6

8

10

12

14

16

18

20

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

Avg Wt Mult Sum Max Min

Pi = 55 + .5 PWL, n = 5, Equal PWLs = 70

0

2

4

6

8

10

12

14

16

18

20

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

Avg Wt Mult Sum Max Min

Pi = 55 + .5 PWL, n = 5, Equal PWLs = 70
 232 

Figure 95a. Comparison of simulation results for various methods for combining 

individual expected payment factors for two populations with PWL = 50.

Figure 95b. Comparison of simulation results for various methods for combining individual 

standard deviation payment factors for two populations with PWL = 50. 

55

60

65

70

75

80

85

90

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

Avg Wt Mult Sum Max Min

Pi = 55 + .5 PWL, n = 5, Equal PWLs = 50

55

60

65

70

75

80

85

90

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Expected Payment, %

Avg Wt Mult Sum Max Min

Pi = 55 + .5 PWL, n = 5, Equal PWLs = 50

0

2

4

6

8

10

12

14

16

18

20

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Correlation Coefficient

Standard Deviation, %Pay

Avg Wt Mult Sum Max Min
 233 

CONCLUSION: TRADITIONAL COMPOSITE QUALITY MEASURES 

The maximum and minimum methods essentially disregard the quality characteristics that do not 

give the maximum or minimum payment value, respectively. This seems to ignore useful 

information from the other quality characteristics. The averaging, multiplying, and summing 

methods for combining individual payment factors implicitly assume that each individual 

payment property is equally important. This is probably, but not always, a valid assumption. 

Weighted average composite payment factors are intuitively appealing since it is very likely that 

all payment quality characteristics do not have an identical impact on pavement performance. A 

drawback to this approach, however, is that there is no obvious methodology for determining the 

appropriate weightings. The weightings, therefore, are subjective in nature and can vary from 

agency to agency depending on the agency or individual preferences. 

Based on the analyses and discussion presented in this chapter, the weighted average method 

appears to be the best of the traditional methods for combining payment factors for individual 

quality characteristics into a single composite payment factor. However, as shown in the next 

chapter, there are other methods, based on predicted performance, for determining the final 

payment factor for a lot. The use of such methods is not widespread at this time. 
 
 235 

11. RELATING PAYMENT TO PERFORMANCE 

BACKGROUND 

Decisions concerning payment relationships are extremely important since experience has shown 

that payment relationships in an acceptance plan are the most important factor from a 

contractor’s perspective. The contractor submits a bid with a certain expectation of the amount of 

payment for the product. Achieving this amount of payment is critical in maintaining a viable 

business. 

By the 1980s, many agencies had included payment reductions as part of the acceptance plans in 

their QA specifications. The intent was to penalize contractors for material or construction that 

did not fully comply with the specification requirements, but which was not of sufficiently poor 

quality to justify removal and replacement. Many agencies have now realized the need to also 

include incentive provisions (i.e., bonuses) to reward contractors for superior quality. These 

incentives and disincentive payment schedules were often based on engineering judgment, and 

similar test results could yield very different payment factors from one agency to another. 

Relating quality and performance to payment is the most desirable form of payment relationship, 

because the relationship supports and defends the decision. This is true because negative 

payment adjustments are typically viewed with skepticism by the contracting industry. However, 

when the payment schedule can be shown to be related to quality and, preferably, to 

performance, it is viewed to be more credible than when it is established arbitrarily. 

Two different approaches for developing performance models were considered during the 

current project. The first approach, which has been developed over many years through FHWA 

and National Cooperative Highway Research Program (NCHRP) support, is what has been 

called performance-related specifications (PRS). These efforts have led to the development of 

PRS guide specifications and to computer programs that use theoretical and/or empirical models 

to predict pavement performance based on test results from the newly placed pavement. The 

software program PaveSpec, now in version 3.0, has been developed for PCC pavements and has 

been available to the general highway industry for a number of years. More recently, HMASpec 

has been developed for HMAC pavements. 

The second approach that was considered is based on concepts similar to those of the PRS 

mentioned above. However, this approach is more general in that it can be applied to any set of 

quality characteristics for which a general relationship to pavement life can be developed or 

approximated. The second approach also considers only initial construction costs and major 

reconstruction or rehabilitation activities, and does not try to include routine or localized 

maintenance and repair costs into the payment determination. 

Each of these is discussed briefly in the following sections. Much of the material on PRS is taken 

from unpublished manuscripts prepared by P.A. Kopac, T.M. Mitchell, and T.P. Teng of FHWA. 
 236 

PERFORMANCE-RELATED SPECIFICATIONS (PRS) 

An extensive research effort has been conducted to develop what are generally known as PRS. 

PRS attempt to use acceptance and payment plans that incorporate payment provisions based on 

pavement performance models. Current PRS are based on mathematical models that attempt to 

quantify pavement performance relationships among a number of important quality 

characteristics. The intent of these models is to provide a clearer understanding and better 

estimate of a pavement’s performance than can be obtained by even the best intuitive 

engineering judgment. These performance models are then used to determine the payment 

adjustments in the acceptance plan. 

The most elaborate PRS incorporate models to predict both pavement performance and 

maintenance costs. The performance prediction models use the pavement’s design and materials 

information to predict when and to what extent the pavement will exhibit various types of 

distress, such as fatigue cracking or joint spalling. The maintenance cost models use predicted 

stress development to estimate post-construction life-cycle costs (LCC), which include the 

predicted costs of maintenance and rehabilitation that will be encountered throughout the 

projected life of the pavement. 

As shown in figure 96, the inputs to the performance model include design variables such as 

traffic loading, climatic factors, drainage, and roadbed soil factors, and materials quality 

characteristics such as asphalt content, concrete strength, and pavement smoothness. These, in 

turn, are converted by the model into predicted occurrence and the extent of the various types of 

distress. This then becomes the input to the maintenance cost models whose output is ultimately 

a predicted LCC for the pavement. 

Figure 96. Flowchart of the PRS process (after Kopac, Mitchell, and Teng). 

Performance Prediction 

Models

Maintenance Cost 

Models

Inputs:

design variables

construction variables

Predicted distress 

occurrence and extent

Output:

life–cycle cost
 237 

The target values for the quality characteristics are input to the models to determine the asdesigned LCC. The actual measured values for the pavement’s quality characteristics are then 

used as input to produce the estimated as-constructed LCC. The difference between the asdesigned and as-constructed LCC values is then used as the basis for any incentive or 

disincentive payment. 

Computer simulation routines are used with PRS to consider variability when determining 

payment adjustments. Mean and standard deviation values for the as-designed quality 

characteristics are entered into the simulation program along with other pavement design 

information. The values for as-designed mean and standard deviation can be determined from the 

design or based on agency policy or past experience. Hundreds, or preferably thousands, of 

iterations of the model are then performed. 

For each iteration, the simulation routine selects from a normal distribution, defined by the asdesigned mean and standard deviation values, for each designated quality characteristic. These 

simulated values are then input into the PRS models to arrive at an estimate for the asconstructed LCC for each iteration. The average of the many simulated LCC estimates is then 

used as the as-designed LCC. 

The same process is then used with the mean and standard deviation from the acceptance sample 

test results to develop an estimated as-constructed LCC for each constructed lot. Payment factors 

are then based on the following equation: 

[ ( )]

Bid

Bid LCC LCC PF + des − con = 100 (30) 

where: PF = payment factor as a percentage of the bid price 

Bid = contractor’s bid price 

LCCdes = as-designed LCC 

LCCcon = as-constructed LCC 

Up to now, most PRS have been developed for individual projects. In that way, project-specific 

values for inputs, such as design, traffic, climate, etc., can be used. This project-by-project 

approach obviously provides the best estimate for pavement performance. An alternative would 

be to develop generic PRS for similar types or groups of projects, such as those within a given 

geographic or climatic region, or those with similar average daily traffic. While this would 

eliminate the need for developing a different PRS for each project, the predicted performance 

would not be as good an estimate as would a project-specific PRS. 

To fully benefit from PRS, the performance models must be accurate and properly applied. The 

current models were developed based on generic national norms. Each agency that wishes to use 

the current PRS models must decide whether or not its local conditions deviate sufficiently from 

those norms to require revised performance models for its pavements. 

PRS offer great promise to radically change the way in which payment plans are developed and 

payment factors are determined. However, they are, to a great extent, still in the pilot project 
 238 

stage and have not yet been widely adopted or employed by highway agencies. There are some 

potential roadblocks to their widespread adoption and implementation. One of these is that they 

can be perceived as being much more complex than any traditional specification with which an 

agency is familiar. This perception may be justified in light of the fact that 126 different inputs 

are possible when using PaveSpec 3.0. This may lead some agencies to be slow to adopt PRS. 

While these perceptions of complexity may be overcome, another serious potential drawback is 

that PRS may require a significant amount of additional testing compared to current QA 

specifications. Regardless of how sophisticated the internal mathematical or empirical models in 

a PRS process are, the limiting factor for the accuracy and validity of the LCC prediction will 

still be the quality of the as-constructed information that is input. No matter how many iterations 

are simulated within a PRS model, the predictions will not be accurate if the as-constructed mean 

and standard deviation are not correct. 

The large variability in the estimates, based on typical sample sizes = 3–5, has been shown in 

numerous places in preceding chapters. No matter how sophisticated and precise the models 

within the PRS, the limiting factor will be the size of the sample that the agency will take to 

determine the as-constructed means and standard deviations. If, because of personnel shortages, 

highway agencies are passing acceptance testing responsibilities to the contractors, it seems 

unlikely that agencies will be willing to increase the amount of acceptance testing that is 

performed. 

Decision Regarding PRS 

For a number of reasons, it was decided that this is not the right time to recommend PRS in a QA 

specifications development manual that is recommended for current use by all highway agencies 

around the country. Some of these reasons include: 

• There is a wide array of input data that must be collected and maintained for input to the 

PRS. 

• There is a possible perception that the PRS is a black box over which the agency has no 

control. 

• Many agencies may not currently have the data or expertise to revise the generic models 

to fit the specific needs of their agencies. 

• It seems unlikely that a large number of agencies would be willing to increase the amount 

of acceptance testing at this time. 

It is believed that PRS are probably the specifications of the future (whether this is the immediate 

future or sometime later is still open for debate). While the promise of PRS was recognized, it 

was decided that a simpler approach for incorporating LCCs into payment determinations would 

be best for inclusion in the QA manual under development. The approach that was selected for 

the QA manual is presented in the next section. 

LCC BASIS FOR PAYMENT ADJUSTMENTS 

The concepts incorporated into the PRS discussed above are sound. The method that was 

selected for the QA manual is based on similar LCC concepts, but is easier to implement and 
 239 

should be easier to modify for a specific agency than is the PRS approach described above. The 

method considers only major repairs (such as resurfacing) and does not attempt to include 

routine or localized repair costs when calculating the LCC. The general approach can be applied 

to any agency, provided that there is some model or method for predicting pavement life before 

an overlay is required. Any existing model, including sophisticated PRS models, can be used to 

develop the predicted lives. 

LCC Approach 

Ordinarily, a pavement is designed to sustain a specified number of load applications before 

major repairs (such as resurfacing) are required. If, because of construction deficiencies, the 

pavement is not capable of withstanding the design load, it will fail prematurely. The need to 

repair this pavement at an earlier date results in an additional expense that must be borne by the 

agency since it usually occurs long after any contractual obligations have expired. Therefore, one 

possible purpose for an adjusted payment schedule might be to withhold sufficient payment at 

the time of construction to cover the extra costs anticipated in the future as a result of work that 

is of deficient quality. 

Pavements are usually designed to withstand a required number of equivalent single-axle loads 

(ESALs). For those quality characteristics used in the design procedure, the as-built values can 

be compared to the design values to estimate the fraction of the design load that the pavement is 

capable of sustaining. As an approximate estimate, this fraction can be multiplied by the design 

life to obtain the expected life of the pavement. If greater precision is desired, a traffic growth 

rate can be assumed, the effect of which is to extend slightly the expected life (since fewer of the 

allowable loads will occur in the early part of a pavement’s life). 

To estimate the cost to the agency of premature pavement failure, it is necessary to determine the 

net present value of the various actions made necessary by early failure. For example, suppose 

that experience has shown that a new pavement typically lasts about 20 years and an overlay 

about 10 years. If the initial surface were to fail 1 or 2 years prematurely, it is not likely that an 

agency would do a minor repair to extend the life of the pavement to the originally expected 

value of 20 years. A much more practical decision would be to reschedule the overlay that was 

planned for the 20th year and do it 1 or 2 years sooner. However, if the overlay that was planned 

for the 20th year is rescheduled to an earlier date and overlays typically last 10 years, then all 

future overlays must be moved to an earlier time as well. 

The procedure involves the calculation of a series of debits and credits and turns out to be 

relatively easy. Moving the overlay that was planned for the 20th year to the 18th year, for 

example, would result in a debit in net present value terms because it represents a cost in the 18th

year that was not planned. However, there will also be a credit for no longer having to do an 

overlay in the 20th year. Since the overlay that was planned for the 20th year is now later in the 

future, the credit for this action is discounted to a greater degree, resulting in a net debit for the 

rescheduling of the overlay that was planned for the 20th year. This is illustrated in figure 97. 
 240 

Figure 97. Illustration of the net impact of rescheduling an overlay 2 years earlier 

than originally planned. 

While it is true that the net debits from the rescheduling of overlays farther in the future are 

discounted to a greater extent, and soon become insignificant, ignoring them altogether would 

substantially underestimate the true cost of pavement failure. Alternatively, selecting a specific 

analysis period would require an assumption about the residual value of a partially depleted 

overlay (information that is not readily available). Fortunately, this is an easy problem to solve 

mathematically to yield equation 31, the derivation of which is presented in the QA manual:(1)

(1 )

( )

O

D E

R

C R R PAYADJ −− = (31) 

where: PAYADJ = appropriate payment adjustment for new pavement or overlay 

 (same units of cost as C) 

 C = present total cost of resurfacing 

 D = design life of pavement or initial overlay 

 E = expected life of pavement or overlay (independent variable) 

 O = expected life of successive overlays 

 R = (1 + INF)/(1 + INT) 

 INF = long-term annual inflation rate in decimal form 

 INT = long-term annual interest rate in decimal form 

Performance Relationships 

To apply the LCC basis for payment schedules in a manner that is both fair to all parties and 

legally defensible, it is necessary to have at least an approximate performance relationship. The 

purpose of the performance relationship is to predict from the quality characteristics measured at 

the jobsite what the expected service life of the construction item will be. This is the independent 

variable to be entered into the LCC equation presented in the previous section. 

20

Net Loss

0 18

Planned overlay

Premature overlay

20

Net Loss

0 18

Planned overlay

Premature overlay
 241 

As noted in previous sections, efforts are proceeding on the development and implementation of 

extremely sophisticated computerized procedures to develop performance relationships and 

appropriate payment schedules. However, the successful completion, validation, and widespread 

adoption of these procedures are still some time away. Even when completed, the data 

requirements and the level of complexity of these procedures may deter their widespread use by 

practitioners seeking more practical methods that are easier to understand and apply. 

Therefore, there is a need for an alternative approach for those agencies that choose to develop 

their own procedures, in their own way, that are tailored to their own specific circumstances. 

Perhaps more importantly, this alternative method needs to be sufficiently straightforward and 

scientifically sound so that agency engineers could not only understand it and use it with 

confidence, but could also modify it when necessary and be able to present it convincingly to the 

contractors whose work it will govern. 

The derivation of such an alternative approach for developing simplified performance models is 

presented in depth in the QA manual along with a number of examples.(1) This derivation 

resulted in equation 32, which is a simple exponential model that is applicable to a wide variety 

of quality characteristics: 

 

 ( PD PD PD ) B1 1 B2 2 Bk k EXPLIF Ae− + + + = L (32) 

where: EXPLIF = expected life, in years 

A = constant to be determined 

 Bk = coefficients to be determined for each of the k quality characteristics 

 PDi = percent defective of individual quality characteristics 

k = number of quality characteristics 

e = base of natural logarithms 

This model has certain important advantages. It tends to produce an “S” shape that is believed to 

be an appropriate form for many performance relationships. Also, because this particular model 

produces a maximum of A and a minimum as close to zero as desired (but not below zero), it can 

easily be made to fit most real-world situations. Finally, it requires relatively straightforward 

data and simple mathematics to accommodate as many acceptance characteristics as are likely to 

be necessary. 

For example, consider a resurfacing project for which historical data have shown the typical 

expected life to be about 10 years. A typical value for the AQL is PD = 10, while RQL values 

tend to vary more widely, depending on what quality level the agency believes justifies removal 

and replacement at the contractor’s expense. For the purposes of this example, suppose that the 

agency has decided to use RQL values of PD = 65, 75, and 85 because it is believed that these 

levels correspond to approximately a 50-percent loss in pavement life (or an expected life of 5 

years). These assumptions lead to the completed data matrix shown in table 55. 
 242 

Table 55. Completed data matrix for the example of an exponential model. 

PDVOIDS PDTHICK PDSMOOTH EXPLIF (years) 

10 (AQL) 10 (AQL) 10 (AQL) 10 (AQL) 

65 (RQL) 10 (AQL) 10 (AQL) 5 (poor voids) 

10 (AQL) 75 (RQL) 10 (AQL) 5 (poor thickness) 

10 (AQL) 10 (AQL) 85 (RQL) 5 (poor smoothness) 

All that remains is to use the information in the data matrix to solve for the unknown coefficients 

in equation 32 (the exponential performance equation for EXPLIF). To accomplish this, it is first 

necessary to take logarithms of both sides, producing equation 33: 

 

 ln(EXPLIF) = ln(A) – B1PDVOIDS – B2PDTHICK – B3PDSMOOTH (33) 

 

where: EXPLIF = expected life, in years 

PDVOIDS = air voids percent defective 

PDTHICK = thickness percent defective 

PDSMOOTH = smoothness percent defective 

A, B1, B2, B3 = unknown coefficients 

ln = natural logarithm operator 

The values from each row in table 55 can be inserted into equation 33 to develop four equations 

that can be used to solve for the four unknown coefficients, leading to the following performance 

model: 

(0.0126PD 0.0107PD 0.00924PD ) 13.8 VOIDS THICK SMOOTH EXPLIF e− + + = (34) 

This model would then be used to develop the expected-life value to be used in equation 31 to 

determine the payment adjustment, which could be either positive or negative depending on the 

expected-life estimate. 

The concepts used in the development of equation 34 have widespread application. The only 

requirement is that the agency creates a data table such as table 55. Any method with which the 

agency is comfortable can be used to develop the values for estimated pavement life that result 

from the various levels of the quality measure. For example, if a performance model is available 

and the highway agency has confidence in the predictive capability of the model, then it could be 

used to develop the estimated expected life of the pavement. 

Composite Quality Measure 

As noted in chapter 10, specifications based on multiple quality characteristics frequently use 

payment equations that include a separate term for each of the quality characteristics so that the 

resultant payment adjustment is a function of the combined effects of all of the quality measures. 

An alternate method to accomplish the same purpose is to base the payment equation on a single 
 243 

quality measure that is a composite of the individual quality measures. This latter approach, 

because it keys the various decisionmaking steps to a single performance indicator, simplifies the 

procedure and offers several practical advantages. 

The benefit of the composite quality measure is shown in a simple example that uses two 

acceptance quality characteristics—air voids and thickness. Suppose that the expected-life model 

in equation 35 has been developed for these two characteristics: 

 EXPLIF = 22.9 – 0.163 PDVOIDS – 0.135 PDTHICK + 0.000961 PDVOIDS × PDTHICK (35) 

Since the quality measure in equation 35 is PD, which ranges from a minimum of zero to a 

maximum of 100, it will be convenient to develop a composite quality measure (PD*) that spans 

that same range. As derived, the value of EXPLIF in equation 35 ranges from 2.71 to 22.9. By 

algebraic operations, equation 35 can be modified to span the range from 0 to 100, thereby 

yielding equation 36: 

 PD* = 0.807 PDVOIDS + 0.669 PDTHICK – 0.00476 PDVOIDS × PDTHICK (36) 

where: PD* = composite quality measure, in units of percent defective 

PDVOIDS = air voids percent defective 

PDTHICK = thickness percent defective 

PD* progresses smoothly from 0 to 100 percent as the individual quality measures (PDVOIDS and 

PDTHICK) vary throughout the same range. Table 56 presents a few selected examples of this: 

Table 56. Examples of computed PD* values for selected individual PD values. 

PDVOIDS PDTHICK EXPLIF (years) PD* 

 0 0 22.9 0.0 

 10 10 20.0 14.3 

 50 50 10.4 61.9 

 25 75 10.5 61.4 

 100 100 2.7 100.0 

As shown in table 56, the case in which PDVOIDS and PDTHICK are both equal to 50 produces 

essentially the same level of expected life as the case in which PDVOIDS = 25 and PDTHICK = 75. 

This result flows directly from the manner in which the EXPLIF equation was derived and is 

realistic because an increase in the quality of one measure might be expected to offset a decrease 

in the quality of the other. Appropriately, both cases produce virtually the same value of PD* in 

the last column of the table, indicating that PD* is well suited as a measure on which a QA 

specification can be based. 
 244 

This property of the composite quality measure, which properly accounts for the combined 

effects of multiple quality characteristics, also makes it possible to develop an RQL provision 

that is far superior to the alternative of defining separate RQL provisions for the individual 

quality measures. For the example in table 57, it is assumed that the agency has defined for air 

voids and thickness separate RQL provisions of PDVOIDS ≥ 75 and PDTHICK ≥ 90. Clearly, case 3 

in table 57 is, by far, the worst case, yet it is not recognized as an RQL condition when using 

individual RQL provisions, while the other two cases are considered as such. 

Table 57. Illustration of the problem with separate RQL provisions.

Quality Level Case 

Air Voids Thickness Reject? PD* 

1 PD = 75 (RQL) PD = 0 (excellent) Yes 60.5 

2 PD = 0 (excellent) PD = 90 (RQL) Yes 60.2 

3 PD = 74 (almost RQL) PD = 89 (almost RQL) No 87.9 

To demonstrate the effectiveness of an RQL provision based on the composite quality measure, 

equation 36 was used to compute the corresponding values for PD* that appear in the last 

column of table 57. In this example, a PD* of 60 or more would be regarded as rejectable and, as 

shown in the last column, case 3 is properly recognized as being well into the rejectable region. 

All of the topics covered in this section are covered in much greater detail and with more 

examples in the QA manual.(1) This information is included in this report merely to show the 

relative simplicity of this method and to highlight some of its benefits. 
 245 

12. BRIEF SUMMARY AND RECOMMENDATIONS 

PROJECT OBJECTIVE 

The objective of the project was to develop a comprehensive QA manual, supported by scientific 

evidence and statistical theory that provides step-by-step procedures and instructions for 

developing effective and efficient QA specifications. This technical report summarizes what 

steps were taken to accomplish this goal and the analyses that were conducted to support the 

recommendations made in the QA manual. 

While the focus and objectives of the two documents are quite different, they are not totally 

stand-alone documents. As such, the technical report should be read in conjunction with and as a 

companion to the QA specifications manual that also resulted from the project. 

QA SPECIFICATIONS DEVELOPMENT PROCESS 

One of the major accomplishments of the project was the development of the three-phase QA 

specifications development process that is presented in chapter 3. The process was presented to 

and approved by the panel of representatives from the pooled fund States that provided the 

funding for this project. The process is presented in flowchart form for ease of understanding. 

RESULTS OF DETAILED ANALYSES 

After seeing and approving the specifications development process flowcharts, the panel was 

surveyed to identify those topics that they thought were most important for further detailed 

analyses. The items selected by the panel also comprise some of the major decisions that are 

necessary during the specifications development process. These items are presented below, along 

with the recommendations that were included in the QA manual:(1)

What quality measure should be used for individual quality characteristics? 

From the many potential quality measures that were identified, three were chosen for detailed 

analyses—PWL (and its complement PD), AAD, and CI. After initial analyses, CI was 

eliminated because it offered no benefits over AAD and because it provided a slightly biased 

estimate for the population CI. Further studies showed that PWL and AAD generally performed 

comparably to one another. PWL was recommended because it provides a direct measure of 

variability, whereas AAD provides, at best, an indirect measure of variability. In fact, the same 

population AAD could apply to very different populations. PWL also works effectively with 

both one-side and two-sided specifications, while AAD only applies to two-sided specifications 

that have a definite target value. 

What payment relationships should be used for individual quality characteristics?

The selection of the payment equation is, to some extent, a subjective decision for each agency 

based on the level of risk that it considers reasonable. A general method was developed to 

determine the equivalent PWL and AAD values for any given value of PWL or AAD. From this 

it was possible to show that separate PWL and AAD payment equations could be developed to 
 246 

provide equivalent EPs for the same population. While no single payment relationship was 

recommended for individual quality characteristics, the extensive discussion in this report 

regarding calculating and evaluating risks provides the agency with the tools and techniques to 

develop payment equations that they believe to be fair and reasonable. 

How should multiple quality characteristics be combined into a single payment factor? 

Analyses were conducted on six different methods for combining payment factors for individual 

quality characteristics into one combined payment factor: 

• Averaging the individual payment factors. 

• Using a weighted average of the individual payment factors. 

• Multiplying the individual payment factors. 

• Summing the individual payment adjustments. 

• Using the maximum individual payment factor. 

• Using the minimum individual payment factor. 

The analyses showed that the variability of the combined payment factors for two characteristics 

was related to whether they were positively correlated, negatively correlated, or not correlated. 

For four of the methods, the EP values did not vary with the correlation between the variables. 

However, the EP did vary with the correlation when using either the maximum or minimum 

individual payment factor. 

Another method—using a single composite quality measure derived from a general performance 

model to predict expected pavement life—was developed and is the recommended approach. If 

an agency wishes to use a simpler method for determining the composite payment factor, the first 

four methods listed above are candidates. One of the two averaging methods might be preferred 

since the averaging process has the effect of reducing the variability in the estimated payment 

factors. 

What procedures should be used to verify the contractor’s test results if they are to be used 

in the acceptance and payment decision?

A detailed discussion and analysis were presented regarding the risks involved in the various 

approaches to verifying the contractor’s test results. OC or power curves, surfaces, and tables 

were presented to illustrate the risks associated with the different procedures and sample sizes. 

The relatively high risks that are associated with typical agency verification testing frequencies 

were pointed out. 

A discussion was presented concerning which verification procedures were appropriate for use 

with split and independent samples. For split samples, the recommended approach is to use a 

maximum allowable limit for the difference between the agency’s results and the contractor’s 

results for each individual pair of split samples, but also to accumulate the results for use in a 

t-test for paired measurements. The best approach for independent samples is to use the F-test to 

compare the variances in the contractor’s tests and the agency’s tests. The result of the F-test 

then determines the manner in which the t-test is conducted to compare the means of the 
 247 

contractor’s tests and the agency’s tests. Each agency can select its own testing frequency based 

on its assessment of the risks provided in the OC curves that were developed during this project. 
 
 249

APPENDIX A: ANNOTATED BIBLIOGRAPHY OF SELECTED 

ITEMS FROM THE INITIAL LITERATURE SEARCH

Afferton, K.C.; Freidenrich, J.; and Weed, R.M. “Managing Quality: Time for a National 

Policy,” Transportation Research Record 1340, 1992, pp. 3-39. 

Statistical quality assurance (SQA) (currently in use or under development in approximately three-quarters of 

the States) has proven to be a very effective tool for encouraging high-quality construction. However, although 

statistical specifications writing must now be recognized as a thoroughly scientific activity, there is great 

disparity in its application on the State level and many current practices and published standards are far from 

optimal. Part 1 of this paper stresses the need for sweeping reforms and suggests that the establishment of a 

uniform national policy on transportation QA is overdue. Part 2 describes a variety of obstacles that must be 

overcome if such a transformation is to be made. Part 3 outlines an extensive series of principles that must be 

understood in order to derive the maximum benefit from a QA program. And finally, Part 4 presents a plan of 

action that, if conscientiously followed, will significantly increase the effectiveness of transportation QA 

practices nationwide. 

Allen, O.B. and Newcombe, P.A. “A Three-Class Procedure for Acceptance Sampling by 

Variables,” Technometrics, Vol. 30, No. 4, 1988, pp. 415-421. 

A three-class procedure for acceptance sampling by variables is introduced as an alternative to both the threeclass attributes plan and the two-class variables plan. The procedure, which requires that the quality 

characteristic be normally distributed, has the advantage of requiring a smaller sample size than a three-class 

attributes plan with approximately the same OC surface. The advantage of the three-class variables plan over 

the two-class variables plan is the ability to discriminate between lots with high and low percentages of 

marginally conforming products. Two equivalent methods of stating the decision rules for the plan are 

suggested. It is shown that the OC surface for the three-class variables procedure may be constructed with a 

special case of bivariate, noncentral t-distribution. The motivation and methods for choosing a plan are 

discussed and illustrated. 

Aminzadeh, M.S. “Inverse-Gaussian Acceptance Sampling Plans by Variables,” 

Communications in Statistics: Theory and Method, Vol. 125, No. 5, 1996, pp. 923-935. 

Variable sampling plans to control the defective fraction are obtained using the Inverse-Gaussian distribution. 

OC curves are obtained and the impact of the sample size and the specification limits on these curves are 

discussed. Simulation studies are used to investigate the sensitivity of the sampling plans under the more 

commonly used normal distribution. 

Amirkhanian, S.N.; Burati, J.L.; and Mirchandani, H.C. “Effect of Testing Variability on 

Contractor Payment for Asphalt Pavements,” Journal of Construction Engineering and 

Management, Vol. 120, No. 3, September 1994, pp. 579-592. 

Acceptance of asphalt pavements on a lot-by-lot basis depends on how well the material conforms to specified 

quality characteristics. If a lot does not fall within the specified tolerance limits for a given characteristic, the 

payment to the contractor for that lot may be adjusted according to a predetermined payment schedule. This 

study evaluated the effects of the variability of the test method on the payment to the contractor. Payment 

factors based on selected State highway agency specifications, typical of the most commonly used type of 

acceptance plans, were estimated using simulation techniques. The results were illustrated by plotting the OC in 

terms of the expected payment. The OC plots show that the contractor might receive a reduced payment even 

when there is no material variability and the process mean is on target. This reduction was a result of the 

relatively large magnitude of the testing variability as compared with the tolerance range specified in the 

simulated acceptance plans. With suitable modifications, the simulation routine developed to estimate 

contractor payment can be used for any acceptance plan. 
 250

Balamurali, S. and Kalyanasundaram, M. “Determination of an Attribute Single Sampling 

Scheme,” Journal of Applied Statistics, Vol. 24, No. 6, 1997, pp. 689-695. 

A procedure for the selection of a new sampling scheme, called the Single Sampling Scheme, is presented. 

Additionally, a table for the selection of a single sampling scheme, which is indexed by various combinations of 

entry parameters, is given. The method of table construction and the advantages of single sampling schemes are 

discussed. 

Barros, R.T.; Weed, R.M.; and Willenbrock, J.H. “Software Package for Design and Analysis of 

Acceptance Procedures Based on Percent Defective,” Transportation Research Record 924, 

1983, pp. 85-93. 

The trend toward statistical end-result specifications (ERS) has led to the development of construction 

specifications based on the concept of PD. To analyze the risks and determine the effectiveness of the 

acceptance procedures associated with these specifications, OC curves must be constructed. However, many 

potential users do not have a working knowledge of the noncentral t- and beta distributions necessary for this 

development. The underlying theory, several useful references, and a conversational computer program that 

greatly simplifies the design and analysis of specifications of this type are presented. 

Benson, P.E. “Comparison of End Result and Method Specifications for Managing Quality,” 

Transportation Research Record 1491, 1995, pp. 3-10. 

The results of a statistically designed experiment in which AC cores and nuclear gauge readings were taken 

from five California projects are reported. Relative compaction for the projects was controlled with a method 

specification. Analysis of variance is used to separate test error and the locational components of variance for 

specific gravity, asphalt content, lift thickness, and grading. Compaction results are compared with similar 

results from 16 ERS jobs studied previously. Relative compaction on the end-result jobs averages 3.1 percent 

higher in value. Findings on test precision, increased lot size, and material variability are discussed. 

Benson, P.E. “Performance Review of a Quality Control/Quality Assurance Specification for 

Asphalt Concrete,” Transportation Research Record 1654, 1999, pp. 88-94. 

A statistical review of 50 jobs recently completed by using California’s QC/QA specification for AC is 

presented. Performance is contrasted to the quality achieved under method specification and ERS. A cost 

analysis is made and issues related to verification are discussed. Improvements to the current specification are 

proposed. The data present clear evidence that the allowable tolerance of ±0.5 percent for asphalt content is too 

broad for current practices. Also, an increase in compaction variability for many QC/QA jobs could be 

controlled by adding an upper specification limit or adopting a two-sided volumetric specification. The cost for 

QC/QA jobs went up approximately 3 percent to pay for bonuses allowed under the specification. An analysis 

of the contractor’s QC test data indicate that this increase is more than compensated for by projected reductions 

in future rehabilitation costs. However, a significant lack of agreement between the contractor QC and agency 

QA tests brings this finding into question. More rigorous verification of contractor-provided test results must be 

incorporated into the specification and the results must be analyzed before cost-effectiveness can be determined. 

Budwig, J.L. “Bituminous Pavement Smoothness: Statistically Based Approach to Acceptance,” 

Transportation Research Record 1544, 1996, pp. 125-134. 

Since 1987, the Federal Lands Highway (FLH) branch of FHWA has been evaluating acceptance of newly 

constructed bituminous pavements using California-type profilograph measurements. California Test Method 

526 and FLH T504, as well as various acceptance plans, have been used in this evaluation. This study 

examined: (1) whether operator trace reduction variability was too large for acceptance testing, (2) which type 

of acceptance plan should be incorporated into the Standard Specifications for Construction of Roads and 

Bridges on Federal Highway Projects (FP 92), and (3) two commercially available computerized trace 

reduction systems. The study concludes that when used in conjunction with statistical evaluation procedures, the 

test method is suitable for acceptance purposes and that computerized trace reduction is superior to manual 

reduction. Also presented are some fundamentals of statistically based acceptance that are not widely known or 

understood by highway engineers. 
 251

Burati, J.L.; Antle, C.E.; and Willenbrock, J.H. “Development of a Bayesian Acceptance 

Approach for Bituminous Pavements,” Transportation Research Record 924, 1983, pp. 64-71. 

Traditional approaches for estimating the percentage of a lot that is within specification limits (PWL) are based 

on random samples taken from the lot being evaluated. These approaches suffer from the small sample size 

necessitated by the destructive and time-consuming tests that are usually used in determining the quality of the 

materials. The development of a Bayesian approach for estimating PWL, which incorporates information 

concerning the contractor’s past performance on the project and the current sample results that determine the 

estimate for the PWL of the current lot, is presented. The procedure assumes that the daily population mean is a 

random variable that follows a normal distribution, that the production process is also normally distributed, and 

that the process variance is constant. These assumptions are confirmed by using goodness-of-fit tests on data 

collected from 13 bituminous runway paving projects. Computer simulation shows that the Bayesian PWL 

estimators are slightly biased in comparison with the traditional quality index method, but that the PWL 

estimators exhibit smaller variances than the traditional method. 

Burati, J.L.; Bridges, W.C.; and Ackerman S.A. “Evaluation of Quality Assurance Programs for 

Bituminous Paving Mixtures,” Report No. FHWA-SC-95-002, South Carolina DOT, May 1995. 

The South Carolina DOT QA program currently consists of the Record Sample, Independent Assurance (known 

as the Green Tag program), and Split-Sample programs. Because these three programs were developed at 

different times and possibly for different reasons, a study was needed to determine how the three programs 

meshed together, if there are overlaps or gaps in these programs, and if they are serving the function or 

functions for which they were intended. The study consisted of: (1) a review of the documentation of the 

existing programs, (2) interviews with FHWA and South Carolina DOT personnel to establish the objectives for 

each program, (3) a survey of other State highway agency independent assurance practices, and (4) an analysis 

of historical data from each of the programs. ASTM precision statements were also considered when evaluating 

the allowable tolerances for comparing test results. This report recommends that the Record Sample and SplitSample programs be maintained with some modification in the procedures, including using independently 

obtained samples (as opposed to split samples) for the Record Sample program and slightly modified allowable 

comparison tolerances for both programs. It was concluded that the Green Tag program is not necessary for the 

purpose of independent assurance. It primarily serves as an enforcement function and is not needed unless the 

South Carolina DOT believes that it is necessary to provide an external psychological inducement to ensure that 

its inspectors properly perform their job functions. 

Burati, J.L. and Willenbrock, J.H. “Acceptance Criteria for Bituminous Surface Course on Civil 

Airport Pavements,” Report No. FAA-RD-79-089, Federal Aviation Administration, 1981. 

Research was undertaken to extend the use of statistically based airport pavement materials specifications that 

incorporate price-adjustment features. During the course of the project, data on the physical characteristics of 

the pavement materials were collected from 13 airport pavement construction projects. A statistical analysis of 

this data permitted the determination of the parameters (mean and standard deviation) of existing airport 

projects, and these parameters were then used to develop acceptance plans and price-adjustment factors. OC 

curves and the curves of expected payment were used to determine the appropriate acceptance plans, which 

were based on the percentage of material falling within the specification limits (PWL). By using a continuous 

rather than a discrete price-adjustment schedule, it was possible to avoid the problem of large differences in 

payment associated with relatively small differences in quality (as measured by the PWL). A computer program 

was developed to approximate the expected payment curves associated with different continuous priceadjustment systems. This program is applicable to one-sided specification limits such as density. For properties 

such as air voids, which require both an upper and lower specification limit, the OC curves were determined by 

computer simulation of 10,000 randomly drawn samples. 

Chang, L-M. and Hsie, M. “Developing Acceptance Sampling Methods for Quality 

Construction,” Journal of Construction Engineering and Management, Vol. 121, No. 2, 1995, 

pp. 246-253. 

The role that the acceptance sampling method plays in designing the QA specification is discussed. The 

acceptance sampling method applies statistics to specify the number of measurements needed and determines 

how to make an acceptance or rejection decision based on measured data. Four acceptance sampling methods 
 252

are presented, including the variable single sampling method, quality index sampling method, attribute double 

sampling method, and attribute proportion sampling method. The theories are derived and the applications of 

the four quality acceptance sampling methods are described. The advantages and disadvantages of the different 

sampling methods are compared. 

Darter, M.I.; Hoerner, T.E.; Smith, K.D.; Okamoto, P.A.; and Kopac, P.A. “Development of 

Prototype Performance-Related Specification for Concrete Pavements,” Transportation Research 

Record 1544, 1996, pp. 109-115. 

The development of a prototype PRS for concrete pavement is summarized. The prototype PRS requires that the 

pavement lot be divided into consistent sublots for the measurement of quality characteristics that are then used 

to estimate future performance and life-cycle costs. The difference between the life-cycle costs of the target (asdesigned) pavement and the actual (as-constructed) pavement lot is used to determine a rational pay adjustment. 

Both the means and the variations of all of the quality characteristics are directly considered in the pay factor 

determination. A Microsoft® Windows®-based computer program, PaveSpec, was developed for use with the 

specification simulation and for generating pay adjustments. However, additional work is required to make this 

a fully practical PRS. 

Dobrowolski, J. and Bressette, T. “Development of Quality Control/Quality Assurance 

Specifications by Using Statistical Quality Assurance for Asphalt Concrete Pavements in 

California,” Transportation Research Record 1632, 1998, pp. 13-21. 

In 1996, Caltrans implemented QC/QA specifications for AC paving. These specifications require contractor 

QC and provide rewards or penalties based on statistical quality analysis of eight quality characteristics. These 

specifications were developed through a joint Caltrans/industry group and are supplemented with a QC manual. 

In March 1996, the first project using the specifications went to bid and, in the first year, six projects were 

completed. Since then, revisions to the specifications have been developed. The specifications and manual 

issues, recommendations based on 1996 projects, changes that Caltrans has made, and anticipated additional 

endeavors in areas of QC/QA and materials testing are discussed. 

Douglas, K.D.; Coplantz, J.; Lehman, R.; and Bressette, T. “Evaluation of Quality 

Control/Quality Assurance Implementation for Asphalt Concrete Specifications in California,” 

Transportation Research Record 1654, 1999, pp. 95-101. 

Caltrans has implemented a new QC/QA specification for AC pavement. As part of this implementation, 

Caltrans realized the need for objective feedback on its design and implementation, and selected Nichols 

Consulting Engineers to perform this study. The study objective, the research approach used, a description of 

the projects included in the study, and recommendations and findings are presented. Lessons learned from the 

Caltrans experience are shared to the benefit of other States implementing QC/QA specifications for AC 

pavement. 

Elliott, R.P. “A Value Concept for Pavement Construction Pay Adjustment Schedules,” 

Transportation Research Record 1040, 1985, pp. 45-48. 

A value concept is presented that can serve as a basis for developing rational payment schedules for pavement 

construction. Provisions are made for incorporating both the average and the standard deviation of the materials 

tests into a payment determination scheme that is based on the relative pavement life effects. The concept is 

based on the recognition that, at the time the pavement is considered to have failed, only a small percentage of 

the surface actually exhibits severe distress. As a result, the life of the pavement is controlled not by the average 

or 50th percentile of the material, but by a lower percentile representative of the actual surface distress. 

Elliott, R.P. and Qiu, Yanjun. “Analysis of Contractor Pay Adjustment Schedule Using 

Simulation,” Transportation Research Record 1544, 1996, pp. 109-115. 

A common provision in QC/QA construction contracts is the adjustment of the contractor’s pay on the basis of 

the quality of the construction. The expected impact of the provision on payment should be examined to ensure 

that the adjustments are neither unduly severe nor excessively lenient. Most pay adjustment plans have been 
 253

developed around a quality index by using a PD approach. Analyses of these plans are complex, but reasonably 

well defined. Other plans are more complex and do not lend themselves to direct analysis. Computer simulation 

can be used to examine these plans. An example is given. It is shown that the simulation process can provide a 

better, more detailed examination of the pay schedule than is possible by simply determining the expected 

payment. In particular, the simulation process can indicate the variability of payment at various quality levels 

and can identify the factors most responsible for the pay adjustment. 

Fan, D-Y. “Bayesian Acceptance Sampling Scheme for Pass-Fail Components,” 

Communications in Statistics: Theory and Method, Vol. 20, No. 8, 1995, pp. 2351-2355. 

An approach is given for determining the choice of a prior distribution for the design of a Bayesian acceptance 

sampling scheme. The appropriate prior distribution is selected from confidence coefficients corresponding to 

classical lower confidence bounds. A numerical example is provided. 

Gentry, C. and Yrjanson, W.A. “Specifications for Quality Control: A Case Study,”

Transportation Research Record 1126, 1987, pp. 37-41. 

One of the essential qualities of a specification is reasonableness. Court decisions and their economic 

consequences demand that specifications be based on reasonable requirements. Specifications that call for 

unnecessary perfection are hardly reasonable; furthermore, they do not ensure performance. Specifications that 

attempt to control quality through extremely limited tolerances may, in fact, be counterproductive. When QC 

efforts are directed to comply with such specifications, quality may be compromised, contract administration 

may be difficult, and additional costs may be incurred, all without improving the performance of the completed 

work. A case study is presented to illustrate the problems created by an excessively restrictive specification. 

Alternatives and comments for improvement of the specification are offered. 

Gibra, I.N. “Recent Developments in Control Chart Techniques,” Journal of Quality Technology, 

Vol. 7, No. 4, 1975, pp. 183-192. 

In the last three decades, several control chart procedures were developed. Prominent among these are the 

Cumulative Sum, Economic Design of X-Control, and Acceptance and Multicharacteristic charts. Descriptions 

and analyses of these charts are presented in an attempt to bring past research into current perspective. 

Govindaraju, K. “Certain Observations on Lot-Sensitive Sampling Plan,” Communications in 

Statistics: Theory and Method, Vol. 19, No. 2, 1990, pp. 617-627. 

A comparison of the sample-size efficiency of the lot-sensitive plan with the double and multiple sampling 

plans is presented. It is shown that a fully curtailed lot-sensitive plan will involve a smaller average sample 

number than the equivalent double and multiple sampling plans. 

Govindaraju, K. and Kuralmani, V. “A Note on the Operating Characteristic Curve of the Known 

Sigma Sampling Variables Plan,” Communications in Statistics: Theory and Method, Vol. 21, 

No. 8, 1992, pp. 2339-2347. 

The OC curves of certain known sigma sampling plans may not be satisfactory in that they have a tendency to 

reject lots of acceptable quality. The theory and a method to identify such known sigma variables plans 

possessing unsatisfactory OC curves are presented. 

Govindaraju, K. and Subramani, K. “Selection of Double Sampling Attributes Plan for Given 

Acceptable Quality Level and Limiting Quality Level,” Communications in Statistics: 

Simulation and Computation, Vol. 21, No. 1, 1992, pp. 221-242. 

Tables and procedures are given for finding the double sampling plan, conditional double sampling plan, link 

sampling plan, and ChSP-4 and ChSP-4A chain sampling plans involving a minimum sum of the producer’s 

and consumer’s risks for the specified acceptable quality level and the limiting quality level. 
 254

Govindaraju, K. and Subramani, K. “Selection of Single Sampling Attributes Plan for Given 

Acceptable Quality Level and Limiting Quality Level Involving Minimum Risk,” 

Communications in Statistics: Simulation and Computation, Vol. 19, No. 4, 1990, pp. 1293-

1302. 

A table and a procedure are given for finding the single sampling attributes plan involving a minimum sum of 

the producer’s and the consumer’s risk for the specified acceptable quality level and the limiting quality level. 

Hamaker, H.C. “Acceptance Sampling for Percent Defective by Variable and by Attributes,” 

Journal of Quality Technology, Vol. 11, No. 3, 1979, pp. 139-148. 

Various methods for adjusting variables and attributes sampling plans so that they possess nearly identical OC 

curves are reviewed and extended. It is demonstrated that the OC curve for the S-method plans can be 

adequately derived from a normal approximation and that the more complicated use of the noncentral tdistribution can be avoided. The relative efficiencies of the different types of single sampling plans are shown to 

be practically functions of the indifference quality alone. The relationships between these efficiencies and the 

choice of the specification limit are discussed in detail. 

Hawkes, C.J. “Curves for Sample Size Determination in Lot-Sensitive Sampling Plans,” Journal 

of Quality Technology, Vol. 11, No. 4, 1979, pp. 205-210. 

Curves are given for determining the required sample size for zero acceptance number sampling plans for 

specified values of lot size and lot tolerance PD. These allow a quick and easy assessment of the minimum 

required sample size to obtain lot PD protection for 1-percent, 5-percent, 10-percent, 20-percent, and 50-percent 

consumer risk. 

Hughes, C.S. “Incentive and Disincentive Specification for Asphalt Concrete Density,” 

Transportation Research Record 986, 1984, pp. 38-42. 

The background for a specification that includes both positive and negative price adjustments for the density of 

AC is presented. The results that have been obtained since the specification was introduced in 1978 are 

described. The incentive features of the specification are emphasized because it is believed that they are unique 

and have been the primary reason that improved densities have been obtained in Virginia for the past 6 years. 

Hughes, C.S. “Variability in Highway Pavement Construction,” NCHRP Synthesis 232, 1996. 

This synthesis addresses variability questions and the importance of defining the variability of materials and 

construction processes. For potential users of these measures of variability, an update of the typical materials 

and construction variability and the use of incentive and disincentive pay schedules for acceptance are 

presented. 

Irick, P.E. “A Conceptual Framework for the Development of a Performance-Related Materials 

and Construction Specification,” Transportation Research Record 1126, 1987, pp. 1-27. 

Pavement design and performance concepts that provide a systematic basis for the development of 

specifications for materials and construction (M&C) are presented. It is assumed that the conceptual framework 

for specifications includes eight sets of relationships among the process variables and nine sets of inputs or 

outputs for the relationships. Independent variables are selected that have predictable effects on performancerelated output variables. From these independent variables, variables appearing explicitly in prediction functions 

(EPF) are selected and subdivided into traffic factors, environmental factors, and pavement structure factors. 

EPF variables can be replaced by surrogate variables (SPF) when M&C control for these secondary variables is 

easier to provide. Other secondary variables are the control factors (CF), which have predictable effects on the 

EPF of SPF variables. EPF variables related to the M&C process are denoted as MCF. In general, a stochastic 

prediction model consists of a prediction function that may be completely known from mechanistic 

considerations; may be partially known except for undetermined constants; or may be assumed to be a linear 

combination of linear, curvilinear, and interaction effects among independent variables. General forms of 

prediction equations for stress and distress, stress-load equivalence relationships, traffic prediction relationships, 

relationships among M&C specification factors, and performance-cost relationships are presented. Pavement 
 255

design criteria and M&C specification factors are added as the initial conditions for the definition of a pavement 

design for a given requirement. 

Kandahl, P.S.; Cominsky, R.J.; Maurer, D.; and Motter, J.B. “Development and Implementation 

of Statistically Based End-Result Specifications for Hot-Mix Asphalt in Pennsylvania,” 

Transportation Research Record 1389, 1993, pp. 9-16. 

In the past, Pennsylvania DOT used the concept of single samples and tests to determine the quality of hot-mix 

asphalt (HMA) mixtures. This study develops a statistically based ERS for HMA pavements that makes the 

contractor responsible for QC and Pennsylvania DOT responsible for QA. Field data from several HMA paving 

projects were analyzed statistically to establish realistic numerical limits for the various test parameters used in 

the specification. Three pay items (asphalt content, percent passing #200, and mat density) are included in the 

specification. The specifications have provided Pennsylvania DOT with a means of evaluating and comparing 

the dollar value of the year-by-year improvement in HMA quality. 

Kirkpatrick, R.L. “Confidence Limits on a Percent Defective Characterized by Two 

Specification Limits,” Journal of Quality Technology, Vol. 2, No. 3, 1970, pp. 150-155. 

A quality problem is to assess the performance of a production process with respect to specifications. A 

solution is obtained by estimating PD and placing confidence limits on the PD for each product characteristic. 

Tables are presented for determining a point estimate of PD and 90-, 95-, and 99-percent confidence limits on 

the PD of a characteristic with both upper and lower or single specification limits. On the assumption of 

normality, only the specification limits, sample size, sample mean, and standard deviation are needed to give a 

solution to the problem. 

Kopac, P.A. “Current Practices in Acceptance of Bituminous Concrete Compaction,” 

Transportation Research Record 986, 1984, pp. 43-46. 

Current procedures employed by State highway agencies to determine the acceptability of bituminous concrete 

compaction are discussed. Both statistical and nonstatistical acceptance plans are covered. Statistical acceptance 

plans can provide a clear indication of the quality levels that are desired with the estimation of the construction 

quality. However, many of the statistical acceptance plans currently in use are inadequate because they are 

inefficient or statistically unsound, or both. Also, there is a considerable lack of uniformity among acceptance 

plans and the price adjustment schedules they contain. Recommendations are made to improve acceptance 

plans. 

Kuralmani, V. and Govindaraju, K. “Modified Tables for the Selection of the Double Sampling 

Attributes Plan Indexed by AQL and LQL,” Communications in Statistics: Theory and Method, 

Vol. 24, No. 7, 1995, pp. 1897-1921. 

Modified tables are presented for the selection of double sampling plans for a given AQL, producer’s risk, LQL, 

and consumer’s risk, giving the minimum sum of the average sample numbers at the AQL and LQL under the 

conditions of the Poisson model for the OC curve. 

Kuralmani, V. and Govindaraju, K. “Selection of Conditional Sampling Plans for Given AQL 

and LQL,” Journal of Applied Statistics, Vol. 20, No. 4, 1993, pp. 467-479. 

Procedures and tables for the selection of conditional sampling plans are given. The sampling plans discussed 

include the conditional double sampling plan, chain-deferred sampling plan, link sampling plan, and ChSP-4A 

chain sampling plan. The procedures and tables for each are provided for a given AQL, producer’s risk, limiting 

quality level (LQL), and consumer’s risk, giving a minimum sample size at AQL under the conditions of the 

Poisson model for the OC curve. 

Moore, R.M.; Mahoney, J.P.; Hicks, R.G.; and Wilson, J.E. “Overview of Pay-Adjustment 

Factors for Asphalt Concrete Mixtures,” Transportation Research Record 821, 1981, pp. 49-56. 

In fall 1979, the Oregon State Highway Division and Oregon State University, with participation from the 

University of Washington, initiated a research project to study the impact of variations in material properties on 
 256

asphalt pavement life. The study was aimed at developing a rational approach to assessing the effects of 

variations from specification limits so that a firm basis could be established for the development of pay factors. 

Analysis of the results indicates the following: (1) most State agencies will accept one or more property 

characteristics that are outside of specification tolerances, (2) most State agencies apply a pay factor for 

accepted material outside of specification tolerances, (3) only 26 percent of the State agencies consider their pay 

factor to be proportional to reduced pavement serviceability, (4) approximately half of the agencies consider 

pay-factor plans to be effective in encouraging compliance with specifications, and (5) there is a wide disparity 

in the pay-adjustment factors used by the different agencies. 

Nachlas, J.A. and Kim, S-I. “Generalized Attribute Acceptance Sampling Plans,” Journal of 

Quality Technology, Vol. 21, No. 1, 1989, pp. 32-40. 

The imposition of acceptance criteria upon realizations of generalized sampling plans is described and has been 

shown to yield a large variety of new attribute acceptance sampling plans. The resulting generalized attribute 

acceptance sampling plans can be constructed to yield the same OC curve as that for conventional single 

sampling plans with lower expected total cost. An example is presented. 

Nelson, L.S. “Factors for Confidence Limits on Standard Deviation,” Journal of Quality 

Technology, Vol. 29, No. 4, 1997, pp. 485-487. 

Tables are presented for easily determining both one-sided and two-sided confidence limits on standard 

deviations, assuming that the sample is random and from a normal population. 

Patel, A.; Thompson, M.; Harm, E.; and Sheftick, W. “Developing QC/QA Specifications for 

Hot-Mix Asphalt Concrete in Illinois,” Transportation Research Record 1575, 1997, pp. 66-74. 

The Illinois DOT has recently undertaken a quality management program to improve the quality of 

construction, allow more innovation, and reduce the department’s management of industry construction 

programs. The AC QC/QA program is a significant part of this quality management program. The Illinois DOT 

credits the success of the AC QC/QA program to gradual implementation, and contractor and industry 

involvement. In 1991, four projects were constructed under a newly developed QC/QA specification. After 

reviewing and evaluating feedback, the specification was revised for 1992. In 1992, 30 projects were 

constructed using the QC/QA specification and, in 1993, 65 projects used the specification. In 1994 and 1995, 

most projects over 225 Mg used the specification. In conjunction with this effort, an aggregate certification 

program was implemented. Training programs for contractor QC and aggregate certification were also 

implemented. Analysis of the data indicates an increase in the uniformity of the HMA, potentially leading to a 

15-percent increase in fatigue life. The Illinois DOT is now examining the implications of developing ERS and 

PRS for the AC QC/QA program. In summer 1996, one QC/QA project was shadowed and evaluated based on 

newly developed ERS/PRS. 

Pham, T.G. and Turkkan, N. “Bayes Binomial Sampling by Attributes With a General-Beta Prior 

Distribution,” IEEE Transactions on Reliability, Vol. 41, No. 2, 1992, pp. 310-316. 

In binomial sampling, the standard beta is frequently used as a prior because of its conjugate property for a 

sample from a Bernoulli distribution. The case where the prior is a general beta is examined. Practical 

advantages in the elicitation of expert opinion (to obtain a prior) and a convenient expression of the posterior 

are presented. A computer program on an IBM® personal computer (or a compatible computer) permits simple 

use of the general beta and is available from the authors. This program is used to solve the numerical examples, 

showing the advantages of using the general beta as a prior. 

Schmitt, R.L.; Russell, J.S.; Hanna, A.S.; Bahia, H.U.; and Jung, G.A. “Summary of Current 

Quality Control/Quality Assurance Practices for Hot-Mix Asphalt Construction,” Transportation 

Research Record 1632, 1998, pp. 22-31. 

State highway agencies and contractors have been implementing QC/QA specifications in recent years to 

advance the quality of HMA construction. During their continued development, the attributes of these QC/QA 

specifications have varied among the States. This paper provides a compilation of state-of-the-art practices in 
 257

QC/QA for HMA construction and provides recommendations for State highway agencies and contractors when 

modifying or developing a QC/QA specification. 

Seeds, S.B.; Basavaraju, R.; Epps, J.A.; and Weed, R.M. “Development of Performance-Related 

Specifications for Hot-Mix Asphalt Pavements Through WesTrack,” Transportation Research 

Record 1575, 1997, pp. 85-91. 

The primary purpose of the FHWA-sponsored WesTrack project is to further the development of PRS for HMA 

construction. This objective is being achieved, in part, through the accelerated loading of a full-scale test track 

facility in northern Nevada. Twenty-six HMA test sections that were constructed to meet the criteria set forth in 

a statistically based experimental design are providing performance data that will be used to improve (or 

develop new) pavement performance prediction relationships that better account for the effects that off-target 

values of asphalt content, air voids, and aggregate gradation have on such distress factors as fatigue cracking, 

permanent deformation, roughness, raveling, and tire-pavement friction. The concept of the planned PRS and 

how it will incorporate the modified pavement performance prediction models are described. The current plan 

for assessing contractor pay adjustments based on data collected from the as-constructed pavement is also 

discussed. 

Semenov, V.A. “Quality Control in Highway Construction and Maintenance When the 

Measurement Parameters Are Highly Nonuniform,” Transportation Research Record 1126, 

1987, pp. 28-36. 

An original method is presented for QC based on Weibull’s law for the distribution of random quantities with 

variable parameters. The nomographs obtained for determining the extreme values of the parameters and the 

defectiveness index can be used for various cases in the statistical reduction of research results. The proposed 

method can be used for QC for both highly uniform data (normal distribution) and highly nonuniform data. The 

method described is widely used in the former Soviet Union for QC in the construction and maintenance of 

roads. 

Sheng, Z. and Fan, D-Y. “Bayes Attribute Acceptance Sampling Plan,” IEEE Transactions on 

Reliability, Vol. 41, No. 2, 1992, pp. 307-309. 

An approach is reviewed for choosing a prior distribution for a Bayes attribute acceptance sampling plan. A 

prior distribution is chosen from confidence levels corresponding to classical lower confidence bounds. Where a 

Bayes plan is acceptable, the sample size can be reduced. 

Soundararajan, V. “Maximum Allowable Percent Defective (MAPD) Single Sampling 

Inspection by Attributes Plan,” Journal of Quality Technology, Vol. 7, No. 4, 1975, pp. 173-182. 

Single sampling attribute plans indexed by maximum allowable percent defective (MAPD) are given. A table 

allowing the transitioning from one set of parameters to match the OC curve to other similar sets is given. 

Soundararajan, V. and Arumainayagam, S.D. “Lot Sensitive Double Sampling Plan,”

Communications in Statistics: Theory and Method, Vol. 21, No. 10, 1992, pp. 2931-2948. 

A sampling plan is derived for compliance testing, which provides consumer protection. 

Suresh, K.K. and Ramkumar, T.B. “Selection of a Sampling Plan Indexed With Maximum 

Allowable Average Outgoing Quality,” Journal of Applied Statistics, Vol. 23, No. 6, 1996, 

pp. 645-654. 

A new concept of maximum allowable average outgoing quality (MAAOQ), which is the average outgoing 

quality at the inflection point, is introduced. The procedure for designing a single sampling plan indexed 

through the MAPD and MAAOQ is stated. Tables are constructed for the selection of parameters for the plan 

and parametric conversions are studied. 
 258

Suresh, R.P. and Ramamathan, T.V. “Acceptance Sampling Plans by Variables for a Class of 

Symmetric Distributions,” Communications in Statistics: Simulation and Computation, Vol. 26, 

No. 4, 1997, pp. 1379-1391. 

The estimation of PD using a normal sampling plan is not appropriate when the assumption of normality is 

violated. A sampling plan based on a more general symmetrical family of distributions with the parameters 

estimated using the modified maximum likelihood (MML) procedures is presented. Some numerical studies are 

also presented, showing that the sampling plan works well for most of the symmetrical non-normal 

distributions. 

Vlatas, D.A. and Smith, R.E. “Implications of Life-Cycle Performance Specifications,” 

Transportation Research Record 1215, 1989, pp. 25-30. 

The professional, managerial, and legal implications of using life-cycle performance specifications are 

presented. Changes in the roles of the parties using life-cycle performance specifications are discussed. This 

approach can improve quality, reduce costs, and expedite the construction process. The basis of the process is 

the development of models of expected performance. These models will be used to predict whether the 

pavement will perform as required over the life of the project. Tests are performed at the end of construction to 

determine whether the expected performance is likely to be achieved. Adjustments in payment can be made, 

based on the performance model predictions. The ramifications of adopting life-cycle performance 

specifications are discussed. 

Wasserman, G.S. “Matching Performance of Continuous Sampling Plans With Single Sampling 

Plans,” Communications in Statistics: Simulation and Computation, Vol. 19, No. 4, 1990, 

pp. 1303-1317. 

Single sampling plans are widely used for appraising product quality. However, for situations where a 

continuous product flow exists, lot-by-lot demarcations may not exist, and it may be necessary to use alternate 

procedures, such as Continuous Sampling Plan 1 (CSP-1), for continuous processes. In this case, one would like 

to understand how the average performance of the continuous sampling procedures compares to the more 

commonly used single sampling plans. A model is devised that can be used to relate plan performance between 

single sample lot acceptance procedures and CSP-1. It is shown that it is generally not possible to match up 

performance based on OC expressions for the two plans. Instead, the plans are matched by equating expressions 

for π (p), the long run proportion of the product that is accepted, under both procedures. This is shown to be 

equivalent to matching properties on an average outgoing quality basis. Tables are generated that may be used 

to look up equivalent CSP-1 plans. The tables may also be used to match up plan performance around two target 

quality levels (po and p1). 

Weed, R.M. “Adjusted Pay Schedules: New Concepts and Provisions,” Transportation Research 

Record 986, 1984, pp. 32-38. 

Shortly after the American Association of State Highway Officials (AASHO) Road Test had furnished a wealth 

of statistical data on pavement construction and performance, highway agencies began to use this data to 

develop ERS based on statistical concepts. These specifications usually included adjusted pay schedules, the 

development of which was sometimes quite arbitrary. More recently, attempts have been made to improve both 

the accuracy with which the pay schedules are established and the fairness with which they are administered. 

The rationale underlying several recent advances in the state-of-the-art is discussed. Included are the use of the 

principal liquidated damages to relate pay reductions to the anticipated monetary loss resulting from 

substandard work, the development of the crediting concept to overcome a basic inequity of many existing pay 

schedules, and the establishment of bonus provisions that provide additional incentive by awarding payment 

slightly in excess of the contract price for superior quality work. 

Weed, R.M. “Analysis and Application of Correlated Compound Probabilities,” Transportation 

Research Record 792, 1981, pp. 49-53. 

Many statistical applications require the calculation of compound probabilities and, frequently, the individual 

probabilities are not independent. The failure to recognize that a correlation exists in cases such as these has 

resulted in numerous errors in published literature. Although an exact analytical solution is not known, 
 259

problems of this type can often be handled effectively by calculating lower and upper bounds for the desired 

probabilities. Bounds for both positively and negatively correlated cases are derived and then applied in the 

analysis of statistical acceptance procedures. The results of several computer simulation tests are presented to 

demonstrate the validity of the theoretically derived results. 

Weed, R.M. “Composite Pay Equations: General Approach,” Transportation Research Record 

1465, 1994, pp. 9-15. 

Highway construction specifications involving the acceptance testing of several different quality characteristics 

are sometime confusing and difficult to administer. A procedure is developed by which multiple quality 

measures may be combined in a rational manner in a single, composite pay equation. This approach is 

scientifically sound and may be applied to almost any construction specification for which a relationship 

between quality and performance is known or can be approximated. An example based on PCC pavement is 

presented to illustrate the practicality of this method. 

Weed, R.M. “Computer-Assisted Random Sampling,” Transportation Research Record 1034, 

1985, pp. 140-152. 

Many State transportation agencies use SQA specifications to govern construction work. A vital step in the 

application of these and other types of specifications is the selection of random samples to obtain a valid 

estimate of the quality received. Random sampling procedures are often tedious and time-consuming, but can be 

considerably simplified with computer assistance, either by using special forms generated by computer or by 

working directly at an interactive terminal. Examples of several applications are presented. 

Weed, R.M. “Development of Air Voids Specification for Bituminous Concrete,” Transportation 

Research Record 1491, 1995, pp. 33-39. 

The New Jersey DOT has been using SQA specifications for various construction products since the 1960s. 

Throughout this period, there has been a continuing process leading to a better understanding of the operation 

and implementation of SQA procedures. The New Jersey DOT specification for air voids in bituminous 

concrete was one of the first to be developed and, as such, is a prime candidate for upgrading. A major change 

is to base the acceptance procedure on the PD rather than on the average of the test values in order to control 

both the level and the variability of the air voids in a statistically efficient way. Doing this required new 

definitions of AQL and rejectable quality level (RQL), and a reexamination of the adjusted pay schedule to be 

applied when other than AQL work is received. It was decided to use a bonus provision for superior quality, an 

approach that has worked well with other recently developed New Jersey DOT specifications. Another change 

is to use a continuous (equation-type) pay schedule to provide a smooth progression of payment as quality 

varies, thus avoiding potential disputes over measurement precision when a quality estimate falls just onto one 

side or the other of a boundary in a stepped pay schedule. The various developmental steps are described, 

including the construction of the OC curve to verify the performance of the specification and the field trials 

leading to its successful implementation. 

Weed, R.M. “Development of Multicharacteristic Acceptance Procedures for Rigid Pavement,” 

Transportation Research Record 885, 1982, pp. 25-36. 

The manner in which the AASHTO design can be used to develop multicharacteristic acceptance procedures for 

rigid pavement is outlined. The AASHTO equation is used to compute both the expected load-bearing capacity 

based on the as-built characteristics of the pavement and the desired load-bearing capacity based on the design 

parameters. The ratio of these two values is then used to determine the appropriate pay adjustment, which may 

be either positive or negative. Sensitivity tests are performed to verify the reliability of this approach and 

computer simulation is used to demonstrate the effectiveness of several different acceptance procedures of this 

type. A secondary study is conducted to determine how the procedure based on the AASHTO equation 

compares with several other methods of treating multiple pay factors to obtain a single overall pay factor. Under 

the assumption that the AASHTO method is the fundamentally correct approach, the method of multiplying 

individual pay factors together is shown to be among the best of the other methods that were tested. 
 260

Weed, R.M. “Method to Establish Pay Schedules for Rigid Pavement,” Transportation Research 

Record 885, 1982, pp. 18-24. 

An equation is derived to compute the appropriate pay factor for any quality level of rigid pavement. The 

measure of quality used in this development is the estimated load-bearing capacity of the pavement, although 

the results may be applied to specifications based on other quality measures. The appropriate pay adjustment is 

considered to be the present worth of any expense or savings expected to occur in the future as the result of a 

departure from the specified level of quality and may be positive or negative. Sensitivity tests demonstrate that 

the method is reliable provided that the input variables are determined with reasonable accuracy. By using input 

values typical of a relatively urbanized area, this procedure indicates that a minimum pay factor of 60 percent is 

appropriate for the poorest quality work and a maximum pay factor of 115 percent is justified for work of truly 

superior quality. Additional factors are cited that, though unquantified, would tend to lower the minimum pay 

factor and raise the maximum pay factor. Finally, pay schedules are developed, the OC curves of which closely 

approximate the theoretically derived relationship. 

Weed, R.M. “OCPLOT: Program to Generate Operating Characteristic Curves for Statistical 

Construction Specifications,” Transportation Research Record 1491, 1995, pp. 18-26. 

The performance of the Nation’s highway system is inexorably linked to the quality of the design and the 

quality of the construction. To control the quality of the construction, transportation agencies have developed 

elaborate QA programs, most of which employ ERS that rely on statistical sampling and acceptance procedures 

to ensure that the work is done in accordance with the plans and specifications. Whether the acceptance 

procedure leads to a simple pass/fail decision or an adjustment in contract price, the proper design of such plans 

is critical to their performance. Poorly conceived plans may be totally ineffective or impractically severe, and 

both extremes have been found in published or proposed national standards. To encourage the design of plans 

that are both effective and fair, an interactive software program has been developed that enables the user to 

construct OC curves to analyze the performance of a wide range of acceptance plans. An example is presented 

to demonstrate the versatility of the program and the ease with which it can be applied. 

Weed, R.M. “Practical Framework for Performance-Related Specifications,” Transportation 

Research Record 1654, 1999, pp. 81-87. 

As highway agencies moved away from the older prescription-type specifications and began to develop ERS 

and PRS, several different statistical measures of quality have been used. These include sample mean, PD or its 

complement, PWL, and AAD. The CI is yet another measure that has been proposed. What has not been 

undertaken during this developmental period is any sort of formal analysis to determine which, if any, of these 

measures accurately reflects the expected performance of the construction products to which they are applied. 

Specialized computer programs were developed to demonstrate the potential weaknesses of the current quality 

measures and to explore alternate approaches that may overcome these weaknesses. It was found that pay 

equations based on the mean and standard deviation computed from the sample can be tailored to closely match 

the value of the constructed product as estimated by life-cycle cost techniques. It is believed that this forms a 

practical starting point for the development of construction acceptance procedures more closely linked to 

quantified performance models. 

Weed, R.M. “Quality Assurance Software for Personal Computer,” Transportation Research 

Record 1544, 1996, pp. 116-124. 

Demonstration Project 89 on Quality Management was created to provide guidance on the use of practical and 

effective procedures to ensure that the level of quality designed into the plans and specifications is accurately 

achieved in the constructed product. One part of this effort is the distribution of a software package consisting 

of several interactive programs developed for use on a personal computer. These extremely user-friendly 

programs make it possible to analyze both pass/fail and pay-adjustment acceptance procedures, construct OC 

curves, plot control charts, experiment with computer simulation, perform statistical comparisons of data sets, 

demonstrate the unreliability of decisions based on a single test result, and explore the effectiveness of stratified 

random sampling. This comprehensive software package provides highway engineers with the necessary tools 

for learning why some statistical procedures are inherently superior to others and how to incorporate this 

knowledge into fair and effective construction specifications. 
 261

Weed, R.M. “A Rational Method for Relating As-Built Quality to Pavement Performance and 

Value,” Transportation Research Record 1632, 1998, pp. 31-39. 

Highway construction specifications routinely use adjusted-payment provisions to award payment in proportion 

to the level of quality received. Work that is defined as acceptable is eligible for 100-percent payment, whereas 

work that fails to meet the desired quality level, but is not sufficiently deficient to warrant removal and 

replacement, typically receives some degree of pay reduction. There is not yet consistency in the practices 

regarding the magnitude of the pay adjustment that is judged as being appropriate for varying levels of as-built 

quality. There needs to be a method for relating as-built quality to expected performance, which can then be 

related to value by engineering economics procedures. The extension and refinement of earlier work are 

described and an example structured around one of the performance relationships in the AASHTO Guide for 

Design of Pavement Structures is provided. 

Weed, R.M. “Revision of a Flawed Acceptance Standard,” Transportation Research Record 

1056, 1986, pp. 21-35. 

A major revision of AASHTO Standard R9-84, Acceptance Sampling Plans for Highway Construction, has 

been completed. The primary goals were to correct a major conceptual error and to reduce the level of 

complexity. In this paper, the flaws in the original version are discussed, the basic changes that were made are 

described, and a significant addition to the new standard is presented. This addition is OC tables that enable the 

user to quickly and easily select acceptance plans that will provide the desired degree of QA. Computer 

simulation is used to demonstrate that single-limit variable OC curves are sufficiently accurate for most doublelimit applications. Two examples are included to illustrate the use of the revised standard. 

Weed, R.M. “Stratified Random Sampling From a Discrete Population,” Transportation 

Research Record 792, 1981, pp. 41-44. 

In the development of statistical acceptance procedures for products whose quantity is measured on a 

continuous scale by using units such as length, area, volume, or weight, QA engineers usually specify stratified 

random sampling plans to ensure a more uniform coverage of the product than is often achieved by pure random 

sampling. Stratified plans divide the total quantity of the product into an appropriate number of equal-sized 

sublots and require that a single random sample be taken from each. Not only is it desirable to develop an 

equivalent procedure for products that are measured in discrete units, but, in many cases, such a procedure will 

prove to be more convenient for continuous products that are delivered in discrete units, such as batches or 

truckloads. However, the development of such a procedure is not as straightforward as might be expected. The 

weaknesses of some of the more obvious approaches are discussed and then a method is presented that achieves 

the desired result. 

Weed, R.M. and Strawderman, W.E. “Method to Exclude the Effect of Testing Error When 

Estimating the Percentage Defective of a Continuous Normal Population,” Transportation 

Research Record 792, 1981, pp. 45-49. 

The quality of a product is often characterized by the percentage of the population that falls outside of specific 

limits. Although established methods for estimating PD are accurate as far as the overall distribution of test 

results is concerned, part of the variability of this distribution is caused by the presence of testing errors that 

cause the PD of the product itself to be overestimated. A method has been developed to overcome this problem 

and computer simulation has been used to demonstrate that it is effective for situations in which the testing error 

is no larger than about one-half of the variability associated with the product. The results of several 

unsuccessful attempts to improve on this technique are also presented and described briefly. 

Weingarten, H. “Confidence Intervals for the Percent Nonconforming Based on Variables Data,” 

Journal of Quality Technology, Vol. 14, No. 4, 1982, pp. 207-210. 

Sampling inspection by variables for percent nonconforming, such as with the use of MIL-STD-414, also 

provides the data for estimating the population percent nonconforming. A method is provided for obtaining a 

confidence interval for the population value by means of an OC curve. This OC curve is achieved from the data 

and is not that of the original sampling plan. 
 262

White, T.D. and Brown, E.R. “Statistical Quality Control Procedures for Airfield Pavement 

Materials and Construction,” Transportation Research Record 652, 1977, pp. 36-42. 

The interaction of materials, construction, and the environment makes projections of pavement performance 

difficult. To increase the confidence with which these projections can be made, a simplified statistical QC plan 

for airfield pavement materials and construction was developed. Test results based on a specific number of 

samples can be extended with the desired confidence limits based on probability theory to evaluate an entire lot 

of material. This statistical QC plan evaluates pavement quality much better than do conventional 

specifications. It also describes how to handle materials or construction of borderline acceptability. When test 

results indicate that the material or construction meets the desired quality, it is accepted at 100-percent payment. 

If it is below the minimum requirements, it is rejected. When material or construction is found to be below the 

desired quality, but above the minimum requirement, it is accepted at a reduced price. The evaluation is done on 

a daily basis so that the engineer and the contractor know the level of acceptability as the project progresses. 

Willenbrock, J.H. and Kopac, P.A. “Development of Price-Adjustment Systems for Statistically 

Based Highway Construction Specifications,” Transportation Research Record 652, 1977, 

pp. 52-58. 

A methodology is presented that can be used to develop price-adjustment systems for use in statistically based 

highway construction specifications. Three approaches are proposed for the development of the priceadjustment system: (1) the serviceability approach, (2) the cost of production approach, and (3) the OC curve 

approach. The three approaches are discussed and compared, and their most appropriate applications are 

recommended. A fourth approach, the cost of QC approach, is also discussed, but is not fully developed because 

of the limited cost data available. 
 263

APPENDIX B: SUMMARY OF AGENCY HMAC SPECIFICATIONS 

RECEIVED

Alaska

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of the results. The frequency of the testing is determined by the 

contractor, but must be approved by the engineer prior to construction. 

Mix design verification is evaluated for the following properties: Marshall stability, Marshall flow, percent voids in 

mix, dust-to-asphalt ratio, gradation, VFA, density, and VMA. 

Quality Assurance 

The engineer is responsible for all assurance testing. Assurance testing is conducted for the following 

characteristics: gradation, in-place density, and asphalt content. The State’s assurance tests cannot be used for QC. 

The results of the tests will be made available to the contractor within 2 working days. 

Acceptance 

Acceptance is based on the results of the assurance tests. Acceptance is based on a lot size of 5000 tons (4535 Mg), 

with tests conducted on 500-ton (454 Mg) sublots. Acceptance of the material and any associated pay factors are 

determined using quality-level analysis. 

Pay Factors 

Pay factors are assigned for gradation, in-place density, and asphalt content. For density, the density pay factor 

(DPF) is determined by calculating the quality level of the lot and then selecting a pay factor from a table according 

to the sample size. The pay factor for gradation and asphalt content (composite pay factor (CPF)) is determined by 

the following equation: 

 CPF = [f19mm(PF19mm) + f12.5mm(PF12.5mm) + ... fac(PFac)] / ∑f (37) 

where: f factors are determined from the Weight Factors table below 

After the DPF and CPF are known, the pay adjustment is determined by the following: 

 Price Adjustment = [(CPF or DPF)* – 1.00] × (Mg in lot) × (PAB) (38) 

where: PAB = price-adjustment base (equal to $33.00/Mg) 
 264

Weight Factors 

Table 58. Alaska weight factors. 

Gradation (mm) f Factor 

19.0 4 

12.5 5 

9.5 5 

4.75 4 

2.36 4 

1.18 4 

0.600 5 

0.300 5 

0.150 4 

0.075 20 

Asphalt Content 40 

QC/QA Tests for Alaska 

Table 59. Alaska QC/QA tests. 

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Assurance 

Testing 

Frequency by 

Alaska DOT 

Acceptance by 

Alaska DOT 

Gradation 

(extracted) Behind paver Determined by contractor One per sublot Based on assurance testing 

Asphalt content Behind paver Determined by 

contractor One per sublot Based on assurance testing 

Density Roadway Determined by 

contractor One per sublot Based on assurance testing 
 265

Arkansas 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. The sampling location and testing frequency are determined 

by the contractor with the exception of gradation (one per 750 tons (680 Mg) minimum), and must be sufficient for 

control of the mix. 

The contractor is responsible for QC for the following properties: gradation, asphalt content, air voids, VMA, 

Marshall stability, Maximum theoretical density, in-place density, and water sensitivity. 

Mix design verification is evaluated for the following properties: gradation, Marshall stability, Marshall flow, air 

voids, VMA, density, asphalt content, dust-to-asphalt ratio, and water sensitivity. 

Quality Assurance 

The Arkansas State Highway and Transportation Department (AHDT) is responsible for all assurance sampling and 

testing. Testing is based on a lot size of 3000 tons (2720 Mg) with 750-ton (680-Mg) sublots. Assurance testing is 

conducted on gradation, asphalt content, air voids, VMA, Marshall stability, and in-place density. 

Acceptance 

The contractor’s control test cannot be used for acceptance. Under the supervision of the engineer, the contractor 

must conduct acceptance testing on the materials. The sampling locations will be independent of the QC locations. 

Acceptance of the material is based on verification of the contractor’s acceptance testing by the AHDT’s assurance 

tests. Acceptance and adjustment is by lot. Lot size is 3000 tons (2720 Mg) with 750-ton (680 Mg) sublots. In 

addition, the engineer will check the smoothness of the pavement by using a rolling straightedge. Variations shall 

not exceed 5 millimeters (mm) for binder courses and 3 mm for surface courses. At least one pass shall be made for 

the full length of each lane. If the surface test is deficient, the surface must be corrected at the contractor’s expense. 

Pay Factors 

Pay factors are assigned for in-place density, asphalt content, Marshall stability, air voids, and VMA. The pay 

factors for each property are determined according to the following schedule and the total reduction for the lot is the 

sum of the reductions for each property of the lot. 

Asphalt Binder: The contract price of the entire lot will be reduced by 12 percent for each deviation outside the 

compliance limits, up to a maximum of three deviations. One deviation is 0.1 percentage point. 

Stability: The contract price of the lot will be reduced by 10 percent for each deviation below the compliance limits, 

up to a maximum of five deviations. One deviation is 0.2 kilonewtons (kN) (45 pounds force (lbf)). 

Air Voids: The contract price of the lot will be reduced by 10 percent for each deviation outside the compliance 

limits, up to a maximum of five deviations. One deviation is 0.1 percentage point. 

VMA: The contract price of the lot will be reduced by 4 percent for each deviation outside the compliance limits, up 

to a maximum of 10 deviations. One deviation is 0.1 percentage point. 

Density: The contract price of the lot will be reduced by 4 percent for each deviation outside the compliance limits, 

up to a maximum of 10 deviations. One deviation is 0.1 percentage points. 
 266

QC/QA Tests for Arkansas 

Table 60. Arkansas QC/QA tests. 

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Assurance 

Testing 

Frequency by 

AHDT 

Acceptance 

Testing 

Frequency by 

Contractor 

Gradation 

(extracted) Behind paver One per 750 tons One per 750 tons One per 750 tons 

Asphalt content Behind paver Contractor’s 

discretion One per 3000 tons One per sublot 

Air voids Behind paver Contractor’s 

discretion One per 3000 tons One per sublot 

VMA Behind paver Contractor’s 

discretion One per 3000 tons One per sublot 

Maximum 

theoretical density 

Laboratory 

sample 

Contractor’s 

discretion NA NA 

In-place density Behind paver Contractor’s 

discretion One per 3000 tons One per sublot 

Water sensitivity Laboratory 

sample 

Contractor’s 

discretion NA NA 

Smoothness Roadway Contractor’s 

discretion NA 

By engineer (one 

pass for full length of 

each lane) 

Marshall stability Behind paver Contractor’s 

discretion One per 3000 tons One per sublot 

NA = Not applicable 

1 ton = 0.907 Mg 
 267

Colorado 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

All QC tests must be independent of assurance and acceptance tests. Acceptable testing standards in order of 

preference include Colorado procedures, AASHTO, and ASTM. 

Quality Assurance 

Colorado DOT ensures QA by conducting check tests to verify that the personnel and equipment conducting the QC 

tests and acceptance tests are within acceptable tolerances. If the check test is outside the allowable differences, 

production will stop and the problem will be identified and corrected. 

Acceptance 

Acceptance testing is the responsibility of the Colorado DOT. A minimum of 5 asphalt content, 3 gradation, and 10 

density tests are required for each project. The contractor, under the direction of the engineer, obtains acceptance 

test samples. Acceptance is based on the State’s tests only. 

Pay Factors 

Pay factors are assigned for gradation, density, and asphalt content. To determine the pay factors, quality-level 

analysis is used to obtain a quality level for a process with three or more test results. When the process has been 

completed, the number of random samples (Pn) that it contains will determine the computation of the element pay 

factor, based on the table and equation below. When the Pn is 3 to 9, or greater than 200, the element pay factor is 

computed using the equations designated in the table below. When the Pn is equal to or greater than 10, and less 

than 201, the element pay factor is computed using the following equation: 

Pay Factor (PF) = [(PF1 + PF2) / 2] + [(PF2 + PF3) / 2 – (PF1 + PF2) / 2] × [(Pn2 – Pnx) / (Pn2 – Pn3)] 

(39) 

where (when referring to the table below): 

PF1 = PF determined at the next lowest Pn equation using the process quality level (QL) 

PF2 = PF determined using the Pn equation shown for the process QL 

PF3 = PF determined at the next highest Pn equation using the process QL 

Pn2 = lowest Pn in the spread of values listed for the process Pn equation 

Pn3 = lowest Pn in the spread of values listed for the next highest Pn equation 

Pnx = actual number of test values in the process 
 268

Table 61. Colorado pay factor equations. 

Pn 

When the Pn is 3 to 9, or greater than 200, use the 

designated equation below to calculate the pay 

factor. 

 

When the Pn is equal to or greater than 10, and less 

than 201, use formula 1. 

Maximum Pay 

Factor 

 3 PF = 0.31177 + 1.57878(QL/100) – 0.84862(QL/100)2 1.025 

 4 PF = 0.27890 + 1.15471(QL/100) – 0.73553(QL/100)2 1.030 

 5 PF = 0.25529 + 1.48268(QL/100) – 0.67759(QL/100)2 1.030 

 6 PF = 0.19468 + 1.56729(QL/100) – 0.70239(QL/100)2 1.035 

 7 PF = 0.16709 + 1.58245(QL/100) – 0.68705(QL/100)2 1.035 

 8 PF = 0.16394 + 1.55070(QL/100) – 0.65270(QL/100)2 1.040 

 9 PF = 0.11412 + 1.63532(QL/100) – 0.68786(QL/100)2 1.040 

10-11 PF = 0.15344 + 1.50104(QL/100) – 0.58896(QL/100)2 1.045 

12-14 PF = 0.07278 + 1.62485(QL/100) – 0.65033(QL/100)2 1.045 

15-18 PF = 0.07826 + 1.55649(QL/100) – 0.56616(QL/100)2 1.050 

19-25 PF = 0.09907 + 1.43088(QL/100) – 0.45550(QL/100)2 1.050 

26-37 PF = 0.07373 + 1.41851(QL/100) – 0.41777(QL/100)2 1.055 

38-69 PF = 0.10586 + 1.26473(QL/100) – 0.29660(QL/100)2 1.055 

70-200 PF = 0.21611 + 0.86111(QL/100) 1.060 

≥ 201 PF = 0.15221 + 0.92171(QL/100) 1.060 

For pay estimates, each individual element will have the average pay factor (PFa), weighted by the quantities as 

follows: 

 PFa = [M1(PF1) + M2(PF2) + ... Mj(PFj)] / ∑M (40) 

where: Mj = quantity represented by the process 

 PFj = process pay factor 

 ∑M = total quantity 

QC/QA Tests for Colorado 

Table 62. Colorado QC/QA tests. 

Test Sample Location 

QC Testing 

Frequency by 

Contractor 

Acceptance Testing 

Frequency by 

Colorado DOT 

Check Testing 

Frequency by 

Colorado DOT 

Asphalt content Behind paver One per 500 metric 

tons 

One per 1000 metric 

tons 

One per 10,000 metric 

tons 

Dry gradation: #4, #8, 

#200 Cold feed One per day One per 2000 metric tons 

One per 20,000 metric 

tons 

In-place density Roadway One per 500 metric 

tons One per 500 metric tons One per 5000 metric tons
 269

Idaho 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation (3/4 inch (19 mm), 3/8 inch (9.5 mm), 

#8, and #200), sand equivalent, fracture count, asphalt content, and density. The lot size is 1 day’s production, 

represented by at least four tests. If the day’s production is represented by fewer than four tests, the tests will be 

combined with the following day’s production to form one lot. One test must be conducted for every 750 tons (680 

Mg) of production (never less than once per day). Asphalt content is the exception to this rule, with testing 

conducted as needed to control production. 

Quality Assurance 

Idaho DOT is responsible for all assurance testing. Assurance testing is conducted on gradation, sand equivalent, 

asphalt content, density, and fracture count. The QA testing frequency is two per lot, not to exceed two per day. 

Acceptance 

Acceptance testing is the responsibility of the contractor for the properties of gradation, sand equivalent, and 

fracture count. These acceptance tests are not to be used as part of the QC testing. The State is responsible for the 

acceptance testing of asphalt content and density. The lot size is 1 day’s production, represented by at least four 

tests. If the day’s production is represented by fewer than four tests, the tests will be combined with the following 

day’s production to form one lot. One test must be conducted for every 750 tons (680 Mg) of production (never less 

than one per day). 

Pay Factors 

Pay factors are assigned for gradation, in-place density, and asphalt content. Quality-level analysis is used to 

determine the PWL for the lot. If any one pay factor for asphalt content, density, or gradation falls below 0.85, the 

pay factor for all quality characteristics represented by the same quantity will be determined by the lowest pay 

factor. 

Once the individual gradation pay factors for the lots are determined, they are averaged on a weighted basis, 

considering the amount of material represented by each lot. This average gradation pay factor is then multiplied by 

0.3 to obtain the composite pay factor for plant-mix aggregate (CPFPM). 

The individual pay factors for asphalt content are also averaged on a weighted basis to obtain the average asphalt 

content pay factor. This factor is then multiplied by 0.3 to obtain the composite pay factor for asphalt content 

(CPFAC). 

The same procedure is then performed to obtain the composite pay factor for density (CPFDens). The bonus to be 

paid or deducted is determined using the following equation: 

 B = (A)[(CPFPM + CPFAC + CPFDens) – 1](Q) (41) 

where: B = total bonus 

A = unit bid price 

Q = total quantity of plant mix accepted 
 270

QC/QA Tests for Idaho 

Table 63. Idaho QC/QA tests. 

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Assurance or 

Verification Testing 

Frequency by State 

Acceptance 

Gradation (washed) 

(3⁄4 inch, 3⁄8 inch, #8, 

#200) 

Crusher for QC, 

cold feed for 

acceptance 

One per 750 tons, 

but not less than one 

per day 

Two per lot, not to 

exceed two per day 

By contractor: one per 

750 tons, but not less 

than one per day 

Asphalt content Hot plant or 

roadway 

As needed to control 

operation 

Two per lot, not to 

exceed two per day 

By State: one per 750 

tons, but not less than 

one per day 

Sand equivalent 

Crusher for QC, 

cold feed for 

acceptance 

One per 750 tons, 

but not less than one 

per day 

Two per lot, not to 

exceed two per day 

By contractor: one per 

750 tons, but not less 

than one per day 

Fracture count 

Crusher for QC, 

cold feed for 

acceptance 

One per 750 tons, 

but not less than one 

per day 

Two per lot, not to 

exceed two per day 

By contractor: one per 

750 tons, but not less 

than one per day 

Density Roadway 

One per 750 tons, 

but not less than one 

per day 

Two per lot, not to 

exceed two per day 

By State: one per 750 

tons, but not less than 

one per day 

1 inch = 25.4 mm, 1 ton = 0.907 Mg 
 271

Illinois 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation (1/2 inch (13 mm), #4, #8, #30, and 

#200), BSG, MSG, air voids, and density. The contractor is responsible for providing split samples to the engineer 

for assurance testing. 

Quality Assurance 

The engineer is responsible for all assurance testing and acceptance of the material. The assurance tests are 

conducted on split samples taken by the contractor for QC testing. Additionally, the engineer will witness the 

sampling and splitting of these samples a minimum of twice a month and will retain these samples for assurance 

testing. 

Acceptance 

The contractor’s control tests are used for acceptance if they are verified by the engineer’s assurance tests. The 

testing frequencies vary, depending on the property being measured. Acceptance is based on the validation of the 

QC program, reviewing the contractor’s control charts, and by the assurance tests for voids and density. Assurance 

testing is conducted on gradation (1/2 inch (13 mm), #4, #8, #30, and #200), BSG, MSG, air voids, and density. 

Pay Factors 

Pay factors are assigned for thickness and smoothness. Smoothness is determined using a California profilograph. 

The pay factor for smoothness is presented in the price-adjustment table for smoothness. The lot size for thickness 

testing shall be 1500 meters (m) (single lane). Each lot is divided into 10 sublots. One core will be taken from each 

sublot. Any pavement found to be less than 10-percent deficient may remain in place with no additional action 

required. After analyzing the cores, the PWL for the lot is determined. The following equations allow for the 

calculation of a bonus or penalty for thickness. 

Thickness: Pay Factor Percentage = 55 + 0.5PWL (42) 

 Total Payment = PF [CUP(SQMPAVT – DEFPAVT)] (43) 

where: PF = total pay factor 

CUP = contract unit price 

SQMPAVT = square meters of pavement placed 

DEFPAVT = square meters of deficient pavement 
 272

Smoothness Pay Factors 

Table 64. Illinois smoothness pay factors. 

Profile Index for Entire Project (mm/km) Percent of Unit Bid Price 

≤ 3 103 

> 3-6 102 

> 6-8 101 

Profile Index for 160-m Section (mm/km) Percent of Unit Bid Price 

 > 8-160 100 

> 160-175 98 

> 175-190 96 

> 190-205 94 

> 205-220 92 

> 220-235 90 

> 235 Corrective work required 

QC/QA Tests for Illinois 

Table 65. Illinois QC/QA tests. 

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Assurance 

Testing 

Frequency by 

Illinois DOT 

Acceptance by 

Illinois DOT 

Dry gradation (1⁄2 inch, #4, 

#8, #30, #200) Cold Feed One per 1⁄2 day ≥ 10% of QC Based on validation of QC 

tests by assurance tests 

MSG Plant (hot mix) 

One per 1⁄2 day for first 

2 days, then one per 

day 

≥ 20% of QC Based on validation of QC 

tests by assurance tests 

BSG Plant (hot mix) 

One per 1⁄2 day for first 

2 days, then one per 

day 

≥ 20% of QC Based on validation of QC 

tests by assurance tests 

Air voids Plant (hot mix) 

One per 1⁄2 day for first 

2 days, then one per 

day 

≥ 20% of QC Based on validation of QC 

tests by assurance tests 

Thickness Roadway One per sublot NA Based on contractor’s tests 

Asphalt content Plant (hot mix) One per 1⁄2 day ≥ 20% of QC Based on validation of QC 

tests by assurance tests 

Smoothness Roadway 

One pass per lane per 

1 day’s paving 

(1000 ft) 

NA Based on contractor’s tests 

Density Roadway One per 1⁄2 mi ≥ 20% of QC Based on validation of QC 

tests by assurance tests 

1 inch = 25.4 mm, 1 foot (ft) = 0.305 meter (m), 1 mile (mi) = 1.61 kilometers (km) 
 273

Iowa 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: laboratory density, laboratory voids, asphalt 

content, gradation, MSG, and field density. The contractor must provide split samples to the engineer for assurance 

testing. One of each control test must be conducted on the first 500 tons (450 Mg) and the remaining day’s 

production is divided into three equal sublots and tested once per sublot. 

Quality Assurance 

Iowa DOT is responsible for all assurance testing and acceptance of the material. Assurance testing frequency is at 

the discretion of the engineer, but must be at least 10 percent of the contractor’s QC testing, except for density and 

thickness, which require a minimum of seven cores per lot. Properties tested include gradation, laboratory density, 

laboratory voids, asphalt content, MSG, air voids, thickness, and in-place density. 

Acceptance 

The contractor’s control test will be used for acceptance when the test results are verified by Iowa DOT’s assurance 

tests. 

Pay Factors 

Pay factors are assigned for in-place density and thickness, based on a quality index for these two properties. The 

pay factors are based on square yards per lot. The pay factors for density are determined using the following tables: 

Pay Factors for Density 

Table 66. Iowa pay factors for density. 

Quality Index Percentage of Full Payment 

0.73 100 

0.40-0.72 95 

0.00-0.39 85 

All negative values 75 maximum or remove 

Pay Factors for Thickness 

Table 67. Iowa pay factors for thickness. 

Quality Index Percentage of Payment 

Previously Adjusted for Density 

0.35+ 100 

0.14-0.34 95 

0.00-0.13 85 

All negative values 75 maximum or remove 
 274

QC/QA Tests for Iowa 

Table 68. Iowa QC/QA tests. 

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Assurance Testing 

Frequency by 

Iowa DOT 

Acceptance by 

Iowa DOT 

Gradation 

(extracted) Behind paver One per day As needed for QC verification 

Based on QC tests verified 

by assurance tests 

Laboratory 

density 

Plant mix 

(laboratory) One per sublot As needed for QC verification 

Based on QC tests verified 

by assurance tests 

Laboratory 

voids 

Plant mix 

(laboratory) One per sublot As needed for QC verification 

Based on QC tests verified 

by assurance tests 

Asphalt content Behind paver One per sublot As needed for QC 

verification 

Based on QC tests verified 

by assurance tests 

MSG Plant mix 

(laboratory) One per sublot As needed for QC verification 

Based on QC tests verified 

by assurance tests 

Thickness Roadway One per sublot Seven cores per lot Based on QC tests verified 

by assurance tests 

Air voids Roadway One per sublot As needed for QC 

verification 

Based on QC tests verified 

by assurance tests 

In-place density Roadway One per sublot Seven cores per lot Based on QC tests verified 

by assurance tests 
 275

Maryland 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer 30 days prior to construction, and providing qualified personnel and equipment to conduct 

QC testing. The contractor must allow the engineer access to the laboratory to observe any and all testing 

procedures, calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation, density, asphalt content, MSG, VTM, 

VMA, Marshall flow smoothness, and Marshall stability. The contractor must provide split samples to the engineer 

for assurance testing. 

Quality Assurance 

Maryland DOT is responsible for all assurance testing. Assurance testing is conducted on split samples provided by 

the contractor. Properties measured include gradation, density, asphalt content, MSG, VTM, VMA, Marshall flow 

smoothness, and Marshall stability. Additional assurance samples may be taken and tested if deemed necessary by 

the engineer. The lot size for all properties is 3000 tons (2720 Mg) with 1000-ton (910-Mg) sublots, except for 

density, which is 1000 tons (910 Mg) with 200-ton (180-Mg) sublots. 

Acceptance 

Independent tests on split samples, observing contractor’s testing, monitoring control charts, additional sampling as 

deemed necessary by the engineer, and monitoring the contractor’s adherence to the QC plan provides acceptance. 

The lot size for all properties is 3000 tons (2720 Mg) with 1000-ton (910-Mg) sublots, except for density, which is 

1000 tons (910 Mg) with 200-ton (180 Mg) sublots. 

Pay Factors 

Pay factors are assigned for gradation, in-place density, asphalt content, and smoothness. A pay factor for density 

and a composite pay factor for binder content and gradation is computed using quality-level analysis—the standard 

deviation method to determine the PWL and the associated pay factor. The lot payment is then determined using the 

following equation: 

 Lot Payment = (CUP)(PFDens)(CPFGAC)(tonnage) (44) 

where: CUP = contract unit price 

 PFDens = density pay factor 

 CPFGAC = composite pay factor for gradation and density

Smoothness is measured by a computerized profilograph that takes measurements in 0.1-mi (0.16-km) increments 

and must conform to the following: 

1. Single-lift construction with a wedge and leveling on tangent alignment and pavement on horizontal curves 

having a centerline radius of 2000 ft (610 m) or more shall have a profile index of ≤ 10 inches/mi (158 

mm/km). 

2. Multiple-lift construction, with or without a wedge and leveling on tangent alignment and pavement on 

horizontal curves having a centerline radius of 2000 ft (610 m) or more shall have a profile index of ≤7 

inches/mi (110 mm/km). 

3. Either of the above on horizontal curves having a centerline radius of curve of 1000 ft (305 m) or more, but 

less than 2000 ft (610 m), and pavement within the superelevation transition of such curves shall have a 

profile index of ≤ 12 inches/mi (190 mm/km). 

4. Single-lift construction without wedge and leveling placed on a tangent alignment pavement after grinding 

shall have a profile index of ≤ 12 inches/mi (190 mm/km). 

5. Single-lift construction without wedge and leveling placed on pavement shall have a profile index of ≤ 15 

inches/mi (240 mm/km). 
 276

Areas that are accepted at a reduced price for a profile index deficiency will be adjusted by the factors shown in the 

following table in conformance with the procedures specified in items 1 through 5 above: 

Table 69. Maryland profile index adjustment (normal projects).

Profile Index Exceeds Specification 

(inches/mi per 0.1-mi section) Percentage of Payment Unit Bid Price 

0.1-1.0 98 

1.1-2.0 96 

2.1-3.0 94 

3.1-4.0 92 

4.1-5.0 90 

 ≥ 5.1 Corrective work required 

1 inch/mi = 16 mm/km, 1 mi = 1.61 km 

This adjustment is made at the end of paving and is based on the overall average profile index for the project. 

Areas that are accepted on incentive projects shall use the following adjustment pay factor schedule and shall be in 

conformance with items 1 through 5 above.

Table 70. Maryland profile index adjustment (incentive projects). 

Final Average Profile Index 

 (5) (3,4) (1) (2) 

Percentage of Pavement Unit Price 

0.0-1.9 0.0-1.9 0.0-1.9 0.0-1.9 105 

2.0-2.9 2.0-2.9 2.0-2.9 2.0-2.9 104 

3.0-3.9 3.0-3.9 3.0-3.9 3.0-3.9 103 

4.0-15.0 4.0-10.0 4.0-10.0 4.0-7.0 100 

QC/QA Tests for Maryland 

Table 71. Maryland QC/QA tests.

Test Sampling Location

QC Testing 

Frequency by 

Contractor 

Assurance 

Testing Frequency 

by Maryland DOT 

Acceptance by 

Maryland DOT 

Density Roadway Five cores per 1000-ton 

lot 

Three cores per 6000 

tons 

Based on verification 

of QC by QA 

Gradation Cold feed One per 1000 tons or 

per day One per lot Based on verification 

of QC by QA 

Asphalt content Plant (hot mix) or 

behind paver 

One per 1000 tons or 

per day One per lot Based on verification of QC by QA 

MSG Plant (hot mix) or 

behind paver 

One per 1000 tons or 

per day One per lot NA 

VTM Plant (hot mix) or 

behind paver 

One per 1000 tons or 

per day One per lot NA 

VMA Plant (hot mix) or 

behind paver 

One per 1000 tons or 

per day One per lot NA 

Flow Plant (hot mix) or 

behind paver 

One per 1000 tons or 

per day One per lot NA 

Stability Plant (hot mix) or 

behind paver 

One per 1000 tons or 

per day One per lot NA 

Smoothness Roadway 0.1-mi segments Randomly at 

engineer’s discretion 

Based on verification 

of QC by QA 

NA = Not applicable, 1 mi = 1.61 km, 1 ton = 0.907 Mg 
 277

Michigan 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation (#4, #30, and #200), air voids, VMA, 

TMD, binder content, and crushed particle content. 

 

Quality Assurance 

Michigan DOT is responsible for all assurance testing and acceptance of the material. QA testing on loose mix and 

on compacted mix will be done at the field laboratory. 

Acceptance 

Acceptance is based on verification of the contractor’s QC tests. If the engineer’s tests and the contractor’s tests are 

within allowable tolerances for the job mix formula, the material is considered to be acceptable. Testing is 

conducted on equal sublot sizes with a maximum size of 1200 tons (1090 Mg). Five consecutive sublots constitute a 

lot. 

Pay Factors 

Pay factors are assigned for gradation, TMD, binder content, crushed particle count, density, and VMA. The criteria 

for each of the pay factors are not available at this time. However, the negative price adjustments for the bituminous 

mixture contract unit price is cumulative according to the following equation: 

 Price Adjustment = (CUP)(Pavement density + Mixture properties + Failure to suspend operations) 

(45) 

where: CUP = contract unit price 

 Mixture properties = gradation, TMD, binder content, crushed particle count, and VMA 

QC/QA Tests for Michigan 

Table 72. Michigan QC/QA tests. 

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Assurance 

Testing Frequency 

by Michigan DOT 

Acceptance by 

Michigan DOT 

TMD Plant One per sublot One per lot Based on acceptable 

tolerances with JMF 

Gradation (dry): #4, 

#30, #200 Cold feed One per day One per lot Based on acceptable tolerances with JMF 

Crushed particle 

content Cold feed One per day One per lot Based on acceptable tolerances with JMF 

Binder content Plant One per sublot One per lot Based on acceptable 

tolerances with JMF 

Air voids Plant One per sublot One per lot Based on acceptable 

tolerances with JMF 

VMA Plant One per sublot One per lot Based on acceptable 

tolerances with JMF 

In-place density Roadway Three per sublot One per lot Based on acceptable 

tolerances with JMF 
 278

Minnesota 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation, asphalt content, Marshall BSG, MSG, 

air voids, VMA, TSR, crushed count, and moisture content. Testing frequency is determined by taking the day’s 

planned production, dividing by 1000, then rounding that figure to the next whole number. 

Quality Assurance 

Minnesota DOT is responsible for all assurance testing and acceptance of the material. At least one set of assurance 

tests is run on one set of production tests per day. Assurance is accomplished by testing, observing the contractor’s 

QC sampling and testing, taking additional samples, and monitoring QC control charts. 

Acceptance 

The contractor’s control test cannot be used for acceptance. Acceptance is based on Minnesota DOT’s assurance 

tests. Pavement smoothness is the exception. The contractor’s testing for smoothness is observed by the engineer 

and is used for acceptance and pay adjustments. 

Pay Factors 

Pay factors are assigned for gradation, VMA, asphalt content, density, profile index, and air voids. Pay factors for 

gradation, VMA, asphalt content, and air voids are determined from the following table, with the lowest single 

payment applied: 

Payment Schedule 

Table 73. Minnesota payment schedule. 

Item Percentage of Payment 

Gradation 90 

VMA 85 

Asphalt content 85 

Air voids (individual) 70 

The pay factor for in-place density is determined on a lot-by-lot basis, with the number of lots determined by the 

following table: 

Determination of Lots for Density 

Table 74. Minnesota determination of lots for density. 

Daily Production (tons) Lots 

 0-300 1 

 301-600 2 

 601-1000 3 

1001-1600 4 

1601-3600 5 

3601-5000 6 

5000+ 7 

1 ton = 0.907 Mg 

When the density of a lot of compacted mixture is less than the specified minimum, payment will be made at an 

adjusted price as specified in the following table: 
 279

Adjusted Payment Schedule for Maximum Density (Disincentive) 

Table 75. Minnesota adjusted payment schedule for maximum density (disincentive). 

Percentage of Density Below 

Specified Minimum 

Payment Factor (percentage of 

contract price)

0.1-1.0 inclusive 98 

1.1-1.5 inclusive 95 

1.6-2.0 inclusive 91 

2.1-2.5 inclusive 85 

2.6-3.0 inclusive 70 

 > 3.0 Remove and replace 

An incentive is available for density exceeding the specified minimum according to the following table: 

Adjusted Payment Schedule for Maximum Density (Incentive) 

Table 76. Minnesota adjusted payment schedule for maximum density (incentive). 

Percentage of Density Above 

Specified Minimum 

Pay Factor (percentage of 

contract price) 

0.1-1.0 inclusive 100 

1.1-1.5 inclusive 102 

 > 1.5 104 

The profile index is determined using a California-type profilograph. Profiles will be made for segments of 0.1 km. 

The following table determines the pay factor for smoothness: 

Payment Schedule for Smoothness 

Table 77. Minnesota payment schedule for smoothness. 

mm per km per 0.1-km segment Dollars per Segment 

 0-13 110 

 14-25 90 

 26-38 65 

 39-79 0 

 80-92 -90 

 93-105 -180 

106-118 -360 

 > 118 Corrective action 
 280

QC/QA Tests for Minnesota 

Table 78. Minnesota QC/QA tests. 

Test Sampling 

Location 

QC Testing by 

Contractor 

Assurance Testing 

Frequency by 

Minnesota DOT 

Acceptance by 

Minnesota DOT 

Gradation 

(extracted) Truck or paver One per 2200 tons One per 2200 tons Based on assurance test results 

Asphalt content Truck or paver One per 1000 tons One per day Based on assurance test 

results 

VMA Truck or paver Two per day One per day Based on assurance test 

results 

TSR Truck or paver One per 11,000 

tons Engineer’s discretion Based on assurance test results 

Density Roadway Three per lot One per day Based on assurance test 

results 

Air voids Truck or paver Two per day One per day Based on assurance test 

results 

Crushed count Cold feed 

Two per day for 2 

days then one per 

day 

One per day Based on assurance test 

results 

MSG Truck or paver Two per day One per day Based on assurance test 

results 

Smoothness Roadway One pass per lane 

segment (0.1 km) Observation of QC Based on QC testing as observed by the engineer 

BSG Truck or paver Two per day One per day Based on assurance test 

results 

1 ton = 0.907 Mg 
 281

Montana 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation, density, asphalt content, and fractured 

faces. 

Quality Assurance 

Montana DOT is responsible for all assurance sampling and testing. Assurance testing is conducted on the properties 

evaluated for QC. Assurance is conducted on a sublot size of 600 tons (540 Mg). 

Acceptance 

The contractor’s control test cannot be used for acceptance. Acceptance is determined by evaluating the QA tests for 

compliance with the JMF tolerances for gradation, in-place density, and asphalt content. Quality incentives and 

disincentives are established by determining if the mix is within the allowable tolerances of the JMF. Acceptance is 

based on a 3000-ton (2720-Mg) lot. 

Pay Factors 

Pay factors are assigned for gradation, in-place density, and asphalt content. The factors are determined according to 

the following equations and the Price Reduction Factors table. All of the individual test results in the lot for the 

element to be evaluated will be averaged and the percentage of price reduction for the lot will be determined by the 

applicable equation: 

1. The equation P = (Xn + aR – Tu) × F will be used if a maximum limit only is specified, or when the 

average of several test values is above the midpoint of a specified band or above a job mix target value.(46) 

2. The equation P = (TL + aR – Xn) × F will be used if a minimum limit only is specified, or when the average 

of several test values is below the midpoint of a specified band or below a job mix target value. (47) 

where: P = percent reduction in the contract price 

Xn = average of several test values from samples from the lot. (n = sample size) 

a = variable factor used as n changes: n = 3, a =0.45; n = 4, a = 0.38; n = 5, a = 0.33; 

 n = 6, a = 0.30; and n = 7, a = 0.28 

R = difference between the highest and lowest values in the group of values 

Tu = upper tolerance limit 

TL = lower tolerance limit 

F = price reduction factor from the following table: 
 282

Price Reduction Factors 

Table 79. Montana price reduction factors. 

Element f Factor 

100% sieve 1 

1⁄2-inch sieve and larger 1 

#100 sieve to 3⁄8-inch sieve inclusive, 

except 100% sieve size 

Cover material: 2 

All other aggregate: 3 

#200 Cover material: 3 

All other aggregate: 6 

Density 12 

Fractured faces 2 

1 inch = 25.4 mm 

If P < 3 or a negative quantity, the lot is accepted as being in conformance. If one or more elements for a contract 

item show a positive P-value, the positive values are added and the resulting sum is used to determine whether the 

lot is in conformance. If P = 3 to 25, corrective action is required or the lot is accepted at a reduced price. If P = 25+, 

remove and replace. An incentive pay factor of 1.05 is allowed when the aggregate gradation for #4, #40, and #200 

sieves is not more than one-half the allowable tolerance from the JMF. Additionally, a pay factor of 1.05 will be 

applied when the average density of the lot is from 97 percent to 98 percent, inclusive, from the target field Marshall 

density and the range is 3 or less. 

QC/QA Tests for Montana 

Table 80. Montana QC/QA tests.

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Assurance 

Testing 

Frequency by 

Montana DOT 

Acceptance by 

Montana DOT 

Gradation (dry) Cold feed One per 600 tons One per 600 tons Based on QA testing 

Density Roadway One per 600 tons One per 600 tons Based on QA testing 

Asphalt content Behind paver One per 600 tons One per 600 tons Based on QA testing 

Fractured faces Cold feed One per 600 tons One per 600 tons NA 

NA = Not applicable 

1 ton = 0.907 Mg 
 283

Nebraska 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation, BSG, ESG, binder content, TMD, air 

voids, VMA, and density. 

Mix design verification is evaluated for the following properties: number of Marshall blows, Marshall stability, 

Marshall flow, percent voids in mix, dust-to-asphalt ratio, binder content, VMA, crushed face, TMD, BSG, and 

ESG. 

Quality Assurance 

Nebraska Department of Roads (DOR) is responsible for all assurance testing and acceptance of the material. All 

assurance test samples are split samples obtained from the contractor under the direction of the Nebraska DOR. 

From each sample, three Marshall samples are made and tested. Cores for density testing are obtained by Nebraska 

DOR. Assurance testing is conducted on the properties evaluated for QC. 

Acceptance 

Acceptance is based on verification of the contractor’s QC tests for air voids, VMA, gradation, BSG, and in-place 

density by Nebraska DOR’s assurance tests. If the results are verified, Nebraska DOR’s tests are used for 

acceptance. 

Pay Factors 

A pay factor is assigned for density. The appropriate pay factor is then multiplied by the contract unit price. The pay 

factor for density is determined according to the following tables: 

Density of Asphalt Concrete (First Lot) 

Table 81. Nebraska density of asphalt concrete (first lot). 

Average Density (Five Samples) Pay Factor 

> 90.0 1.00 

> 89.5-90.0 0.95 

> 89.0-89.5 0.70 

≤ 89.0 0.40 or reject 

Density of Asphalt Concrete (Subsequent Lots) 

Table 82. Nebraska density of asphalt concrete (subsequent lots). 

Average Density (Five Samples) Pay Factor 

> 92.4 1.00 

> 91.9-92.4 0.95 

> 91.4-91.9 0.90 

> 90.9-91.4 0.85 

> 90.4-90.9 0.80 

> 89.9-90.4 0.70 

≤ 89.9 0.40 or reject 
 284

QC/QA Tests for Nebraska 

Table 83. Nebraska QC/QA tests. 

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Assurance Testing 

Frequency by 

Nebraska DOR 

Acceptance by 

Nebraska DOR 

Marshall test Plant mix One per 1100 tons One per 1 day’s 

production 

Based on verification of 

contractor’s QC tests by 

Nebraska DOR’s QA tests 

BSG Plant mix One per 1100 tons One per 1 day’s 

production 

Based on verification of 

contractor’s QC tests by 

Nebraska DOR’s QA tests 

TMD Plant mix One per 1100 tons One per 1 day’s 

production NA 

ESG Plant mix One per 1100 tons One per 1 day’s 

production NA 

Asphalt content Plant mix or 

behind paver One per 1100 tons One per 1 day’s production 

Based on verification of 

contractor’s QC tests by 

Nebraska DOR’s QA tests 

Air voids Plant mix or 

behind paver One per 1100 tons One per 1 day’s production 

Based on verification of 

contractor’s QC tests by 

Nebraska DOR’s QA tests 

VMA Plant mix One per 1100 tons One per 1 day’s 

production 

Based on verification of 

contractor’s QC tests by 

Nebraska DOR’s QA tests 

Density Roadway Five per 2750 

tons 

Five per 1 day’s 

production 

Based on verification of 

contractor’s QC tests by 

Nebraska DOR’s QA tests 

Gradation Cold feed One per 1100 tons One per 1 day’s 

production 

Based on verification of 

contractor’s QC tests by 

Nebraska DOR’s QA tests 

NA = Not applicable 

1 ton = 0.907 Mg 
 285

Nevada 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor must conduct QC tests on the following properties: gradation, liquid limit and plastic index, in-place 

density, absorption, fractured faces, theoretical MSG, moisture content, binder temperature, and mix temperature. 

The contractor is responsible for providing split samples to the engineer for verification and referee testing. 

Nevada DOT is responsible for conducting the QC tests for Hveem stability, ITS, and TSR. 

Quality Assurance 

The contractor is responsible for all assurance testing. The testing frequencies vary, depending on the property being 

measured. Nevada DOT conducts verification testing to determine if the contractor’s assurance test results are 

within acceptable limits. If the test values are not within the tolerances, referee testing is conducted. 

Acceptance 

Acceptance is based on the results of the QC tests, assurance tests, verification tests, and referee tests (if required). A 

production lot is four equal sublots. The sublot size is agreed upon by the contractor and the engineer prior to 

construction and may be 250, 500, or 750 tons (230, 450, or 680 Mg). 

Pay Factors 

Pay factors are assigned for gradation, in-place density, asphalt content, and profile index. The pay factors for each 

of these properties is determined from the following tables: 

Pay Factors for Profile Index 

Table 84. Nevada pay factors for profile index. 

mm/km Pay Factor Type A Type B Type C 

1.05 0-45 0-80 0-85 

1.04 46-60 81-94 86-110 

1.02 61-70 95-100 111-140 

1.00 71-80 101-110 141-160 

0.98 81-95 111-125 161-165 

0.96 96-110 126-140 166-175 

0.94 111-125 141-160 176-180 

0.92 126-140 161-175 181-190 

0.90 141-160 176-190 — 

Corrective work > 160 > 190 > 190 
 286

Pay Factors for Gradation 

Table 85. Nevada pay factors for gradation. 

Mean Absolute Deviation From JMF Target Pay Factor Pass 2 mm Pass 75 μm 

1.05 0.00-0.99 0.00-0.50 

1.02 1.00-1.90 0.51-0.90 

1.00 1.91-3.00 0.91-1.50 

0.95 3.01-4.00 1.51-2.00 

0.90 4.01-5.00 2.01-2.50 

0.85 5.01-6.00 2.51-3.00 

0.80 6.01-7.00 3.01-3.50 

0.75 7.01-8.00 3.51-4.00 

0.70 8.01-9.00 4.01-4.50 

Remove > 9.00 > 4.50 

Pay Factors for Asphalt Content 

Table 86. Nevada pay factors for asphalt content. 

Pay Factor Mean Absolute Deviation From JMF Target 

1.05 0.00-0.19 

1.02 0.20-0.24 

1.00 0.25-0.30 

0.95 0.31-0.35 

0.90 0.36-0.40 

0.85 0.41-0.45 

0.80 0.46-0.50 

0.75 0.51-0.60 

0.70 0.61-0.65 

Remove > 0.65 

Pay Factors for In-Place Air Voids 

Table 87. Nevada pay factors for in-place air voids. 

Pay Factor Measured Air Voids 

(average of five locations per sublot) 

1.05 4.0-5.9 

1.02 6.0-6.9 

1.00 7.0-8.0 

0.95 3.5-3.9 or 8.1-9.5 

0.90 3.0-3.4 or 9.6-10.0 

0.80 2.6-2.9 or 10.1-11.0 

0.70 11.1-12.0 

Note: If the average in-place air void for a sublot is less than 2.5 or greater then 12.0, the entire sublot 

 must be removed. 

Once the pay factors have been determined, the pay calculation is as follows: 

A = HMA set price × Placement quantity × Production pay factor 

B = HMA set price × Placement quantity tested for air voids × Placement pay factor 

C = HMA set price × Placement quantity tested for rideability × Ride pay factor 

The total pay quantity (TPQ) shall be based on the applicable pay adjustment factors for production, placement, and 

ride quality: 
 287

For production only: TPQ = A

For production and placement: TPQ = 0.40A + 0.60B

For production, placement, and ride quality: TPQ = 0.30A + 0.40B + 0.30C

The incentive or disincentive pay amount is the difference between the standard pay quantity (SPQ) and the TPQ. 

QC/QA Tests for Nevada 

Table 88. Nevada QC/QA tests. 

Test Sample Location QC Testing 

Frequency 

Assurance 

Testing 

Frequency by 

Nevada DOT 

Verification Testing 

Frequency by 

Nevada DOT 

Gradation 

(except 2 mm and 75 μm) Cold feed One per sublot by 

contractor NA 1 per 12 sublots by Nevada DOT 

Gradation 

(2 mm and 75 μm) Behind paver NA One per sublot by 

contractor 

1 per 12 sublots by 

Nevada DOT 

Asphalt content Behind paver NA One per sublot by 

contractor 

1 per 12 sublots by 

Nevada DOT 

HVEEM stability Behind paver 

One per eight 

sublots by Nevada 

DOT 

NA NA 

Theoretical MSG Behind paver 

Average of two 

random tests per lot 

by contractor 

NA 1 per 12 sublots by 

Nevada DOT 

Moisture content Behind paver 

One per four 

sublots by 

contractor 

NA 1 per 12 sublots by 

Nevada DOT 

Binder temperature Plant Continuous by 

contractor NA Continuous by Nevada DOT 

Mix temperature Truck bed Continuous by 

contractor NA Continuous by Nevada DOT 

In-place air voids Roadway NA Five per sublot by 

contractor 

1 per 12 sublots by 

Nevada DOT 

Liquid limit Stockpile 

One per four 

sublots by 

contractor 

NA 1 per 12 sublots by 

Nevada DOT 

Plastic index Stockpile 

One per four 

sublots by 

contractor 

NA 1 per 12 sublots by 

Nevada DOT 

Fractured faces Stockpile 

One per four 

sublots by 

contractor 

NA One per four sublots 

by Nevada DOT 

Absorption Stockpile 

One per four 

sublots by 

contractor 

NA 1 per 12 sublots by 

Nevada DOT 

ITS Behind paver 1 per 20 sublots by 

Nevada DOT NA NA 

Smoothness Roadway One pass per lane 

of travel per 0.1 km

One pass per lane 

of travel per 0.1 km 

One pass per lane per 

lot 

TSR Behind paver 1 per 20 sublots by 

Nevada DOT NA NA 

NA = Not applicable 
 288

North Dakota 

Quality Control 

The contractor must use a mix design developed by North Dakota DOT. It is the contractor’s responsibility to meet 

the design requirements of the JMF. In addition, the contractor is responsible for providing a QC plan that must be 

approved by the engineer prior to construction and for providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation, plastic index, lightweight aggregate, 

fractured faces, MSG, BSG, air voids, asphalt content, and density. The contractor must provide split samples for the 

engineer to conduct assurance testing. 

Quality Assurance 

North Dakota DOT is responsible for all assurance testing and acceptance. The engineer may select any or all of the 

split samples for assurance testing. Assurance testing is conducted on the properties evaluated for QC. 

Acceptance 

Acceptance is based on the contractor’s QC tests if the results are verified by North Dakota DOT’s assurance 

testing. The testing frequencies vary, depending on the property being measured. For density, the lot size is 1 day’s 

production, with two core samples for each equal sublot being averaged. 

Pay Factors 

Pay factors are assigned for gradation, asphalt content, and density. The determination of the pay factors was not 

available; however, the total pay adjustment is determined by successively multiplying the pay factors for gradation, 

asphalt content, and density by the contract unit price. 

QC/QA Tests for North Dakota 

Table 89. North Dakota QC/QA tests. 

Test Sampling 

Location 

QC Testing by 

Contractor 

Assurance Testing 

Frequency by 

North Dakota DOT 

Acceptance by 

North Dakota DOT 

Gradation Cold feed One per 1500 tons Engineer’s discretion, minimum 

of 10% QC testing 

Based on QC tests if 

verified by assurance test

Plastic index Stockpiles Three per 10,000 tons Engineer’s discretion, minimum 

of 10% QC testing 

Based on QC tests if 

verified by assurance test

Lightweight 

aggregate Stockpiles Three per 10,000 tons Engineer’s discretion, minimum of 10% QC testing 

Based on QC tests if 

verified by assurance test

Fractured faces Stockpiles Three per 10,000 tons Engineer’s discretion, minimum 

of 10% QC testing 

Based on QC tests if 

verified by assurance test

MSG Plant mix One per 1500 tons Engineer’s discretion, minimum 

of 10% QC testing 

Based on QC tests if 

verified by assurance test

BSG Plant mix One per 1500 tons Engineer’s discretion, minimum 

of 10% QC testing 

Based on QC tests if 

verified by assurance test

Air voids Plant mix or 

behind paver One per 1500 tons Engineer’s discretion, minimum 

of 10% QC testing 

Based on QC tests if 

verified by assurance test

Asphalt content Plant mix or 

behind paver 

Four per 1 day’s 

production 

Engineer’s discretion, minimum 

of 10% QC testing 

Based on QC tests if 

verified by assurance test

Density Roadway Two per sublot per 1 

day’s production 

Engineer’s discretion, minimum 

of 10% QC testing 

Based on QC tests if 

verified by assurance test

1 ton = 0.907 Mg 
 289

Ohio 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation, asphalt content, air voids, MSG, and 

density. The contractor must provide split samples to the engineer for assurance testing. 

Quality Assurance 

Ohio DOT is responsible for all assurance testing and acceptance of the material. Assurance testing is conducted on 

the properties evaluated for QC. 

Acceptance 

The contractor’s control test can be used for acceptance for gradation, density, and asphalt content if the results are 

verified by Ohio DOT’s assurance tests. 

Pay Factors 

A pay factor is assigned for gradation, density, and asphalt content. The determination of the pay factors was not 

provided. However, the pay factors are cumulative. The sum is then multiplied by the contract unit price to 

determine the amount of the bonus or penalty. 

QC/QA Tests for Ohio 

Table 90. Ohio QC/QA tests.

Test Sample 

Location 

QC Testing 

Frequency by 

Contractor 

Assurance Testing 

Frequency by 

Ohio DOT 

Acceptance by 

Ohio DOT 

Asphalt 

content Behind paver Two per day or one for each 1400 tons One per 2800 tons Based on QC tests as verified by QA tests 

Gradation 

(dry) Cold feed Two per day or one for 

each 1400 tons One per 2800 tons Based on QC tests as 

verified by QA tests 

Air voids Behind paver Two per day or one for 

each 1400 tons One per 2800 tons NA 

MSG Behind paver Two per day or one for 

each 1400 tons One per 2800 tons NA 

Density Roadway Two per sublot 

(700 tons) One per 2800 tons Based on QC tests as verified by QA tests 

NA = Not applicable 

1 ton = 0.907 Mg 
 290

Ontario 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the contract administrator prior to construction, and providing qualified personnel and equipment to 

conduct QC testing. The contractor must allow assurance samples to be taken to a certified laboratory at the request 

of the owner. 

The contractor is responsible for QC for the following properties: gradation, asphalt content, air voids, and density. 

Quality Assurance 

The owner is responsible for all assurance testing. Owner assurance testing is conducted on gradation, air voids, 

VMA, Marshall stability, Marshall flow, and asphalt content. 

Acceptance 

The contractor is responsible for all acceptance testing. The contractor’s control test can be used for acceptance if 

the results agree with the owner’s assurance results. In the event that the two results do not agree, additional samples 

will be taken for referee testing by an independent laboratory. Acceptance testing is conducted for the following 

characteristics: gradation, air voids, VMA, Marshall stability, Marshall flow, and asphalt content. Acceptance is 

based on the PWL for each lot. Sublot sizes vary and are decided on by the owner and the contractor, but usually 

consist of 500-metric ton sublots. 

Pay Factors 

Pay factors are assigned for gradation, in-place density, asphalt content, and air voids. The individual pay factors are 

weighted and summed to arrive at the composite pay factor. The composite pay factor is then multiplied by the 

contract unit price to determine the amount of the bonus or penalty. 

QC/QA Tests for Ontario 

Table 91. Ontario QC/QA tests. 

Test Sampling 

Location 

QC Testing Frequency 

by Contractor 

Assurance Testing 

Frequency by 

Owner 

Acceptance Testing 

Frequency by 

Contractor 

Gradation Cold feed As needed for control One per 500 metric tons One per 500 metric 

tons 

Asphalt content Behind paver As needed for control One per 500 metric tons One per 500 metric 

tons 

Air voids Behind paver As needed for control One per 500 metric tons One per 500 metric 

tons 

VMA Behind paver As needed for control One per 500 metric tons One per 500 metric 

tons 

Marshall stability Behind paver As needed for control One per 500 metric tons One per 500 metric 

tons 

Marshall flow Behind paver As needed for control One per 500 metric tons One per 500 metric 

tons 

Density Roadway As needed for control Three per sublot Three per sublot 
 291

Oregon

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation, asphalt content, MSG, moisture 

content, air voids, VMA, TSR, dust-to-asphalt ratio, and density. The contractor must supply the engineer with split 

samples for assurance testing. 

Quality Assurance 

Oregon DOT is responsible for all assurance testing. Assurance testing is conducted on the properties evaluated for 

QC. All control and assurance tests are conducted on 1000-Mg sublots. 

Acceptance 

The contractor’s control test cannot be used for acceptance. Acceptance testing is conducted for the following 

characteristics: gradation, in-place density, asphalt content, and moisture content. Acceptance is on a lot-by-lot basis 

where a lot is represented by all the material produced under the JMF. Test results are used to establish an 

acceptable quality level using quality-level analysis. 

Pay Factors 

Pay factors are assigned for gradation, in-place density, asphalt content, and moisture. Once the PWL for an element 

is determined, the pay factor is taken from a table and then weighted according to the following equation: 

 WPF = (PF) × fi (48) 

where: fi = weighting factor 

The following table determines these factors: 

Weighting Factors 

Table 92. Oregon weighting factors. 

Constituent Weighting f Factor 

Aggregate passing 37.5-, 31.5-, 25.0-, 19.0-, and 12.5-mm sieves 1 each 

Aggregate passing 6.3-mm sieve 5 

Aggregate passing 2.00-mm sieve 5 

Aggregate passing 425-μm sieve 3 

Aggregate passing 75-μm sieve 10 

Asphalt content 26 

Moisture content 8 

Density 40 

The composite pay factor is then determined by the following equation: 

 CPF = ∑WPF / ∑fi (49) 

The composite pay factor is then multiplied by the quantity in the lot and by the contract unit price to determine the 

total amount of payment for the lot. The following table is a summary of the QC/QA tests: 
 292

QC/QA Tests for Oregon 

Table 93. Oregon QC/QA tests. 

Test Sampling 

Location 

QC Testing by 

Contractor 

Assurance 

Testing Frequency 

by Oregon DOT 

Acceptance Testing 

Frequency by 

Oregon DOT 

Gradation Cold feed One per 1000 Mg One per 1000 Mg One per lot 

Dust-to-asphalt 

ratio Cold feed One per 1000 Mg One per 1000 Mg NA 

Asphalt content Truck or behind 

paver One per 1000 Mg One per 1000 Mg One per lot 

Moisture Truck One per 1000 Mg One per 1000 Mg One per lot 

Air voids Truck or behind 

paver One per 1000 Mg One per 1000 Mg NA 

VMA Truck or behind 

paver One per 1000 Mg One per 1000 Mg NA 

TSR Truck or behind 

paver One per 1000 Mg One per 1000 Mg NA 

MSG Truck or behind 

paver One per 1000 Mg One per 1000 Mg NA 

Density Roadway One per 1000 Mg One per 1000 Mg Five per lot 

NA = Not applicable 
 293

Pennsylvania 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation, stability, flow, voids, asphalt content, 

mix temperature, and density. These properties must be tested once per 250-ton (230 Mg) sublot. 

Quality Assurance 

Pennsylvania DOT is responsible for all assurance sampling and testing. The assurance tests are independent of the 

contractor’s control tests. Assurance testing is conducted for gradation, asphalt content, and density. 

Acceptance 

Acceptance is based on certification at the plant when QC and assurance tests conform to the JMF tolerances, except 

for in-place density. Acceptance for density is accomplished by verification of the QC tests by the QA tests in the 

field. Acceptance properties are asphalt content, gradation (#200), stability, flow voids, and density. 

Pay Factors 

Pay factors are assigned for gradation, in-place density, and asphalt content. The pay factors are based on the 

deviation from the specification limits. The following tables provide the pay factor for asphalt content, gradation, 

and density: 

Adjustment of Contract Price Relative to the Specification Limits 

Table 94. Pennsylvania adjustment of contract price relative to the specification limits. 

Test Value Payment Factor Percentage 

±0.07% 100 

±0.8-1.0% 75 Asphalt Content 

> ±1.0% * 

±3.0% 100 

±3.1-4.0% 75 Percentage Passing #200 

Sieve 

> ±4.0% * 

≥ 92% or < 97% of TMD 100 

Density 90-91% or 97-99% of TMD 98 

≤ 89% or > 99% of TMD * 

*Indicates remove and replace.
 294

QC/QA Tests for Pennsylvania 

Table 95. Pennsylvania QC/QA tests. 

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Assurance 

Testing 

Frequency by 

Pennsylvania 

DOT 

Acceptance by 

Pennsylvania DOT 

Gradation (dry) Cold feed One per sublot 

(250 tons) 

One per lot 

(1000 tons) 

Based on plant 

certification by QC and 

assurance tests 

Mixture temperature Truck One per sublot 

(250 tons) NA NA 

Flow Plant One per sublot 

(250 tons) NA 

Based on plant 

certification by QC and 

assurance tests 

Air voids Plant One per sublot 

(250 tons) NA 

Based on plant 

certification by QC and 

assurance tests 

Asphalt content Plant One per sublot 

(250 tons) 

One per lot 

(1000 tons) 

Based on plant 

certification by QC and 

assurance tests 

Density Roadway One per sublot 

(250 tons) 

One per lot 

(1000 tons) 

Based on QC as 

verified by QA 

VMA Plant One per sublot 

(250 tons) NA 

Based on plant 

certification by QC and 

assurance tests 

NA = Not applicable 

1 ton = 0.907 Mg 
 295

South Carolina 

Quality Control 

The contractor is responsible for all QC testing. Prior to the start of construction, the contractor must submit a QC 

program for approval by the engineer. This plan must include, at a minimum, the name of the QC manager, 

sampling frequency and procedure, testing procedures, method of recording results, and the procedure for managing 

variable test results. The contractor must provide a certified level 1 technician to monitor gradation and binder 

content. The contractor is also required to develop and submit a JMF that conforms to all South Carolina DOT 

requirements. The technician must make all test results available to the engineer and the engineer must be permitted 

to observe all testing of material. 

The JMF is verified based on gradation, asphalt content, Marshall stability, Marshall flow, air voids, TSR, VMA, 

and dust-to-asphalt ratio. 

The contractor shall use the test methods identified in the table below to perform the plant QC tests and verifications 

at a frequency not less than that indicated. 

Required QC Tests and Verifications 

Table 96. South Carolina required QC tests and verifications. 

Parameter Minimum Frequency Sampling Method Test Method 

Bituminous mixture gradation One per lot SC-T-62 SC-T-76 

MSG (Rice method) Two per lot SC-T-62 SC-T-83 

Marshall stability One per lot SC-T-62 SC-T-66 

Lime rate verification Two per lot SC-T-71 

SC-T-71 

SC-T-78 

Mixture temperature verification Four per lot — SC-T-84 

Aggregate stockpile gradation One per 10,000 tons SC-T-1, SC-T-2 SC-T-4 

Temperature 

Ambient air 

Mat 

Before paving starts, 

then two per lot 

 

Four per lot 

— 

— 

SC-T-84 

SC-T-84 

Calculated lay-down rate One per 200 tons — SC-T-85 

Tack rate, type, and dilution One per application — SC-T-86 

1 ton = 0.907 Mg 

Compaction control and the number and types of rollers shall be the contractor’s responsibility. The contractor shall 

have a nuclear density gauge at the site during all HMA placement and compaction operations and shall use the 

gauge to assist in QC of the compaction process. 

Quality Assurance 

The contractor is responsible for conducting all acceptance testing. South Carolina DOT personnel may witness the 

sampling and testing being performed by the contractor. South Carolina DOT will conduct its own tests to verify the 

contractor’s test results. The verification tests for asphalt binder content will be on the split samples retained by the 

contractor. For most paving, the verification tests for in-place density will be on cores retained by the contractor. For 
 296

thin-lift surface courses, verification tests for in-place density analysis will be at independent locations determined 

and tested by South Carolina DOT using the same type of nuclear density gauge as used by the contractor. 

In general, the frequency of South Carolina DOT’s verification tests will be equal to or greater than 10 percent of the 

tests required for the contractor. 

Acceptance 

The contractor shall use the test methods identified in the table below to perform the plant acceptance tests at a 

frequency not less than that indicated. 

Required Acceptance Tests 

Table 97. South Carolina required acceptance tests. 

Test Parameter Typical Frequency Sampling Method Test Method 

Asphalt binder content, percent Four per lot SC-T-100 

SC-T-62 

SC-T-64 or

SC-T-75 

Voids analysis 

 Air voids, percent 

 VMA, percent 

Four per lot 

Four per lot 

SC-T-100, SC-T-62 

SC-T-100, SC-T-62 

SC-T-68 

SC-T-68 

In-place density (percentage of 

maximum theoretical) 

Note: Requirements apply to 

intermediate courses and to 

surface courses other than type 3 

and type 4. 

One per 1500-ft sublot SC-T-100, SC-T-87 SC-T-87 

In-place density (percentage of 

target nuclear control-strip 

density) 

Note: Requirements apply to 

type 3 and type 4 surface 

courses and for all thin-lift 

surface courses regardless of 

type. 

10 per lot SC-T-100 SC-T-65 

1 ft = 0.305 m 

The engineer is responsible for determining the air voids and VMA from Marshall samples. The technician is 

responsible for determining the binder content. Density is determined by coring and is the responsibility of the 

contractor at the direction of the engineer. Lot size is 1 day’s production. Test frequency is at the discretion of the 

engineer; however, the minimum requirements are listed in the table. 

Pay Factors 

For asphalt binder content and the Marshall volumetric properties of air voids and VMA, a lot is normally defined as 

1 day’s production. The acceptance and pay factors for asphalt binder content and the Marshall volumetric 

properties of air voids and VMA will be based on the percentage of the lot that is within the specification limits 

(PWL) based on the quality index calculated using the test results from the lot. 

The pay factor for each property is determined based on the total percent within limits (TPWL) value from the 

following equation: 

 PF = 55 + 0.5(TPWL) (50) 
 297

The percent pay factor for the lot (LPF) will be determined by multiplying the percent pay factors for asphalt binder 

content, air voids, VMA, and in-place density by weighted coefficients as shown in the equation below. The LPF

will be calculated to the nearest tenth (0.1) and rounded up to the next whole percent (1.0). The LPF will be 

determined from the following equation: 

LPF = 0.2(PFAC) + 0.35(PFAV) + 0.1(PFVMA) + 0.35(PFDen) 

 (51) 

where: LPF = percent pay factor for the lot 

 PFAC = percent pay factor for asphalt binder content 

PFAV = percent pay factor for air voids 

PFVMA = percent pay factor for VMA 

PFDen = percent pay factor for in-place density 
 298

Texas 

Quality Control 

The contractor is responsible for providing a QC plan that must be approved by the engineer prior to construction 

and providing qualified personnel and equipment to conduct QC testing. The contractor must allow the engineer 

access to the laboratory to observe any and all testing procedures, calculations, test documentation, and plotting of 

results. To complete the required testing, the contractor must provide technicians certified at levels I through IV, 

depending on the test being conducted. The standard lot size is 6000 square yards (5000 m2

) of surface area of PCC 

pavement of the same thickness. A sublot is equal to one-fifth of the surface area of a lot. If the final lot is 3000 

square yards (2500 m2 ) or greater, it will be considered a full lot. QC must be exercised on all properties contained 

in the table entitled QC/QA Tests for Texas. 

Quality Assurance 

QA is accomplished through verification testing conducted by Texas DOT. Independent assurance testing is 

conducted to review the QC testing and to check the accuracy of the equipment used for verification testing. 

Acceptance 

Acceptance of the material is based on the QC tests and the verification tests. If the contractor’s QC test results and 

the engineer’s verification test results are not within tolerances, referee testing will be conducted. If referee tests are 

requested, the referee test results shall govern. 

Pay Factors 

Pay factors are assigned for ride quality, thickness, and flexural strength. The pay factor for flexural strength is 

based on the following table: 

Table 98. Texas pay factors for flexural strength. 

Average Flexural Strength 

(Three Beams) (lbf/inch2

) 

Strength Adjustment Factor 

(SF) 

≥ 555 1.0000 

550 0.9695 

545 0.9397 

540 0.9106 

535 0.8820 

530 0.8542 

525 0.8269 

< 525 Remove or remain with no 

payment 

 1 pound force per square inch (lbf/inch2) = 6.89 kilopascals (kPa) 

The pay factor for ride quality was not provided. The pay factor for thickness is determined using the following 

table: 

Table 99. Texas pay factors for thickness. 

Thickness Deficiency (inch) Thickness Adjustment Factor 

(TF) 

> 0.00-0.20 1.00 

> 0.20-0.30 0.80 

> 0.30-0.40 0.72 

> 0.40-0.50 0.68 

> 0.50-0.75 0.57 

 1 inch = 25.4 mm 

The pay adjustment factor for each lot is calculated as AF = (SF)(TF). The pay adjustment for each lot is then 

calculated to be Payment = (BP)(AF)(Q), where BP = bid price and Q = lot quantity of acceptable pavement. 
 299

QC/QA Tests for Texas

Table 100. Texas QC/QA tests. 

Test Sample 

Location 

QC Testing by 

Contractor 

Verification Testing 

by Texas DOT 

Acceptance by 

Texas DOT 

Mix temperature Plant One per sublot One per day Based on QC, verification, 

and referee, if required 

Air content (plastic) Roadway after 

discharge One per sublot 1 per 10 QC tests Based on QC, verification, 

and referee, if required 

Concrete unit weight Roadway after 

discharge One per sublot 1 per 10 QC tests Based on QC, verification, 

and referee, if required 

Making/curing 

strength specimens 

Roadway after 

discharge One per sublot 1 set per 10 sublots NA 

7-day flexural 

strength 

Roadway after 

discharge NA One per sublot Based on verification 

Coarse aggregate 

gradation Stockpile One per stockpile 

per day 1 per 10 QC tests Based on QC, verification, 

and referee, if required 

Coarse aggregate loss 

by decantation Stockpile One per 5 PCC 

production days 1 per 10 QC tests Based on QC, verification, 

and referee, if required 

Fine aggregate 

gradation Stockpile One per stockpile per day 1 per 10 QC tests Based on QC, verification, and referee, if required 

Fineness modulus Stockpile One per stockpile 

per day 1 per 10 QC tests Based on QC, verification, and referee, if required 

Fine aggregate 

organic impurity Stockpile One per 5 PCC production days 1 per 10 QC tests Based on QC, verification, and referee, if required 

Sand equivalent Stockpile One per 5 PCC 

production days 1 per 10 QC tests Based on QC, verification, and referee, if required 

Aggregate moisture 

content Stockpile One per stockpile 

per day 1 per 10 QC tests Based on QC, verification, 

and referee, if required 

Water-to-cement ratio Roadway after 

discharge One per batch Review plant printout 

and calibration 

Based on QC, verification, 

and referee, if required 

Cement factor Roadway after 

discharge One per batch Review plant printout 

and calibration 

Based on QC, verification, 

and referee, if required 

Admixture dosage Roadway after 

discharge One per batch Review plant printout 

and calibration 

Based on QC, verification, 

and referee, if required 

Coring for thickness Roadway One per sublot NA NA 

Measuring pavement 

thickness 

Roadway 

(plastic) NA One per sublot Based on QC, verification, and referee, if required 
 300

Virginia 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

Mix design verification is evaluated for the following properties: Marshall stability, Marshall flow, MSG, gradation, 

BSG, VMA, VFA, VTM, dust-to-asphalt ratio, and asphalt viscosity. 

The contractor is responsible for QC for the following properties: air voids, VFA, VMA, dust-to-asphalt ratio, 

density, temperature, and fractured faces. The contractor must cooperate with the engineer to obtain samples for 

assurance testing. The contractor must perform at least one set of tests per day or one set per 1000 metric tons. 

Quality Assurance 

QA is achieved through the QC testing by the contractor and by independent monitoring (assurance) tests conducted 

by Virginia DOT. Assurance testing may be conducted on air voids, VFA, VMA, dust-to-asphalt ratio, density, 

temperature, and fractured faces. The frequency of testing is at the engineer’s discretion. 

Acceptance 

The contractor’s control test can be used for acceptance if the results are verified by the assurance tests conducted by 

Virginia DOT. The testing frequencies vary, depending on the property being measured. 

Pay Factors 

Pay factors are assigned for gradation and asphalt content. The pay factors are assigned on an adjustment point 

system. Adjustment points are assigned for each sieve size and for asphalt content. Any lot receiving 25 or more 

points must be removed. Any lot with fewer than 25 adjustment points may be left in place with a 1-percent 

reduction in the unit bid price per point. A summary of the QC/QA tests follows: 

QC/QA Tests for Virginia 

Table 101. Virginia QC/QA tests. 

Test Sampling 

Location 

QC Testing by 

Contractor 

Assurance Testing 

Frequency by 

Virginia DOT 

Acceptance by 

Virginia DOT 

Gradation 

(extracted) Truck One per 500 tons or four per lot (2000 tons)

At the discretion of the 

engineer for QC verification 

Based on observed and verified 

results of QC by assurance tests 

Asphalt 

content Truck One per 500 tons or four per lot (2000 tons)

At the discretion of the 

engineer for QC verification 

Based on observed and verified 

results of QC by assurance tests 

Air voids Truck One Marshall test per 

day or 1000 tons 

At the discretion of the 

engineer for QC verification 

Based on observed and verified 

results of QC by assurance tests 

VFA Truck One Marshall test per 

day or 1000 tons 

At the discretion of the 

engineer for QC verification 

Based on observed and verified 

results of QC by assurance tests 

VMA Truck One Marshall test per 

day or 1000 tons 

At the discretion of the 

engineer for QC verification 

Based on observed and verified 

results of QC by assurance tests 

Fractured 

faces Cold feed One Marshall test per day or 1000 tons 

At the discretion of the 

engineer for QC verification 

Based on observed and verified 

results of QC by assurance tests 

Dust-toasphalt ratio Cold feed One Marshall test per 

day or 1000 tons 

At the discretion of the 

engineer for QC verification 

Based on observed and verified 

results of QC by assurance tests 

Density Roadway Based on nuclear gauge 

testing of control strip 

At the discretion of the 

engineer for QC verification 

Based on nuclear gauge testing 

of control strip 

VTM Truck One Marshall test per 

day or 1000 tons 

At the discretion of the 

engineer for QC verification 

Based on observed and verified 

results of QC by assurance tests 

1 ton = 0.907 Mg 
 301

Washington 

Quality Control 

The contractor must submit materials to be used in the mix to Washington State DOT for mix design development. 

The contractor must allow 15 working days for the development and approval of the design. In addition, the 

contractor is responsible for providing a QC plan that must be approved by the engineer prior to construction and 

providing qualified personnel and equipment to conduct QC testing. The contractor must allow the engineer access 

to the laboratory to observe any and all testing procedures, calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation, asphalt content, density, fractured faces, 

and sand equivalent. 

The frequency of the QC testing is established and agreed upon by the contractor and the engineer in a 

preconstruction conference establishing the control plan. 

Quality Assurance 

Washington State DOT is responsible for all assurance sampling and testing. The assurance testing frequencies vary, 

depending on the property being measured and are established by the engineer. Assurance testing is conducted on 

the properties evaluated for QC. 

Acceptance 

Acceptance testing is the responsibility of Washington State DOT. The contractor’s control test cannot be used for 

acceptance. Acceptance is based on the following characteristics: gradation, in-place density, smoothness, and 

asphalt content. A lot is considered to be all material produced using the same JMF. Sublot sizes vary, but cannot 

exceed 800 tons (730 Mg). Acceptance and the corresponding pay factors are based on the PWL. 

Pay Factors 

Pay factors are assigned for gradation, in-place density, and asphalt content. The individual pay factors for an 

element are determined using quality-level analysis. These individual pay factors are weighted with an f-factor. The 

composite pay factor is then determined according to the following equation: 

 CPF = f1(PF1) + f2(PF2) + ... fi(Pfi) / ∑fi (52) 

QC/QA Tests for Washington 

Table 102. Washington QC/QA tests. 

Test Sample 

Location 

QC Testing Frequency 

by Contractor 

Assurance Testing 

Frequency by 

Washington 

State DOT 

Acceptance Testing 

Frequency by 

Washington State DOT 

Extracted gradation 

(5⁄8 inch, 1⁄2 inch, 3⁄8 inch, 

1⁄4 inch, #10, #40, #200) 

Truck Engineer’s discretion Engineer’s discretion Five per lot minimum 

Asphalt content Truck Engineer’s discretion Engineer’s discretion Five per lot minimum 

Density Roadway Engineer’s discretion Engineer’s discretion Five per lot minimum 

Smoothness Roadway Engineer’s discretion Engineer’s discretion Based on one lot 

Fractured faces Cold feed Engineer’s discretion Engineer’s discretion Five per lot minimum 

1 inch = 25.4 mm 
 302

Wisconsin 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor must conduct QC tests for the following properties: gradation (1/2 inch (13 mm), 3/8 inch (10 mm), 

#4, #8, #30, and #200), asphalt content, Marshall BSG, MSG, air voids, and VMA. 

The contractor must supply the engineer with split samples for assurance testing. The number of samples required 

depends on the amount of material produced. The number of required tests is as follows: 

Sampling Frequencies 

Table 103. Wisconsin sampling frequencies. 

Total Daily Plant Production (Mg) Number of Samples per Day 

45-550 1 

551-1360 2 

1361-2450 3 

2451-3810 4 

3811+ Add one sample for each additional 1360 Mg or part thereof 

Quality Assurance 

Wisconsin DOT is responsible for all assurance testing and acceptance. The assurance testing frequencies must be at 

least 10 percent of the contractor’s QC frequency. Assurance testing is conducted on the properties evaluated for 

QC. 

Acceptance 

The contractor’s control tests are used for acceptance if the results are verified by Wisconsin DOT’s assurance tests. 

Pay Factors 

Pay factors are assigned for gradation, asphalt content, air voids, and VMA. The pay factors are assigned if the 

contractor’s test results and the assurance test results are not within specified tolerances. The pay factors are 

presented in the following table: 

Percent Payment for Mixture 

Table 104. Wisconsin percent payment for mixture. 

Item Produced Within Warning Bands Produced Outside JMF Limits 

Gradation 90 75 

Asphalt content 85 75 

Air voids 70 50 

VMA 90 75 

The minimum single payment applies. 
 303

QC/QA Tests for Wisconsin 

Table 105. Wisconsin QC/QA tests. 

Test Sampling 

Location 

QC Testing by 

Contractor 

Assurance Testing 

Frequency by 

Wisconsin DOT 

Acceptance by 

Wisconsin DOT 

Gradation 

(extracted) Truck See Sampling Frequency table 10% of QC frequency Based on verification of QC tests by assurance tests 

Asphalt content Truck See Sampling 

Frequency table 10% of QC frequency Based on verification of QC tests by assurance tests 

BSG Truck See Sampling 

Frequency table 10% of QC frequency Based on verification of QC 

tests by assurance tests 

MSG Truck See Sampling 

Frequency table 10% of QC frequency Based on verification of QC tests by assurance tests 

Air voids Truck See Sampling 

Frequency table 10% of QC frequency Based on verification of QC tests by assurance tests 

VMA Truck See Sampling 

Frequency table 10% of QC frequency Based on verification of QC tests by assurance tests 
 304

Wyoming 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

The contractor is responsible for QC for the following properties: gradation, liquid limit and plastic index of virgin 

material, moisture content of mix, in-place density, test strip, and mix verification. 

Mix design verification is evaluated for the following properties: LA abrasion, number of Marshall blows, Marshall 

stability, Marshall flow, percent voids in mix, dust-to-asphalt ratio, minimum asphalt percentage, minimum TSR, 

film thickness, and VMA. 

Quality Assurance 

Wyoming DOT is responsible for all assurance testing. Assurance testing is conducted on the properties evaluated 

for QC. 

Acceptance 

The contractor’s control test cannot be used for acceptance. The testing frequencies vary, depending on the property 

being measured. Acceptance testing is conducted on the following characteristics: gradation, in-place density, and 

asphalt content. 

Pay Factors 

Pay factors are assigned for gradation, in-place density, and asphalt content. Determination of the pay factors uses 

quality-level analysis. Once the quality level and corresponding pay factor are determined, the pay adjustments are 

calculated using the following equations: 

PAA = 0.67 × PMP × (PFA – 1) × LSA (53) 

PAD = 1.33 × PMP × (PFD – 1) × LSD (54) 

PAAC = 0.67 × PMP × (PFAC – 1) × LSAC (55) 

where: PAA = pay adjustment for aggregate gradation 

PFA = pay factor for gradation 

PAD = pay adjustment for density 

PFD = pay factor for density 

PAAC = pay adjustment for asphalt content 

PFAC = pay factor for asphalt 

LSA = lot size for aggregate gradation evaluation 

LSD = lot size for density evaluation 

LSAC = lot size for asphalt content evaluation 

PMP = bituminous mixture unit price 
 305

QC/QA Tests for Wyoming 

Table 106. Wyoming QC/QA tests. 

Test Sample 

Location 

QC Testing by 

Contractor 

Assurance 

Testing 

Frequency by 

Wyoming DOT 

Acceptance Testing 

Frequency by 

Wyoming DOT 

Stockpile gradation Stockpile One per 1000 tons One per 1000 tons One per lot or 5000 tons 

Liquid limit and 

plastic index (virgin 

aggregate) 

Individual 

stockpiles One per 1000 tons One per 1000 tons NA 

Virgin aggregate 

gradation Cold feed One per 1000 tons One per 1000 tons One per lot or 5000 tons 

Mix verification Truck 

One per day for 4 

days, then one per 

20,000 tons 

One per day for 4 

days, then one per 

20,000 tons 

NA 

Moisture content of 

mix Truck One per day One per day NA 

Test strip Roadway Required NA NA 

In-place density Roadway One per 200 tons One per 200 tons Seven per lot or 1500 

tons 

Asphalt content Truck One per day One per day One per day 

NA = Not applicable 

1 ton = 0.907 Mg 

 307

APPENDIX C: SUMMARY OF AGENCY SUPERPAVE 

SPECIFICATIONS RECEIVED

Connecticut 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The contractor must allow the engineer access to the laboratory to observe any and all testing procedures, 

calculations, test documentation, and plotting of results. 

Mix design verification is evaluated for the following properties: gradation, asphalt content, voids @ Ndes, VMA, 

VFA, Gmm @ Ndes and Nmax, Gmb, Gse, TSR, Ndes, Nini, Nmax, recycled asphalt pavement (RAP) gradation, fine 

aggregate angularity (FAA), coarse aggregate angularity (CAA), sand equivalent, and dust-to-asphalt ratio. 

The contractor must exercise control over the following properties: asphalt content, gradation, air voids, mix 

moisture, VMA, VFA, BSG of mix, MSG of mix, RAP gradation, aggregate moisture, RAP moisture, and density. 

In addition, the contractor must provide split samples to the engineer for verification testing. 

Quality Assurance 

Connecticut DOT is responsible for QA. Assurance is accomplished through verification testing by Connecticut 

DOT and through independent assurance testing. Verification tests are conducted on split samples. A lot is 

considered 1 day’s production. Sublots are five equal divisions of any lot. 

Acceptance 

Acceptance is based on the QC tests as verified by Connecticut DOT’s verification tests. Density acceptance is 

based on the PWL of a lot. The PWL is calculated using 10 tests (2 per sublot). 

Pay Factors 

Pay factors are assigned for gradation, asphalt content, joint density, and mat density. The pay factors for each can 

be determined from the following tables. Total payment is determined by successively multiplying the percent 

payment by the contract price per metric ton of material. 
 308

Pay Factors for Gradation and Binder Content 

Maximum Allowable Deviation (M.A.D.) From Job-Mix Formula for Consecutive Tests 

Table 107. Connecticut M.A.D. from job-mix formula for consecutive tests. 

Item M.A.D. Percent Payment 

Binder ±0.4 90 

 75 ±2 90 

150 ±3 90 

300 ±3 90 

600 ±4 90 

 1.18 ±4 90 

 2.36 ±6 90 

 4.75 ±6 90 

 9.5 ±6 90 

 12.5 ±6 90 

 19.0 ±6 90 

 25.0 ±6 90 

 37.5 ±6 90 

 50.0 ±6 90 

Pay Factors for Joint and Mat Density 

Table 108. Connecticut pay factors for joint and mat density. 

PWL Percent Payment Mat 

Density 

Percent Payment Joint 

Density 

90-100 100 100 

80-90 0.5PWL + 55 0.5PWL + 55 

65-80 2.0PWL – 65 2.0PWL – 65 

 < 65 0 or remove 50 
 309

QC/QA Tests for Connecticut 

Table 109. Connecticut QC/QA tests. 

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Verification Testing 

Frequency by 

Connecticut DOT 

Acceptance by 

Connecticut DOT 

based on ... 

Gradation Conveyor prior to 

binder addition Five per lot One per lot QC verified by Connecticut DOT’s test 

Moisture content 

of RAP Stockpile Two per day One per lot QC verified by Connecticut DOT’s test 

Moisture content 

of aggregate 

Conveyor prior to 

binder addition Two per day One per lot QC verified by Connecticut 

DOT’s test 

Asphalt content 

of RAP Stockpile Two per day One per lot QC verified by Connecticut DOT’s test 

Rap gradation Stockpile Two per day One per lot QC verified by Connecticut 

DOT’s test 

Asphalt content Behind paver Five per lot One per lot QC verified by Connecticut 

DOT’s test 

Superpave molds 

@ Nmax

Behind paver Five sets (two 

molds) per lot One per lot QC verified by Connecticut DOT’s test 

BSG Behind paver Five per lot One per lot QC verified by Connecticut 

DOT’s test 

Air voids Behind paver Five per lot One per lot QC verified by Connecticut 

DOT’s test 

VMA Behind paver Five per lot One per lot QC verified by Connecticut 

DOT’s test 

VFA Behind paver Five per lot One per lot QC verified by Connecticut 

DOT’s test 

MSG of mix Plant Five per lot One per lot QC verified by Connecticut 

DOT’s test 

Mat density 

(nuclear) Roadway 10 per lot NA Based on PWL for lot 

Joint density 

(nuclear) Roadway 10 per lot NA Based on PWL for lot 

NA = Not applicable 
 310

Kansas 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. All testing equipment and procedures must conform to the Kansas DOT Superpave Mix Design and 

Superpave Field Laboratory Technician Certification Training Manual. The contractor must allow the engineer 

access to the laboratory to observe any and all testing procedures, calculations, test documentation, and plotting of 

results. 

Mix design verification is evaluated for the following properties: gradation, asphalt content, theoretical MSG, BSG, 

air voids, VMA, VFA, dust-to-asphalt ratio, flat or elongated particles, percent moisture, and TSR. 

The contractor must exercise control over the following properties: asphalt content, gradation, air voids, VMA, 

VFA, CAA, sand equivalent, FAA, TSR, density, and dust-to-asphalt ratio. Samples for mixture properties, 

aggregate gradation, and binder must be taken from behind the paver and transported to the field laboratory for 

testing. 

Quality Assurance 

Kansas DOT is responsible for all assurance and verification testing. Kansas DOT assurance tests are independent of 

the contractor’s QC tests. The lot size for verification testing is 3000 tons (2720 Mg) with 750-ton (680 Mg) sublots. 

Acceptance 

The contractor’s control tests are used for acceptance unless the results do not compare favorably with Kansas 

DOT’s verification test results. In this case, Kansas DOT’s tests are used for acceptance unless the contractor 

requests referee testing. The testing frequencies vary, depending on the property being measured. 

Pay Factors 

Pay factors are assigned for in-place density, air voids, and smoothness. Pay factors for density and air voids can be 

determined from the following tables. Information concerning methods for determining a smoothness pay factor was 

not included. 
 311

Pay Factors for Specified Density 

Table 110. Kansas pay factors for specified density. 

Pay Factor A(b) Percentage of MSG, 

Pay Factor A(b)

(Average of 10 Density 

Tests(a)) 

Mixes SM-2A, 

SR-2A, SM-2C, and 

SR-2C 

Mix SM-1T Mixes SM-1B and 

SR-1B 

 ≥ 94.0 1.020 

 ≥ 93.0 1.020 1.020 

93.0-93.9 A4 

92.0-92.9 A1 A1 1.000 

91.0-91.9 1.000 1.000 

90.0-90.9 1.000 

88.0-91.9 A5 

87.0-90.9 A2 

86.0-89.9 A3 

 < 88.0 (c) 

 < 87.0 (c) 

 < 86.0 (c) 

Lowest Average of Any 

Sublot Within the Lot Pay Factor B(b)

 ≥ 92.0 1.020 

 ≥ 91.0 1.020 1.020 

91.0-90.9 B4 

90.0-90.9 B1 B1 1.000 

89.0-89.9 1.000 1.000 

88.0-88.9 1.000 

86.0-89.9 B5 

85.0-88.9 B2 

84.0-87.9 B3 

 < 86.0 (c) 

 < 85.0 (c) 

 < 84.0 (c) 

(a) For low daily production rates of less than 1000 tons (910 Mg) or when the engineer’s verification tests are to be 

used for density pay determination, the lot sample size is as determined under subsection 603.05(b) for 

compaction testing. 

(b) For shoulders less than 1.8 m wide and containing rumble strips, the density pay factors will not apply for the top 

lift. Density will be controlled by using an approved rolling procedure. 

(c) Engineer will determine whether the material may remain in place. The pay factor for such material remaining in 

place is 0.700 for pay factor A and 0.800 for pay factor B. 

Calculations for Pay Factors A1, A2, A3, A4, A5, B1, B2, B3, B4, and B5: 

A1 = [100 + 2 (percentage of lot MSG – 92.0)] ÷ 100 

A2 = [90 + 2.5 (percentage of lot MSG – 87.0)] ÷ 100 

A3 = [90 + 2.5 (percentage of lot MSG – 86.0)] ÷ 100 

A4 = [100 + 2 (percentage of lot MSG – 93.0)] ÷ 100 

A5 = [90 + 2.5 (percentage of lot MSG – 88.0)] ÷ 100 

B1 = [100 + 2 (percentage of sublot MSG – 90.0)] ÷ 100 

B2 = [90 + 2.5 (percentage of sublot MSG – 85.0)] ÷ 100 

B3 = [90 + 2.5 (percentage of sublot MSG – 84.0)] ÷ 100 

B4 = [100 + 2 (percentage of sublot MSG – 91.0)] ÷ 100 

B5 = [90 + 2.5 (percentage of sublot MSG – 86.0)] ÷ 100 
 312

Pay Factor Calculation: Density Pay Adjustment Factor (PD)* = [(Pay factor A)(Pay factor B)] – 1.000 (56) 

* PD will be rounded to the nearest 0.001 

Air Void Payment 

Payment adjustment for air voids will be computed by multiplying the air voids pay adjustment factor (PV) times the 

number of tons included in the lot times the bid price per ton. Calculate PV to 0.001. 

A lot will normally be comprised of the results of four contiguous individual air void tests performed on Superpave 

gyratory compacted samples of a given mix design. Lot size is defined in subsections 5.0(f) and 5.0(g) of this special 

provision. (Air void lots and density lots are normally of different sizes.) 

The absolute value of the deviation from the target at Ndes for each individual air void test in a lot will be computed 

as shown: 

 Deviation From Target = 4.0 – Test value (57) 

The average deviation (D) from the target will be computed as follows: 

 D = Sum of deviations from target ÷ number of tests in lot (58) 

The average air void deviation (D) will be used to select the air void pay factor from the appropriate table below. 

Calculate D to 0.01. 

Pay Factor Table for Air Voids (Lot Size of Four Tests) 

Table 111. Kansas pay factors for air voids (lot size of four tests). 

Air Void Average Deviation Pay Factor 

0.00 ≤ D4 ≤ 0.35 1.030 

0.36 ≤ D4 ≤ 0.55 1.000 + 0.15(0.55 – D) 

0.56 ≤ D4 ≤ 1.05 1.000 

1.06 ≤ D4 ≤ 1.40 1.000 – 0.44(D – 1.05) 

1.41 ≤ D4 (a) 

(a) Engineer will determine whether the material may remain in place. The pay factor for such material 

remaining in place is 0.8000. 

When the testing rate does not provide a complete four-test lot (e.g., at the end of a production run), the following 

applies: Combine the three tests into a lot and use the following table for D3. When there are one or two tests 

remaining, combine them with the previous four tests to create a five- or six-test lot, respectively. The pay factors 

for D5 and D6 are also shown below. 

Pay Factor Table for Air Voids (Lot Size of Three Tests) 

Table 112. Kansas pay factors for air voids (lot size of three tests). 

Air Void Average Deviation Pay Factor 

0.00 ≤ D3 ≤ 0.37 1.030 

0.38 ≤ D3 ≤ 0.58 1.000 + 0.14(0.58 – D) 

0.59 ≤ D3 ≤ 1.11 1.000 

1.12 ≤ D3 ≤ 1.48 1.000 – 0.35(D – 1.11) 

1.49 ≤ D3 (a) 

(a) Engineer will determine whether the material may remain in place. The pay factor for such material 

remaining in place is 0.8000. 
 313

Pay Factor Table for Air Voids (Lot Size of Five Tests) 

Table 113. Kansas pay factors for air voids (lot size of five tests). 

Air Void Average Deviation Pay Factor 

0.00 ≤ D5 ≤ 0.29 1.030 

0.30 ≤ D5 ≤ 0.53 1.000 + 0.125(0.53 – D) 

0.54 ≤ D5 ≤ 1.01 1.000 

1.02 ≤ D5 ≤ 1.35 1.000 – 0.41(D – 1.01) 

1.36 ≤ D5 (a) 

(a) Engineer will determine whether the material may remain in place. The pay factor for such material 

remaining in place is 0.8000. 

Pay Factor Table for Air Voids (Lot Size of Six Tests) 

Table 114. Kansas pay factors for air voids (lot size of six tests). 

Air Void Average Deviation Pay Factor 

0.00 ≤ D6 ≤ 0.28 1.030 

0.29 ≤ D6 ≤ 0.51 1.000 + 0.13(0.51 – D) 

0.52 ≤ D6 ≤ 0.98 1.000 

0.99 ≤ D6 ≤ 1.31 1.000 – 0.39(D – 0.98) 

1.32 ≤ D6 (a) 

(a) Engineer will determine whether the material may remain in place. The pay factor for such material 

remaining in place is 0.8000. 

Pay Factor Calculation: Air Void Pay Adjustment Factor (PV) = Pay factor – 1.000 (59) 
 314

QC/QA Tests for Kansas 

Table 115. Kansas QC/QA tests. 

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Verification Testing 

Frequency by 

Kansas DOT 

Assurance Testing 

Frequency by 

Kansas DOT 

Binder sampling Plant One per three loads One per project NA 

Binder content Behind paver One per sublot One per three lots 

Witness and test once per 

year by certified 

technician 

Mix gradation 

(extraction) Behind paver One per sublot One per three lots 

Witness and test once per 

year by certified 

technician 

Theoretical MSG Behind paver One per sublot One per lot 

Witness and test once per 

year by certified 

technician 

Air voids Behind paver One per sublot One per lot 

Compact and test split 

sample once per week or 

15,000 tons 

RAP binder content Stockpile One per 1000 tons One per 20,000 tons NA 

RAP gradation Stockpile One per 1000 tons One per project NA 

Moisture damage to 

mix Behind paver 

One on first lot, then 

one per week or 

10,000 tons 

One per project 

Witness and test once per 

year by certified 

technician 

Sand equivalent From conveyor One per lot One per project 

Witness and test once per 

year by certified 

technician 

CA angularity Stockpile One per 1000 tons One per week or 

10,000 tons 

Witness and test once per 

year by certified 

technician 

FA angularity Stockpile One on first lot, then 

one per 10,000 tons One per project 

Witness and test once per 

year by certified 

technician 

Gradation of 

individual aggregate Stockpile One per 1000 tons One per project per aggregate 

Witness and test once per 

year by certified 

technician 

Percent moisture in 

mix Behind paver One per lot One per project NA 

Percent moisture in 

combined aggregate From conveyor One per lot One per project NA 

Percent moisture in 

RAP Stockpile One per lot One per project NA 

Plasticity index and 

gradation of mineral 

filler 

Stockpile One per 250 tons One per project 

Witness and test once per 

year by certified 

technician 

In-place density Roadway 10 tests per lot Five companion tests 

per lot 

Witness and test once per 

year by certified 

technician 

Smoothness Roadway NA NA NA 

Flat or elongated 

particles Behind paver One on first lot One per project 

Witness and test once per 

year by certified 

technician 

NA = Not applicable 

1 ton = 0.907 Mg 
 315

Louisiana 

Quality Control 

The contractor is responsible for developing the mix design for the project. The JMF must include the recommended 

formula, extracted gradation, and supporting design data. In addition, the contractor is responsible for providing a 

QC plan that must be approved by the engineer prior to construction and providing qualified personnel and 

equipment to conduct QC testing. The contractor must allow the engineer access to the laboratory to observe any 

and all testing procedures, calculations, test documentation, and plotting of results. 

Mix design verification is evaluated on the following properties: gradation, asphalt content, percent crushed 

aggregate, MSG, corrected BSG @ Ndes, percent air voids @ Ndes, percent VMA @ Ndes, percent VFA @ Ndes, 

gyratory compaction curve @ Nmax, slope of the compaction curve, and percent anti-strip. 

Upon validation of the JMF, the validation parameters shall be used to control the process. The quality of the 

mixtures will be evaluated during two phases: (1) production of the mixture at the plant, and (2) hauling, laying, and 

compacting of the mixture. The quality of both phases must be evaluated continuously. 

Acceptance 

The Louisiana Department of Transportation and Development (DOTD) is responsible for all acceptance testing. 

Louisiana DOTD’s acceptance tests are independent of the contractor’s QC tests. The lot size is 5000 Mg, with 

1000-Mg sublots unless otherwise specified by the engineer. Acceptance is based on the validation parameters used 

to verify the JMF. Acceptance testing for VMA, VFA, voids, maximum theoretical gravity, Nini, Ndes, Nmax, percent 

anti-strip, quality of asphalt cement, extracted aggregate gradation, percent crushed aggregate, percent asphalt 

cement, and percent moisture in the mix will be conducted on the total lot quantity. 

Pay Factors 

Payment adjustments will be calculated based on specification limits and specified maximum variances from the 

JMF: 

Plant Acceptance: 

For: Percent voids 

 Percent VMA 

 Extracted aggregate gradation 

 #8 sieve 

 #200 sieve 

PWL will be calculated for each of the parameters listed above. A mixture with any parameter below 40 PWL is 

subject to removal as directed by the engineer. The percent payment for each of the parameters will be calculated 

using the formula: 

 Percent Payment = 55 + 0.5PWL (60) 

For: Percent anti-strip additive 

Payment adjustments for anti-strip additive will be based on the table below. 

For: Asphalt cement 

Payment adjustments for asphalt cement will be based on section 1012 of the specifications. 

For: Percent payment for plant acceptance 

All payment parameters will be averaged. The maximum payment factor shall be 100 percent. 
 316

Roadway Acceptance: 

For: Pavement density 

PWL will be calculated for the pavement density. A mixture with density results below 40 PWL is subject to 

removal as directed by the engineer. The percent payment for pavement density will be calculated using the formula: 

 Percent Payment = 55 + 0.5PWL (61) 

For: Surface tolerance (final wearing course travel lanes) 

Payment adjustments for surface tolerance will be based on the table below. 

For: Percent payment for roadway acceptance 

All payment parameters will be averaged. The maximum payment factor shall be 100 percent. 

Total Payment Factor: 

The total payment factor for the mixture used for the project will be calculated using the following formula: 

Total Percent Payment = (Percent payment for plant acceptance) + (Percent payment for roadway acceptance) – 100 

Payment Adjustment Schedule: Smoothness and Percent Anti-Strip 

Table 116. Louisiana payment adjustment schedule. 

Surface Tolerance Percentage of Contract Unit Price per Lot 

(inches/mi per lot) 100 98 95 80 50 or remove 

Multi-lift new construction 0.0-3.0 — 3.1-4.0 4.1-6 More than 6.0 

One- or two-lift overlays over planed 

surfaces 0.0-5.0 — 5.1-6.0 6.1-10 More than 10.0 

Single-lift overlays over existing 

surface 0.0-8.0 — 8.1-10 10.1-15 More than 15.0 

Anti-strip, percent below JMF Within JMF — 0.2 or less More than 

0.2 — 

1 inch/mi = 15.8 mm/km 

QC/QA Tests for Louisiana 

Table 117. Louisiana QC/QA tests. 

Test Sampling Location QC Testing Frequency 

by Contractor 

Acceptance Testing 

Frequency by 

Louisiana DOTD 

Gradation (extracted) Mix from truck Two per sublot One per sublot 

Percent crushed Mix from truck Two per sublot One per sublot 

Asphalt content Mix from truck Two per sublot One per sublot 

Percent VMA Mix from truck Two per sublot Average of five per lot 

Percent VFA Mix from truck Two per sublot Average of five per lot 

Air voids Mix from truck Two per sublot Average of five per lot 

Maximum theoretical gravity Mix from truck Two per sublot One per lot 

Nini Mix from truck Two per sublot One per lot 

Ndes Mix from truck Two per sublot One per lot 

Nmax Mix from truck Two per sublot One per lot 

Percent anti-strip Plant Two per sublot Two per lot 

Asphalt properties Plant Two per sublot One per lot 

Percent moisture in mix Mix from truck Two per sublot One per lot 

Density Roadway Obtain samples for engineer Five per sublot 

Profile index Roadway One per day’s placement One per lot 
 317

Maine 

Quality Control 

The contractor is responsible for developing the mix design for the project, providing a QC plan that must be 

approved by the engineer prior to construction, and providing qualified personnel and equipment to conduct QC 

testing. The QC inspections and tests must be documented and provided to Maine DOT. The contractor must 

maintain records of all inspections and tests. These records shall indicate the nature, number, and type of 

deficiencies found; the quantities approved and rejected; and the nature of the corrective actions taken. The 

contractor must allow the engineer access to the laboratory to observe any and all testing procedures, calculations, 

test documentation, and plotting of results. 

Mix design verification is evaluated on the following properties: gradation, asphalt content, theoretical MSG, CA 

angularity, FA angularity, air voids, VMA, VFA, dust-to-asphalt ratio, flat or elongated particles, percent moisture, 

and TSR. 

The contractor must exercise control over the following properties: asphalt content, temperature of mix, temperature 

of mat, percent TMD, gradation, air voids, VMA, VFA, Rice specific gravity, CA angularity, and dust-to-asphalt 

ratio. 

Quality Assurance 

Maine DOT is responsible for QA and acceptance of the material. QA is accomplished using verification testing. 

Maine DOT’s verification tests are independent of the contractor’s QC tests. Lot size is usually 6000 Mg or less, 

with sublots of 1500 Mg or less. 

Acceptance 

The contractor’s control tests are not used for acceptance. Acceptance is based on Maine DOT’s verification tests. If 

the QC and QA test results do not agree, acceptance is based on referee testing for asphalt content, density, air voids, 

and VMA. The testing frequencies vary, depending on the property being measured. 

Pay Factors 

Pay factors are assigned for gradation, asphalt content, in-place density, air voids, and VMA. The determination of 

these factors uses quality-level analysis and price adjustment or f factors listed in the specification for the applicable 

property. Pay factors are assigned on a lot-by-lot basis. The composite pay factor is then computed by: 

 CPF = [f1(PF1) + f2(PF2) + .... + fj(PFj)] / ∑f (62) 
 318

QC/QA Tests for Maine 

Table 118. Maine QC/QA tests. 

Test Sampling 

Location 

Quality Control 

Testing 

Frequency by 

Contractor 

Verification 

Testing 

Frequency by 

Maine DOT 

Acceptance by Maine DOT 

Gradation Truck body or 

paver hopper 

One per day 

minimum One per 1500 Mg Based on verification tests 

unless referee testing is required

Percent 

TMD 

Mat behind all 

rollers One per 125 Mg One per 250 Mg Based on verification tests unless referee testing is required

Dust-toasphalt ratio Truck body or 

paver hopper One per 1500 Mg One per 1500 Mg Based on verification tests unless referee testing is required

Temperature 

of mix 

Truck body or 

paver hopper 

Six per day at plant 

and street NA NA 

Temperature 

of mat Roadway Four per day NA NA 

Asphalt 

content 

Truck body or 

paver hopper One per 750 Mg One per 1500 Mg Based on verification tests unless referee testing is required

Air voids Truck body or 

paver hopper One per 750 Mg One per 1500 Mg Based on verification tests unless referee testing is required

VMA Truck body or 

paver hopper One per 750 Mg One per 1500 Mg Based on verification tests unless referee testing is required

Rice specific 

gravity 

Truck body or 

paver hopper One per 750 Mg NA NA 

CA 

angularity Stockpile One per 5000 Mg NA NA 

Flat or 

elongated 

particles 

Stockpile One per 5000 Mg NA NA 

Density 

(nuclear) Roadway One per 250 Mg One per 250 Mg Based on verification tests unless referee testing is required

NA = Not applicable 
 319

Minnesota 

Quality Control 

The contractor is responsible for developing the mix design in accordance with AASHTO TP-4, June 1997, or the 

Asphalt Institutes’ Superpave Mix Design Manual (SP-2), such that it meets the requirements of the specification. In 

addition, the contractor is responsible for providing a QC plan that must be approved by the engineer prior to 

construction and providing qualified personnel and equipment to conduct QC testing. The contractor must allow the 

engineer access to the laboratory to observe any and all testing procedures, calculations, test documentation, and 

plotting of results. 

Mix design verification is evaluated on the following properties: gradation; asphalt content; voids; VMA; TSR; 

compaction @ Ndes, Nini, and Nmax; VFA; FA angularity; CA angularity; Gmm; Gmb; Gsb; mix moisture; and dust-toasphalt ratio. 

The contractor must exercise control over the following properties: asphalt content, gradation, voids, mix moisture, 

VMA, FA angularity, density, CA angularity, and percent Gmm @ Ndes. In addition, the contractor must provide split 

samples to the engineer for verification testing. Testing frequency will be equal to the day’s planned production 

divided by 1000, then rounded to the next whole number. 

Quality Assurance 

Minnesota DOT is responsible for QA and for the acceptance of the material. The assurance process is achieved 

through the use of verification tests conducted on split samples, observation of sampling and testing performed by 

the QC personnel, the taking of additional samples for testing, and the monitoring of the required QC summary 

sheets and control charts. Lot size varies based on the contract quantity. Verification tests are conducted once per 

day on all production parameters. 

Acceptance

Acceptance is based on the QC tests as verified by Minnesota DOT’s tests for gradation, asphalt content, maximum 

specific gravity, Gmb, air voids, VMA, and density. 

Pay Factors 

Pay factors are assigned for nonconforming mixes for the following properties: gradation, FA angularity, CA 

angularity, VMA, asphalt content, and Gmm @ Ndes. For these characteristics, the lowest single payment applies 

according to the following table: 

Payment Schedule 

Table 119. Minnesota payment schedule. 

Item Percent Payment 

FAA 95 

CAA 95 

Gradation (extracted) 90 

VMA 85 

Asphalt binder content 85 

Gmm @ Ndes 70 

The pay factor for in-place density is determined on a lot-by-lot basis, with the number of lots determined by the 

following table: 
 320

Determination of Lots for Density 

Table 120. Minnesota determination of lots for density. 

Daily Production (tons) Lots 

 0-300 1 

 301-600 2 

 601-1000 3 

1001-1600 4 

1601-3600 5 

3601-5000 6 

5000+ 7 

 1 ton = 0.907 Mg 

The pay factor is then computed using the following table: 

Payment Schedule for Density 

Table 121. Minnesota payment schedule for density. 

Percentage of Gmb 

(< 100 mm from surface) 

Percentage of Gmb 

(> 100 mm from surface) Percent Payment 

 ≥ 93.6 ≥ 94.6 104 

93.1-93.5 94.1-94.5 102 

92.0-93.0 93.0-94.0 100 

91.0-91.9 92.0-92.9 98 

90.5-90.9 91.5-91.9 95 

90.0-90.4 91.0-91.4 91 

89.5-89.9 90.5-90.9 85 

89.0-89.4 90.0-90.4 70 

 < 89.0 < 90.0 Remove and replace 

The profile index is determined using a California-type profilograph. Profiles will be made for segments of 0.1 km. 

The following table determines the pay factor for smoothness: 

Payment Schedule for Smoothness 

Table 122. Minnesota payment schedule for smoothness. 

mm/km per 0.1-km Segment Dollars per Segment 

 0-13 110 

 14-25 90 

 26-38 65 

 39-79 0 

 80-92 -90 

 93-105 -180 

106-118 -360 

 > 118 Corrective action 
 321

QC/QA Tests for Minnesota 

Table 123. Minnesota QC/QA tests. 

Test Sampling 

Location 

QC Testing Frequency 

by Contractor 

Verification Test 

by Minnesota 

DOT 

Acceptance by 

Minnesota DOT 

Gradation 

(extracted) Truck or paver 

One day’s planned 

production, divided by 1000, 

then rounded to the next 

whole number 

One per day Based on QC as verified 

by Minnesota DOT’s tests 

Asphalt content Truck or paver 

One day’s planned 

production, divided by 1000, 

then rounded to the next 

whole number 

One per day Based on QC as verified 

by Minnesota DOT’s tests 

Air voids Truck or paver 

One day’s planned 

production, divided by 1000, 

then rounded to the next 

whole number 

One per day Based on QC as verified 

by Minnesota DOT’s tests 

VMA Truck or paver 

One day’s planned 

production, divided by 1000, 

then rounded to the next 

whole number 

One per day Based on QC as verified 

by Minnesota DOT’s tests 

Gmm @ Ndes Truck or paver 

One day’s planned 

production, divided by 1000, 

then rounded to the next 

whole number 

One per day Based on QC as verified 

by Minnesota DOT’s tests 

TSR Plant One per 10,000 tons NA NA 

Gsb Plant One per 10,000 tons NA NA 

CAA Conveyor One per day One per day Based on QC as verified 

by Minnesota DOT’s tests 

FAA Conveyor One per day One per day Based on QC as verified 

by Minnesota DOT’s tests 

Moisture 

content Truck or paver Engineer’s discretion Engineer’s discretion NA 

Smoothness Roadway One pass per lane segment 

 (0.1 km) Observation of QC Based on QC testing as verified by observation 

Density 

(percentage of 

Gmb) 

Roadway Three cores per lot One companion 

core per lot 

Based on QC as verified 

by Minnesota DOT’s tests 

NA = Not applicable 

1 ton = 0.907 Mg 
 322

Mississippi 

Quality Control 

The contractor is responsible for developing the mix design for the project. In addition, the contractor is responsible 

for providing a QC plan that must be approved by the engineer prior to construction. The contractor must also 

provide qualified personnel and equipment to conduct QC testing. The contractor must allow the engineer access to 

the laboratory to observe any and all testing procedures, calculations, test documentation, and plotting of results. 

Mix design verification is evaluated on the following properties: gradation, asphalt content, asphalt properties, voids 

@ Ndes, VMA @ Ndes, fractured faces, TSR, Ndes, Nini, Nmax, stripping, RAP gradation, FA angularity, MSG, and 

dust-to-asphalt ratio. 

The contractor must exercise control over the following properties: asphalt content, gradation, voids @ Ndes, mix 

moisture, VMA @ Ndes, FA angularity, density, fractured faces, MSG, stripping, and RAP gradation. In addition, the 

contractor must provide split samples to the engineer for verification testing. Sampling frequency for QC of voids, 

VMA, gradation, MSG, FA angularity, CA angularity, RAP gradation, stripping, and asphalt content is as follows: 

Sampling Frequency 

Table 124. Mississippi sampling frequency. 

Total Estimated Production (tons) Number of Tests 

 50-600 1 

 601-1500 2 

1501-2700 3 

2701+ 4 

 1 ton = 0.907 Mg 

Determination of the lots for density testing is as follows: 

Lot Determination for Density 

Table 125. Mississippi lot determination for density. 

Daily Production (tons) Number of Lots 

 0-300 1 

 301-600 2 

 601-1000 3 

1001-1500 4 

1501-2100 5 

2101-2800 6 

2800+ 7 

 1 ton = 0.907 Mg 

Quality Assurance 

Mississippi DOT is responsible for QA and acceptance of the material. QA is accomplished by verification testing 

on split samples, testing additional samples at any time, and observing the QC tests. Lot size varies based on the 

contract quantity. 

Acceptance 

Acceptance is based on the QC tests as verified by Mississippi DOT’s tests for total voids, VMA @ Ndes, gradation, 

asphalt content, and density. Acceptance for smoothness is based on the observed contractor QC tests. Should the 

QC tests and the verification tests fail to meet the allowable differences, the engineer’s test results will be used for 

acceptance. 
 323

Pay Factors 

Pay factors are assigned for gradation, in-place density, total voids @ Ndes, smoothness, and VMA @ Ndes. Pay 

factors for these properties can be determined from the following tables: 

Pay Factors for Mixture Quality 

Table 126. Mississippi pay factors for mixture quality.*

Item Produced in Warning 

Bands 

Outside JMF limits 

(allowed to remain in place) 

Gradation 0.90 0.75 

Asphalt content 0.85 0.75 

Total voids @ Ndes 0.70 0.50 

VMA @ Ndes 0.90 0.75 

* Minimum single pay factor applies. 

Pay Factors for Density 

Table 127. Mississippi pay factors for density. 

Pay Factor Lot Density (percentage of maximum 

density) 

1.00 ≥ 92.0 

0.90 91.0-91.9 

0.80 90.0-90.9 

Pay Factors for Smoothness 

Table 128. Mississippi pay factors for smoothness.

Profile Index (inch/mi per lot) Pay Factor (percentage of unit bid price) 

≤ 3 105 

> 3-5 102 

> 5-7 100 

> 7-8 95 

> 8-10 90 

> 10 Unacceptable 

 1 inch/mi = 16 mm/km 
 324

QC/QA Tests for Mississippi 

Table 129. Mississippi QC/QA tests. 

Test Sample 

Location 

QC Testing Frequency 

by Contractor 

Verification 

Testing 

Frequency by 

Mississippi DOT 

Acceptance by 

Mississippi DOT 

Binder sampling Plant tank One per 200,000 gal ≥ 10% of QC tests NA 

Binder content Truck See QC table 1 ≥ 10% of QC tests Based on verified QC by 

Mississippi DOT’s tests 

Mix gradation 

(extracted) Truck See QC table 1 ≥ 10% of QC tests Based on verified QC by Mississippi DOT’s tests 

MSG Truck See QC table 1 ≥ 10% of QC tests NA 

Voids @ Ndes Truck See QC table 1 ≥ 10% of QC tests Based on verified QC by 

Mississippi DOT’s tests 

Fractured faces Stockpiles One per day ≥ 10% of QC tests NA 

RAP gradation Stockpiles See QC table 1 ≥ 10% of QC tests NA 

Moisture damage to mix Plant mix One per 2 weeks ≥ 10% of QC tests NA 

VMA @ Ndes Truck See QC table 1 ≥ 10% of QC tests Based on verified QC by 

Mississippi DOT’s tests 

CA angularity Stockpiles See QC table 1 ≥ 10% of QC tests NA 

FA angularity Stockpiles See QC table 1 ≥ 10% of QC tests NA 

Density (nuclear) Roadway Two readings per lot ≥ 10% of QC tests Based on verified QC by 

Mississippi DOT’s tests 

Smoothness Roadway One wheel path per lane per 

segment (0.1 mi) 

Observation of QC 

tests 

Verification by 

observation 

Percent moisture in mix Truck Two per day ≥ 10% of QC tests NA 

NA = Not applicable 

1 mi = 1.61 km, 1 gallon (gal) = 3.785 liters (L) 
 325

New York 

Quality Control 

The contractor is responsible for developing the mix design for the project. In addition, the contractor is responsible, 

through the mix manufacturer, for providing a QC plan that must be approved by the engineer prior to construction. 

The contractor must allow the engineer access to all testing procedures, calculations, test documentation, and 

plotting of results. 

Mix design verification is evaluated on the following properties: gradation, asphalt content, compaction @ Nini, Ndes, 

Nmax, VMA, VFA, dust-to-asphalt ratio, CA angularity, FA angularity, sand equivalent, and flat or elongated 

particles. 

The contractor must exercise control over the following properties: asphalt content, gradation, MSG, BSG, air voids, 

VMA, VFA, moisture content, RAP extraction, mix temperature, and density. Additionally, the contractor must 

provide split samples for verification testing by the New York State DOT. The lot size is 1 day’s production of a 

JMF with equal sublots not to exceed 1150 metric tons. 

Quality Assurance 

New York State DOT is responsible for QA and acceptance of the material. New York State DOT’s assurance tests 

are independent of the contractor’s QC tests. The contractor’s control tests are used to assign a quality adjustment 

factor (QAF) for pay factors if the results agree with the verification test results. If New York State DOT’s tests and 

the contractor’s tests are not within specified tolerances, referee testing is conducted on independent samples and 

these results will be used for the determination of the QAF. 

Acceptance 

Acceptance for density is based on the PWL of the percent MTD. Acceptance for gradation, air voids, mix moisture, 

asphalt content, VMA, VFA, MSG, and BSG is based on the contractor’s QC tests if the results are within specified 

tolerances of New York State DOT’s tests. 

Pay Factors 

Pay factors are assigned for gradation, asphalt content, in-place density, and air voids. The method for determining 

the pay factors is currently being researched. 
 326

QC/QA Tests for New York 

Table 130. New York QC/QA tests. 

Test QC Testing by 

Contractor 

Assurance Testing 

by New York State 

DOT 

Acceptance by 

New York State DOT 

Gradation (extracted) 

One per every other 

sublot, minimum of one 

per day 

One per lot minimum 

Based on verification of QC 

by New York State DOT’s 

tests 

Aggregate moisture 

One per every other 

sublot, minimum of two 

per day 

One per lot minimum NA 

Air voids One per sublot One per lot minimum 

Based on verification of QC 

by New York State DOT’s 

tests 

Wet analysis of percent 

passing #200 One per week NA NA 

Mix moisture As required by 

New York State DOT One per lot minimum 

Based on verification of QC 

by New York State DOT’s 

tests 

Asphalt content Four times per day 

minimum One per lot minimum 

Based on verification of QC 

by New York State DOT’s 

tests 

RAP moisture Two per week NA NA 

RAP extraction Two per week NA NA 

Binder test Two per day One per lot minimum 

Based on verification of QC 

by New York State DOT’s 

tests 

VMA One per sublot One per lot minimum 

Based on verification of QC 

by New York State DOT’s 

tests 

VFA One per sublot One per lot minimum 

Based on verification of QC 

by New York State DOT’s 

tests 

Density Four cores per 

1800 lane-meters Average of four per lot 

Based on verification of QC 

by New York State DOT’s 

tests 

MSG One per day One per lot minimum 

Based on verification of QC 

by New York State DOT’s 

tests 

BSG One per day One per lot minimum 

Based on verification of QC 

by New York State DOT’s 

tests 

Mix temperature Four times per day 

minimum NA NA 

NA = Not applicable 
 327

North Carolina 

Quality Control 

The contractor is responsible for developing the mix design for the project and providing a QC plan that must be 

approved by the engineer prior to construction. The QC program will include process control inspection, sampling 

and testing, and necessary adjustments in the process that are related to the production of a pavement that meets all 

of the requirements of the specification. The contractor must also provide qualified personnel and equipment to 

conduct QC testing. The contractor must allow the engineer access to the laboratory to observe any and all testing 

procedures, calculations, test documentation, and plotting of results. 

The contractor must exercise control over the following properties: asphalt content, gradation, air voids @ Ndes, 

VMA @ Ndes, VFA @ Ndes, Gmm, Gmb @ Ndes, dust-to-asphalt ratio, percentage of MSG @ Nini, and aggregate 

moisture content. In addition, the contractor must provide split samples to the engineer for assurance testing. Testing 

frequency is as follows: 

Testing Frequency 

Table 131. North Carolina testing frequency. 

Daily Production (metric tons) Number of Tests 

 80-1000 One 

1001-2500 Two 

2500+ (in 1500 increments) One per 1500 metric tons 

Quality Assurance 

North Carolina DOT is responsible for QA and for acceptance of the material. The verification tests are conducted 

on split samples. Testing frequency is equal to or greater than 10 percent of the contractor’s control testing. QA is 

accomplished by conducting verification tests on split samples, observing tests conducted by the contractor, 

monitoring control charts, and conducting assurance testing on samples independent of the QC tests. 

Acceptance 

Acceptance is based on the QC tests as verified by North Carolina DOT’s tests for asphalt content, Gmm, Gmb @ Ndes, 

and density. 

Pay Factors 

Pay factors are assigned for gradation, binder content, air voids, VMA @ Ndes, density, and profile index. Pay factors 

are determined as follows: 

Payment for Mix Produced in Warning Bands* 

Table 132. North Carolina payment for mix produced in warning bands. 

Item Percentage of Bid Price for Mix 

Gradation 90 

Binder content 85 

Air voids 70 

VMA @ Ndes 90 

* Minimum single payment applies. 

The pay factor for density is determined using the following formula: 

 PF = 100 – 10(D)

1.465 (63) 

where: PF = pay factor computed to the nearest 0.1 percent 

D = deficiency of the lot average density, not to exceed 3.0 percent 
 328

The pay factor for smoothness is determined on a lot-by-lot basis, where a lot is 25 test sections of 100 ft (30 m) of 

single lane. Measurements are made using a North Carolina Hearne Straightedge. The pay factor is assigned as 

follows: 

Table 133. North Carolina pay factor. 

Smoothness Index Acceptance Category Corrective Action Pay Adjustments 

0.0-9.9 Acceptable None +$300 

10-20 Acceptable None +$100 

30-40 Acceptable None No adjustment 

50-60 Acceptable Allowed -$300 

Any other number Unacceptable Required -$600 

QC/QA Tests for North Carolina 

Table 134. North Carolina QC/QA tests. 

Test Sampling 

Location 

QC Frequency 

by Contractor 

Verification 

Frequency by 

North Carolina 

DOT 

Acceptance by 

North Carolina DOT 

Gradation 

(extracted) Truck at plant See QC table 1 ≥ 10% of QC 

testing 

Based on QC as verified 

by North Carolina 

DOT’s tests 

Asphalt content Truck at plant See QC table 1 ≥ 10% of QC 

testing 

Based on QC as verified 

by North Carolina 

DOT’s tests 

Air voids Truck at plant See QC table 1 ≥ 10% of QC 

testing 

Based on QC as verified 

by North Carolina 

DOT’s tests 

VMA Truck at plant See QC table 1 ≥ 10% of QC 

testing 

Based on QC as verified 

by North Carolina 

DOT’s tests 

Gmm Truck at plant See QC table 1 NA NA 

Gmb Truck at plant See QC table 1 NA NA 

VFA Truck at plant See QC table 1 NA NA 

Dust-to-asphalt 

ratio Truck at plant See QC table 1 NA NA 

Percent MSG Truck at plant See QC table 1 NA NA 

Aggregate 

moisture 

content 

Cold feed One per day NA NA 

Smoothness Roadway Test each 100 ft of 

single lane Observe QC test As verified by observation 

Density Roadway One test per 1500 

linear-feet 

≥ 10% of QC 

testing 

Based on QC as verified 

by North Carolina 

DOT’s tests 

NA = Not applicable 

1 ft = 0.305 m 
 329

APPENDIX D: SUMMARY OF AGENCY 

PCC SPECIFICATIONS RECEIVED

Kansas 

Quality Control 

The contractor is responsible for providing a QC plan that must be approved by the engineer prior to construction 

and providing qualified personnel and equipment to conduct QC testing. The contractor must allow the engineer 

access to the laboratory to observe any and all testing procedures, calculations, test documentation, and plotting of 

results. At a minimum, values for percent air, slump, unit weight, and gradation must be plotted on control charts. 

QC tests include aggregate gradation, slump, air content, unit weight/yield, compressive strength, flexural strength, 

material passing #200, percent moisture in aggregate, temperature, and density of fresh concrete. Cores are to be 

tested once per each sublot, where a sublot is the day’s lot production divided into five sublots of equal surface area. 

Quality Assurance 

QA is the responsibility of the engineer and is accomplished through verification testing by Kansas DOT. The 

engineer conducts verification testing on aggregate gradation, slump, air content, unit weight/yield, compressive 

strength, flexural strength, material passing #200, percent moisture in aggregate, temperature, and density of fresh 

concrete. 

Acceptance 

Acceptance or rejection of material is based on the contractor’s QC tests as verified by the engineer’s tests. Should 

the results of the two tests disagree, the engineer’s results will be used to determine acceptance or rejection. If the 

test results are disputed, referee testing will be conducted on additional samples by an independent laboratory. The 

referee test results will then be used for acceptance. 

Pay Factors 

Pay factors are assigned for smoothness, compressive strength, and thickness. The smoothness pay factor is based on 

the California profilograph. Values for this pay factor were not provided. Pay factors for thickness and compressive 

strength are based on a quality index for each characteristic. The quality index (Qi) is calculated as follows: 

 Qi = (Xavg – LSL)/S (64) 

where: Xavg = average measured thickness or compressive strength of all cores in the lot 

LSL = lower specification limit 

 S = standard deviation of all thickness or compressive strength samples for the lot 

The quality index is then used to calculate a pay adjustment factor according to the following table: 

Concrete Thickness and Compressive Strength Pay Adjustment 

Table 135. Kansas concrete thickness and comprehensive strength pay adjustment. 

Quality Index Pay Adjustment Factor 

 > 1.71 103% 

1.56-1.71 102% 

1.41-1.55 101% 

1.26-1.40 100% 

0.00-1.25 * 

 < 0 50% 

*Calculated as P = 60 + (Q × 32) 

A composite pay factor for the lot is then calculated as Pc = (Pt × Ps)/100. 
 330

QC/QA Tests for Kansas 

Table 136. Kansas QC/QA tests. 

Test Sampling 

Location 

QC Testing by 

Contractor 

Verification Testing 

by Kansas DOT 

Aggregate gradation Feed bins One test per 1000 tons One per 30,000 tons 

Slump Truck One per 500 yd3

 or 

minimum of one per day One per day 

Air content 

(plastic concrete) Truck One per 500 yd3

 or 

minimum of one per day One per day 

Cores Roadway One per sublot One per lot 

Temperature Truck One per 500 yd3 One per day 

Beams Truck As required for opening to 

traffic One set per week 

Unit weight Truck One per 500 yd3

 or 

minimum of one per day One per day 

Material passing #200 

(washed) Feed bins One test per 1000 tons One per week 

Smoothness Roadway NA NA 

Thickness Roadway One per sublot One per lot 

Density of fresh 

concrete Roadway 

One complete transverse 

profile initially, then one 

per 1⁄2 day 

One per week 

NA = Not applicable 

1 cubic yard (yd3

) = 0.765m3, 1 ton = 0.907 Mg 
 331

Illinois 

Quality Control 

The contractor is responsible for providing a QC plan that must be approved by the engineer prior to construction 

and providing qualified personnel and equipment to conduct QC testing. The contractor must allow the engineer 

access to the laboratory to observe any and all testing procedures, calculations, test documentation, and plotting of 

results. At a minimum, the contractor must have a level 1 PCC technician at the job site, as well as a level 2 PCC 

technician at the plant during mixture production and placement. QC tests include aggregate gradation, slump, air 

content, unit weight/yield, temperature, and compressive or flexural strength. 

Quality Assurance 

The engineer is responsible for all QA testing on split samples. The split sample is one of two equal portions of a 

field sample. At the engineer’s discretion, he/she may also obtain independent field samples for testing. The results 

of the QA tests will be made available to the contractor as soon as they are completed. In the event that the 

contractor’s or the engineer’s tests for air content and strength are not within specification limits, referee testing will 

be conducted by Illinois DOT or by a mutually agreed upon laboratory at the contractor’s option. 

Acceptance 

Acceptance of the material is based on: contractor’s compliance with all contract documents for QC, validation of 

the contractor’s QC test results by comparison with the engineer’s QA tests, comparison of the engineer’s QA test 

results with the specification limits using samples independently obtained by the engineer, and referee test results for 

unresolved failing tests obtained by the contractor or the engineer. 

Pay Factors 

Pay factors are assigned for smoothness and thickness. 
 332

QC/QA Tests for Illinois 

Table 137. Illinois QC/QA tests. 

Test Sampling 

Location 

QC Testing Frequency 

by Contractor 

Verification Testing 

Frequency by Illinois DOT 

(split sample) 

Aggregate 

gradation 

At plant from 

bins or stockpiles One per 2500 yd3

First test performed by the 

contractor at the beginning of the 

project, and then a minimum of 

10% of the tests required by the 

contractor 

Slump Job site One per 500 yd3

 or 

minimum of one per day 

First three tests performed by the 

contractor at the beginning of the 

project, and then a minimum of 

10% of the tests required by the 

contractor 

Air content 

(plastic 

concrete) 

Job site One per 100 yd3

 or 

minimum of one per day 

First three tests performed by the 

contractor at the beginning of the 

project, and then a minimum of 

10% of the tests required by the 

contractor 

Strength Laboratory One per 1250 yd3

 or 

minimum of one per day 

First test performed by the 

contractor at the beginning of the 

project, and then a minimum of 

10% of the tests required by the 

contractor 

Smoothness Job site NA NA 

Thickness Job site NA NA 

Temperature Plant As needed to control 

production NA 

NA = Not applicable 

1 yd3

 = 0.765 m3
 333

Iowa 

Quality Control 

The contractor is responsible for providing a QC plan that must be approved by the engineer prior to construction 

and providing qualified personnel and equipment to conduct QC testing. The contractor must allow the engineer 

access to the laboratory to observe any and all testing procedures, calculations, test documentation, and plotting of 

results. At a minimum, the contractor must perform QC tests on the unit weight of plastic concrete, aggregate 

gradation, flexural strength, air content, slump, and water-to-cement ratio. 

Quality Assurance 

The engineer is responsible for QA and acceptance of the material. QA is accomplished by conducting verification 

sampling and testing, observing the QC sampling and testing, taking additional samples for testing at any time, and 

monitoring control charts. Results of the verification tests will be made available to the contractor as soon as they 

are completed. Verification testing frequency will be at the discretion of the engineer, but must not be less than 10 

percent of the QC tests. Verification testing will be conducted on those properties tested for QC. 

Acceptance 

Acceptance of the material will be based on the contractor’s QC test results as long as the results agree with Iowa 

DOT’s test results. Acceptance for thickness will be based on samples tested by the engineer. Coring for thickness 

will be on a lot-by-lot basis, where 1 day’s production is considered a lot, which is divided into three equal sublots 

for testing. 

Pay Factors 

Pay factors are assigned for smoothness, thickness, and flexural strength. The pay factor for smoothness was not 

provided. The pay factor for thickness is determined using the following table: 

Table 138. Iowa pay factors for thickness. 

Thickness Index* 

Range (mm) Percent Payment Thickness Index* Range (mm) Percent Payment 

 0.00 or more 103 –13.98 to –15.24 91 

 –0.01 to –1.27 102 –15.25 to –16.51 90 

 –1.28 to –2.54 101 –16.52 to –17.78 89 

 –2.55 to –3.81 100 –17.79 to –19.05 88 

 –3.82 to –5.08 99 –19.06 to –20.32 87 

 –5.09 to –6.35 98 –20.33 to –21.59 86 

 –6.36 to –7.62 97 –21.60 to –22.86 85 

 –7.63 to –8.89 96 –22.87 to –24.13 84 

 –8.90 to –10.16 95 –24.14 to –25.40 83 

–10.17 to –11.43 94 –25.41 to –26.67 82 

–11.44 to –12.70 93 –26.68 to –27.94 81 

–12.71 to –13.97 92 –27.95 or less 80 

 * The thickness index is determined as follows: TI = (Xavg – S) – T, where Xavg = mean core length for the section, 

S = core length standard deviation for the section, and T = design thickness. Pavement, represented by cores, 

which is deficient from the design thickness by 25 mm or greater shall be replaced. Payment for thickness will be 

based on the percentage of the contract price. 

The pay factor for flexural strength is determined by subtracting 1 standard deviation from the mean strength 

according to the following table. The average strength is based on three beams. 
 334

Table 139. Iowa pay factors for flexural strength. 

Strength (lbf/inch2

) Pay Factor 

 < 450 Remove 

450-474 70 

475-499 75 

500-524 80 

525-549 85 

550-559 90 

560-569 92 

570-579 94 

580-589 96 

590-599 98 

600-624 100 

625-649 101 

650-674 102 

 ≥ 675 103 

1 lbf/inch2 = 6.89 kPa 

The composite pay factor was not provided; however, it appears to be the individual pay factors multiplied together 

times the contract price. 

QC/QA Tests for Iowa 

Table 140. Iowa QC/QA tests. 

Test Sampling 

Location 

QC Testing 

Frequency by 

Contractor 

Verification 

Testing Frequency 

by Iowa DOT 

Acceptance by 

Iowa DOT 

Aggregate 

gradation Feed bins Two per day At least 10% of QC 

tests 

Verification of QC by 

Iowa DOT’s tests 

Air content 

(plastic concrete) Truck First load and then 

one per 500 yd3

At least 10% of QC 

tests 

Verification of QC by 

Iowa DOT’s tests 

Water-to-cement 

ratio Truck Two per day At least 10% of QC tests 

Verification of QC by 

Iowa DOT’s tests 

Unit weight of 

plastic concrete Truck One per day At least 10% of QC tests 

Verification of QC by 

Iowa DOT’s tests 

Thickness Roadway NA NA Five cores per sublot 

28-day flexural 

strength Truck One per 1000 yd3 At least 10% of QC 

tests 

One set of three beams as 

determined by engineer 

NA = Not applicable 

1 yd3

 = 0.765 m3
 335

New Jersey 

Quality Control 

All plants producing concrete for New Jersey DOT projects shall have a QC plan in place, as outlined in 

“Requirements for a Portland Cement Concrete Quality Control Plan,” June 25, 1998. A QC technician who is 

certified by the American Concrete Institute as grade I must be available during the production of concrete for any 

New Jersey DOT project. QC tests for slump, air content, compressive strength, aggregate gradation, and 

temperature are conducted at the plant. Concrete provided by any producer who has met applicable design, control, 

and acceptance testing requirements will be presumed to be in compliance with New Jersey DOT’s standards at the 

time of delivery. However, the presumption will not waive the engineer’s right to impose pay adjustments. 

Acceptance 

Acceptance of the material is determined through acceptance testing by New Jersey DOT. Acceptance will be based 

on the results of the slump, strength, and air content tests. Acceptance for compressive strength, slump, and air 

content will be on a lot-by-lot basis, with a lot being equal to 1 day’s production. Acceptance tests will be conducted 

at a frequency of five per lot. 

Pay Factors 

Pay factors are assigned for air content and compressive strength. The pay factor for compressive strength and air 

content will be determined by the percent defective within the lot. Percent pay adjustments (PPA) are calculated 

using the following two equations: 

 For PD < 50: PPA = 3.0 – 0.3PD (65) 

 For PD ≥ 50: PPA = 26.0 – 0.76PD (66) 
 336

North Carolina 

Quality Control 

The contractor is responsible for providing a QC plan that must be approved by the engineer prior to construction. 

During the preconstruction meeting, the contractor and the engineer will determine the types and frequencies of 

testing that are required to produce materials that meet the specifications. The contractor must also provide qualified 

personnel and equipment to conduct QC testing. The contractor must allow the engineer access to the laboratory to 

observe any and all testing procedures, calculations, test documentation, and plotting of results. At a minimum, the 

contractor must have a technician who is certified to develop the concrete design, control the gradation and quality 

of the aggregates, and perform the required tests. QC testing must be conducted for aggregate gradation, air content, 

slump, compressive strength, profile index, and thickness. 

Acceptance 

Acceptance of the material is determined through acceptance testing by North Carolina DOT. Acceptance will be 

based on the results of the slump, flexural strength, thickness, profile index, and air content tests. Acceptance for 

slump and air content will be determined at the point of placement at a frequency determined by the engineer. A 

sample will be obtained immediately after the concrete has been discharged onto the road. Acceptance for thickness 

and flexural strength will be determined on a lot-by-lot basis, where a lot is 5333.3 yd3

 and subsequently divided 

into four equal sublots. For each sublot, a set of two beams shall be used to test flexural strength. Additionally, two 

cores shall be taken from each lot to test for thickness. Acceptance testing of the longitudinal profile of the finished 

pavement must be performed by the contractor in the presence of the engineer. 

Pay Factors 

Pay factors are assigned for profile index, thickness, and flexural strength. The pay factor for the profile index was 

not provided. The pay factor for thickness is determined from the following table: 

Table 141. North Carolina pay factors for thickness. 

Deficiency (inches) Percentage of Price Allowed 

0.00-0.20 100 

0.21-0.30 80 

0.31-0.40 72 

0.41-0.50 68 

0.51-0.75 57 

0.76-1.00 50 

 1 inch = 25.4 mm 

The pay factor for flexural strength is determined from the following table: 

Table 142. North Carolina pay factors for flexural strength. 

Lower Acceptance Limits 

Number of Tests in Lot Pay Factor 

Level 3 4 5 6 

1.05 550 + 0.60R 550 + 0.66R 550 + 0.66R 550 + 0.66R

1.00 550 + 0.58R 550 + 0.54R 550 + 0.50R 550 + 0.46R

0.95 550 + 0.36R 550 + 0.27R 550 + 0.23R 550 + 0.21R

0.70 550 550 550 550 

Note: R is the difference between the high and low tests in the lot. If multiple deficiencies occur, the payment is 

determined by successively multiplying the contract price by the appropriate factor indicated for each 

deficiency. 
 337

QC/QA Tests for North Carolina 

Table 143. North Carolina QC/QA tests. 

Test Sampling 

Location 

QC Testing Frequency 

by Contractor 

Acceptance Testing 

Frequency by 

North Carolina DOT 

Aggregate gradation Feed bins Contractor’s discretion as 

needed for control NA 

Slump Roadway after 

discharge Two per day Engineer’s discretion 

Air content (plastic) Roadway after 

discharge Two per day Engineer’s discretion 

Profile index Roadway 

Three profiles: Two at 

3.5 ft inside the outer 

wheel path and one along 

the longitudinal joint 

Based on QC test 

Thickness Roadway 

(coring) 

Contractor’s discretion as 

needed for control Two cores per lot 

Flexural strength Truck NA Two beams per sublot 

(average) 

Compressive strength Roadway 

(coring) 

Contractor’s discretion as 

needed for control NA 

NA = Not applicable 

1 ft = 0.305 m 
 338

Oregon 

Quality Control 

The contractor is responsible for providing a QC plan that must be approved by the engineer prior to construction 

and providing qualified personnel and equipment to conduct QC testing. The contractor must allow the engineer 

access to the laboratory to observe any and all testing procedures, calculations, test documentation, and plotting of 

results. At a minimum, the contractor must have a concrete control technician (CCT) at the plant or job site and a 

QC technician (QCT) at the job site. QC tests include aggregate gradation, slump, air content, water-to-cement ratio, 

unit weight/yield, temperature, and compressive strength. QC tests will be conducted on a lot-by-lot basis, where a 

lot is the total amount of concrete produced for each mix design. This lot will then be divided and tested by the 

sublot, where a sublot is equal to 75 m3

 of material. 

Quality Assurance 

The engineer is responsible for all QA. Assurance is accomplished through verification testing, observation of 

contractor sampling and testing, and additional sampling and testing conducted by the engineer. The results of the 

verification tests will be made available to the contractor as soon as they are completed. In the event that the 

contractor’s or the engineer’s tests are not within specification limits, they shall immediately work together to 

resolve the difference to avoid having the material rejected as not meeting specifications. Verification testing will be 

conducted on a lot-by-lot basis, where a lot is the total amount of concrete produced for each mix design. 

Acceptance 

Acceptance of the material is based on verification of the QC tests by Oregon DOT’s tests. For aggregate 

acceptance, verification of the contractor’s test values through check tests completed by the engineer will be used. 

For plastic concrete, acceptance is based on the QC tests performed by the contractor and on tests conducted by the 

engineer. The engineer may observe the QC tests performed by the contractor and/or conduct tests on all plastic 

concrete. Hardened concrete will be accepted based on the statistical analysis of the 28-day strength tests of 

cylinders cast by the engineer. 

Pay Factors 

Pay factors are assigned for smoothness, thickness, and compressive strength. The pay factor for smoothness is 

determined by the following equation: 

 Bonus = 0.00038(80 – PI)(Quantity)(Unit price) (67) 

where: PI = average of the two wheel path profiles for the segment 

 Quantity = area in m2 represented by the segment 

 Unit price = price for the concrete shown in the bid schedule 

The pay factor for thickness is determined using the following table. No additional payment will be made for 

pavement exceeding the minimum specification. 

Table 144. Oregon pay factors for thickness. 

Deficiency in Thickness 

(mm) 

Proportional Part of Contract 

Unit Price Allowed 

 0.0-5.0 100% 

 5.1-7.6 83% 

 7.7-10.1 76% 

10.2-12.7 73% 

12.8-19.0 63% 

19.1-25.0 59% 
 339

The pay factor for compressive strength is based on statistical analysis to determine the total percent within the 

specification limits. This is then used to determine the pay factor for strength. In no case shall payment exceed the 

contract price. For pay factors less than 1.00, a pay reduction is calculated as follows: 

 Payment = 0.3(PF – 1.0)(Contract bid price) 

 (68) 

A weighted pay factor (WPF) is determined by multiplying PF by a weighting factor fI provided in the contract. The 

method for determining a composite pay factor (CPF) for these items uses the following formula: 

∑

∑=

I f

WPF

CPF (69)

where: WPF = weighted pay factor 

 fI = weighting factor 

QC/QA Tests for Oregon 

Table 145. Oregon QA/QC tests. 

Test Sample 

Location 

QC Testing Frequency 

by Contractor 

Verification 

Testing by 

Oregon DOT 

Acceptance by 

Oregon DOT 

Aggregate 

gradation Feed bins One per shift or 500 Mg One per shift or 500 Mg 

Verification of QC by 

Oregon DOT’s test 

Slump Roadway after 

discharge One per sublot One per lot Verification of QC by Oregon DOT’s test 

Air content 

(plastic) 

Roadway after 

discharge One per sublot One per lot Verification of QC by 

Oregon DOT’s test 

Water-tocement ratio Roadway after 

discharge One per sublot One per lot Verification of QC by 

Oregon DOT’s test 

Yield Roadway after 

discharge One per sublot One per lot Verification of QC by 

Oregon DOT’s test 

Thickness Roadway One sticking of plastic 

concrete per 60 lane-meters 

Observation of QC 

test 

Observation of QC 

test 

Compressive 

strength 

Cast in the 

field NA NA One set of four 

cylinders per sublot 

Smoothness Roadway One pass per wheel path 

per 200-m segment 

Observation of QC 

test 

Observation of QC 

test 

Temperature Roadway after 

discharge One per sublot One per lot NA 

 NA = Not applicable 
 340

Pennsylvania 

Quality Control 

The contractor is responsible for providing a QC plan that must be approved by the engineer prior to construction 

and providing qualified personnel and equipment to conduct QC testing. The contractor must allow the engineer 

access to the laboratory to observe any and all testing procedures, calculations, test documentation, and plotting of 

results. At a minimum, the contractor must have a certified technician to develop the concrete design, control the 

gradation and quality of the aggregates, and perform the required tests. QC testing must be conducted for aggregate 

gradation, air content, slump, compressive strength, profile index, and thickness. 

Quality Assurance 

QA is accomplished through independent sampling and testing by Pennsylvania DOT for air content, strength, and 

pavement thickness. Independent assurance testing is conducted to review the QC testing and to check the accuracy 

of the equipment used for acceptance testing. 

Acceptance 

Acceptance of the material is determined through acceptance testing by Pennsylvania DOT. Acceptance will be 

based on the results of the slump, strength, thickness, and air content tests. Acceptance for compressive strength and 

air content will be on a lot-by-lot basis, with testing conducted on sublots of 1400 yd2

 (1170 m2). Acceptance for 

smoothness will be based on the QC profilograph test. 

Pay Factors 

Pay factors are assigned for profilograph, thickness, air content, and compressive strength tests. The pay factor for 

compressive strength and air content will be determined by the PWL for the lot. The pay factor for thickness will be 

determined by coring. The profile index pay factor is determined using a profilograph and assessing the index in 

inches per mile per lot. After determining the individual pay factors for each of the above characteristics, the lot 

payment is calculated using the following equation: 

 Lp = Cp([(2Ps + 2Pd + Pa)/500] + [(Pp – 100)/100]) (70) 

where: Lp = lot payment 

Cp = contract price per lot 

Ps = payment percentage of the contract price for strength 

Pd = payment percentage of the contract price for depth 

Pa = payment percentage of the contract price for air content 

Pp = payment percentage of the contract price for the profile 

QC/QA Tests for Pennsylvania 

Table 146. Pennsylvania QA/QC tests. 

Test Sample 

Location QC Testing Assurance Testing Acceptance 

Testing 

Aggregate 

gradation Feed bins Varies: Established 

in QC plan NA NA 

Slump Roadway after 

discharge One per 200 yd3

 NA One per sublot 

Air content 

(plastic) 

Roadway after 

discharge One per 200 yd3

 One per two lots One per sublot 

Profilograph Roadway Test as soon as 

concrete is cured NA Based on QC test 

Thickness Roadway One core per sublot One core per two lots Four cores per lot 

Compressive 

strength 

Cast in the 

field 

Two cylinders per 

sublot 

One cylinder per two 

lots 

Four cylinders per 

lot 

NA = Not applicable 

1 yd3

 = 0.765 m3 
 341

Texas 

Quality Control 

The contractor is responsible for providing a QC plan that must be approved by the engineer prior to construction 

and providing qualified personnel and equipment to conduct QC testing. The contractor must allow the engineer 

access to the laboratory to observe any and all testing procedures, calculations, test documentation, and plotting of 

results. To complete the required testing, the contractor must provide technicians certified at levels I through IV, 

depending on the test being conducted. The standard lot size is 6000 yd2

 (5000 m2) of surface area of PCC pavement 

of the same thickness. A sublot is equal to one-fifth of the surface area of a lot. If the final lot is 3000yd2

 (2500 m2) 

or greater, it will be considered a full lot. QC must be exercised on all properties contained in the table entitled 

QC/QA Tests for Texas. 

Quality Assurance 

QA is accomplished through verification testing conducted by Texas DOT. Independent assurance testing is 

conducted to review the QC testing and to check the accuracy of the equipment used for verification testing. 

Acceptance 

Acceptance of the material is based on the QC tests and the verification tests. If the contractor’s QC test results and 

the engineer’s verification test results are not within tolerances, referee testing will be conducted. If referee tests are 

requested, the referee test results shall govern. 

Pay Factors 

Pay factors are assigned for ride quality, thickness, and flexural strength. The pay factor for flexural strength is 

based on the following table: 

Table 147. Texas pay factors for flexural strength. 

Average Flexural Strength 

(Three Beams) (lbf/inch2

) 

Strength Adjustment Factor 

(SF) 

≥ 555 1.0000 

550 0.9695 

545 0.9397 

540 0.9106 

535 0.8820 

530 0.8542 

525 0.8269 

< 525 Remove or remain with no payment 

 1 lbf/inch2 = 6.89 kPa 

The pay factor for ride quality was not provided. The pay factor for thickness is determined using the following 

table: 

Table 148. Texas pay factors for thickness. 

Thickness Deficiency 

(inches) 

Thickness Adjustment Factor 

(TF) 

> 0.00-0.20 1.00 

> 0.20-0.30 0.80 

> 0.30-0.40 0.72 

> 0.40-0.50 0.68 

> 0.50-0.75 0.57 

The pay adjustment factor for each lot is calculated as AF = (SF)(TF). The pay adjustment for each lot is then 

calculated as: 

 Payment = (BP)(AF)(Q) (71) 

where: BP = bid price 

Q = lot quantity of acceptable pavement 
 342

QC/QA Tests for Texas 

Table 149. Texas QC/QA tests. 

Test Sample 

Location 

QC Testing by 

Contractor 

Verification Testing 

by Texas DOT 

Acceptance by 

Texas DOT 

Mix temperature Plant One per sublot One per day 

Based on QC, 

verification, and 

referee, if required 

Air content 

(plastic) 

Roadway after 

discharge One per sublot 1 per 10 QC tests 

Based on QC, 

verification, and 

referee, if required 

Concrete unit 

weight 

Roadway after 

discharge One per sublot 1 per 10 QC tests 

Based on QC, 

verification, and 

referee, if required 

Making/curing 

strength specimens 

Roadway after 

discharge One per sublot 1 set per 10 sublots NA 

7-day flexural 

strength 

Roadway after 

discharge NA One per sublot Based on verification 

Coarse aggregate 

gradation Stockpile One per stockpile 

per day 1 per 10 QC tests 

Based on QC, 

verification, and 

referee, if required 

Coarse aggregate 

loss by decantation Stockpile One per 5 PCC production days 1 per 10 QC tests 

Based on QC, 

verification, and 

referee, if required 

Fine aggregate 

gradation Stockpile One per stockpile per day 1 per 10 QC tests 

Based on QC, 

verification, and 

referee, if required 

Fineness modulus Stockpile One per stockpile 

per day 1 per 10 QC tests 

Based on QC, 

verification, and 

referee, if required 

Fine aggregate 

organic impurity Stockpile One per 5 PCC 

production days 1 per 10 QC tests 

Based on QC, 

verification, and 

referee, if required 

Sand equivalent Stockpile One per 5 PCC 

production days 1 per 10 QC tests 

Based on QC, 

verification, and 

referee, if required 

Aggregate moisture 

content Stockpile One per stockpile per day 1 per 10 QC tests 

Based on QC, 

verification, and 

referee, if required 

Water-to-cement 

ratio 

Roadway after 

discharge One per batch Review plant printout 

and calibration 

Based on QC, 

verification, and 

referee, if required 

Cement factor Roadway after 

discharge One per batch Review plant printout and calibration 

Based on QC, 

verification, and 

referee, if required 

Admixture dosage Roadway after 

discharge One per batch Review plant printout and calibration 

Based on QC, 

verification, and 

referee, if required 

Coring for 

thickness Roadway One per sublot NA NA 

Measuring 

pavement thickness 

Roadway 

(plastic) NA One per sublot 

Based on QC, 

verification, and 

referee, if required 

NA = Not applicable 
 343

Wisconsin 

Quality Control 

The contractor is responsible for providing a QC plan that must be approved by the engineer prior to construction 

and providing qualified personnel and equipment to conduct QC testing. The contractor must allow the engineer 

access to the laboratory to observe any and all testing procedures, calculations, test documentation, and plotting of 

results. At a minimum, the contractor must have the following certified personnel: aggregate technician 1, PCC 

technician 1A, PCC technician 2, concrete compressive strength tester, and profilograph operator 1. QC testing must 

be conducted on the following: aggregate gradation, material passing #200, aggregate moisture, air content, slump, 

temperature, thickness, profilograph, and compressive strength. 

Quality Assurance 

QA is accomplished through verification testing by Wisconsin DOT for air content, strength, and pavement 

thickness. Independent assurance testing is conducted to review the QC testing and the verification testing. The 

independent assurance review will be done in accordance with Wisconsin DOT’s Independent Assurance program 

and may include: split-sample testing, proficiency sample testing, witnessing of sampling and testing, review of 

control charts, and requesting additional samples for testing. 

Acceptance 

Acceptance of the material is based on the contractor’s QC test results. This method of acceptance and payment will 

continue until it can be shown through verification or the dispute resolution process that the contractor’s test results 

are in error. Acceptance will be based on the results of the strength, thickness, and air content tests. 

Pay Factors 

Pay factors are assigned for profilograph, thickness, and compressive strength. The pay factor for compressive 

strength will be determined by the lot and will be based on the contractor’s QC cylinders fabricated for each sublot. 

A lot is considered to be 1 day’s production divided into equal sublots not to exceed 500 yd3

 (380 m3). The pay 

adjustment will be based on the lot’s average strength minus 1 standard deviation. The factor is a dollars per yd2

adjustment. The factor is determined from the following table: 

Table 150. Wisconsin pay factors for compressive strength.

Strength (lbf/inch2

) Pay Adjustment ($/yd2) 

≤ 2850 –0.552 

2850 to 2950 –0.527 

2950 to 3050 –0.452 

3050 to 3150 –0.385 

3150 to 3250 –0.309 

3250 to 3350 –0.234 

3350 to 3450 –0.167 

3450 to 3550 –0.109 

3550 to 3650 –0.050 

3650 to 3750 –0.000 

3750 to 3850 0.067 

3850 to 3950 0.125 

3950 to 4050 0.167 

4050 to 4150 0.201 

4150 to 4250 0.226 

4250 to 4350 0.242 

4350 to 4450 0.259 

4450 to 4550 0.268 

4550 to 4650 0.268 

≥ 4650 0.276 

1 lbf/inch2 = 6.89 kPa, 1 yd2 = 0.836 m2
 344

The pay factor for the profile index will be determined according to the following table: 

Table 151. Wisconsin pay factors for the profile index.

Profile Index (inches/mi) Pay Adjustment per 0.1-mi 

Section per Lane 

 < 19.0 +$585 

19.0-25.2 +$350 

25.3-44.3 $0 

44.4-50.6 –$230 

 ≥ 50.7 –$940 

1 mi = 1.61 km, 1 inch/mi = 15.8 mm/km 

The pay factor for thickness will be determined according to the following table: 

Table 152. Wisconsin pay factors for thickness. 

Average Thickness 

Deficiency (inches) 

Pay Adjustment per 250-ft 

Lane Length Unit 

 0-3⁄8 $0 

> 3⁄8-1⁄2 –$1143 

> 1⁄2-3⁄4 –$2095 

> 3⁄4-1 –$2667 

1 inch = 25.4 mm, 1 ft = 0.305 m 

These factors are applied according to the units listed in the tables. 

QC/QA Tests for Wisconsin 

Table 153. Wisconsin QC/QA tests.

Test QC Testing by 

Contractor 

Verification 

Testing by 

Wisconsin DOT 

Acceptance by 

Wisconsin DOT 

Material passing #200 Two per day NA NA 

Aggregate gradation 

0-1000 tons: One per day 

1000-2000 tons: Two per day 

2000+ tons: Three per day 

NA NA 

Slump One per sublot NA NA 

Aggregate moisture Two per day NA NA 

Air content One per sublot One per lot Based on QC test 

Profilograph As soon as concrete has cured 

enough for testing NA Based on QC test 

Thickness Two per 250 ft Two per day Based on QC test 

Compressive strength Three cylinders per sublot One per five lots Based on QC test 

Temperature One per sublot NA NA 

NA = Not applicable 

1 ft = 0.305 m, 1 ton = 0.907 Mg 
 345

APPENDIX E: MINUTES FROM THE FIRST PANEL MEETING

Minutes of the March 1, 1999, Meeting of the Pooled Fund States for 

Optimal Acceptance Procedures for Statistical Specifications

Present: Chris Abadie Louisiana 

Roger Apple Pennsylvania 

Ataur Bacchus Ontario 

Jim Burati Clemson University (project principal investigator) 

Steve DeWitt North Carolina 

Doug Dirks Illinois 

Steve Gage Connecticut 

Chuck Hughes Consultant (project consultant) 

Kurt Johnson Wisconsin 

Peter Kopac FHWA 

Rick Kreider Kansas 

Bill Maupin Virginia 

David Miller Minnesota 

Tom Reis Iowa 

Deniz Sandhu New York 

Jeff Seiders Texas 

Chris Williams FHWA 

Unable to Attend: Milt Fletcher South Carolina 

Rudy Malfabon Washington 

Gary Selmi Nevada 

Al Stanley Idaho 

Ken Stoneman Oregon 

Richard Weed New Jersey 

(1) The meeting began at 8:10 a.m. with a few introductory comments by Peter Kopac. 

(2) Jim Burati then went through some administrative issues, including the agenda for the 

meeting (see attachment 1), participant introductions, and travel reimbursement procedures. 

(3) Jim Burati then briefly summarized the progress of the project to date, including the literature 

search, the summary of State specifications, and the flowcharts of the specification 

development process. 

Action Item: All participants are asked to review the specification summary 

information for their State and send any corrections or additions to Reagan Prince by 

fax (864-656-2670) or e-mail (jprince@clemson.edu).

(4) Chuck Hughes then reviewed the phase I flowchart of the specification development process. 

Comments and suggestions that were made included: 
 346

• Peter Kopac indicated that he thought there was a need to add an item, “Establish criteria 

for success,” in phase I, possibly as item 2.3. 

• Jeff Seiders indicated that item 5, “Confirm interest and commitment,” was particularly 

important. 

• Ataur Bacchus indicated that an executive summary document should be available for 

review throughout the specification development process. He also indicated that there 

were three objectives for the specification development process: 

1. Obtain effective specifications. 

2. Be able to explain the specifications at the practitioner level. 

3. Be able to convince executives (even if management changes) of the benefits of the 

specification. 

Action Item: Jim Burati will revise the phase I flowchart to incorporate “Establish 

criteria for success” as an item. Chuck Hughes will then make any corresponding 

changes to the text that accompanies the flowchart.

(5) Chuck Hughes then reviewed the phase II flowchart of the specification development 

process. Comments and suggestions that were made included: 

• Jeff Seiders indicated that it is necessary to retain method provisions and stipulated 

engineering practices for cases where good test procedures or measures are not available. 

• Rick Kreider asked in how much detail “Dispute Resolution” would be covered in the 

manual. It was indicated that only the general concepts would be presented. 

• It was decided that an effort should be made to ensure that the terminology in the 

flowcharts and the manual should be consistent with the Code of Federal Regulations, 

23 CFR Part 637. 

• Steve DeWitt asked whether it would be possible to put all of the phase II flowchart on a 

single page. Options will be explored to try to make it easier to follow the flow through 

the chart. 

• Chris Abadie suggested that flowchart items similar to boxes 20 through 25 should be 

incorporated into the QC section (i.e., boxes 11 through 13). 

• It was suggested that a decision diamond should be incorporated after box 25, “Obtain 

data,” to check whether the data warrant use of the property for acceptance purposes. 

Action Item: Jim Burati will investigate revising the phase II flowchart to incorporate 

the items suggested in the bullets above. Chuck Hughes will then make any 

corresponding changes to the text that accompanies the flowchart.

(6) Chuck Hughes then reviewed the phase III flowchart of the specification development 

process. Comments and suggestions that were made included: 

• Peter Kopac pointed out that it would be necessary to modify phase III to be consistent 

with the changes in phase I. 
 347

Action Item: Jim Burati will revise the phase III flowchart to ensure that it is 

consistent with phase I. Chuck Hughes will then make any corresponding changes to 

the text that accompanies the flowchart.

(7) Jim Burati then reviewed the preliminary specification analyses that have been conducted. 

These included: 

• Bias in OC curves for PWL/PD. 

• Precision in OC curves for PWL/PD. 

• Precision of average project PWL/PD. 

• Precision of individual payment estimates. 

• Effect of bimodal distributions. 

• Effect of skewed distributions. 

• OC/expected payment curves for AAD plans. 

• Comparison of expected payment between AAD and PWL plans. 

In the interest of clarity, it was suggested that either PWL or PD, but not both, be used in the 

analyses that are done. 

Action Item: The analyses that are performed on the project will all be reported in 

terms of PWL.

(8) Jim Burati then presented some additional analysis topics that might be considered during the 

project. These included: 

• Use of Bayesian procedures for acceptance. 

• Determination of the lot pay factors for individual properties. 

• Determination of the composite pay factor for the lot. 

• Validation/verification procedures for contractor/agency tests. 

• Effect of non-normal populations. 

• Effect of increased sampling and testing variability. 

• Miscellaneous “bells and whistles,” such as payment based on PWL, but with no price 

reductions if all individual tests are within the limits. 

Action Item: Since there was not sufficient time to decide on the above items, it was 

decided that Jim Burati would distribute the items to the individual panel members for 

prioritizing and the items would then be returned to him for summarization. 

(9) The meeting adjourned at 3:30 p.m. 

Prepared and distributed by: 

________________________________ 

Jim Burati, Principal Investigator
 
 349

APPENDIX F: ILLUSTRATIONS OF POSSIBLE RANGES FOR 

PD OR PWL ESTIMATES

The quality index, which is used to estimate PD (or PWL), is calculated from the sample mean 

and standard deviation. Let us assume that the sample standard deviation is constant. This is not 

true; however, it allows us to show a simplified illustration of the effect that the variability of the 

sample mean can have on the estimated PD (or PWL) value. The top plot in figure 98 shows a 

normal distribution (skewness coefficient = 0) with 10 percent below the lower specification 

limit. 

The two solid-line plots in the figure illustrate the distribution of the sample means for a sample 

size of n = 3. The distribution of the sample means will be centered at the population mean, 

while the standard deviation of the distribution of the sample means (known as the standard 

error) will be equal to n X σ = σ / , where σ is the population standard deviation. Therefore, the 

sample means will vary less than the original population from which they are drawn. 

The distribution with the dashed line in the middle plot represents, for a sample size of n = 3, a 

normal distribution with the smallest mean that can be obtained from the original population 

(once again simplifying by saying that the normal distribution only extends 3 standard deviations 

on either side of the mean). It shows that it is possible to get an estimated PD value of greater 

than 50 (or a PWL of less than 50) even though the actual PD value for the population is 10 (the 

actual PWL is 90). The distribution with the dashed line in the bottom plot represents, for a 

sample size of n = 3, a normal distribution with the largest mean that can be obtained. It shows 

that it is possible to get an estimated PD value as small as zero (a PWL of 100) even though the 

actual PD value for the population is 10 (the PWL is 90). 

These results are relatively consistent with the values shown in the bias histograms in chapter 6 

and appendix G. The histograms in figure 32 indicate that a bias value of +90 was obtained. This 

indicates that a PD estimate of 100 was obtained during the simulation. The reason that the plot 

in figure 98 does not allow for a value this large is that, in developing figure 98, it was assumed 

that the standard deviation was constant. Therefore, the dashed-line distributions have the same 

spread as the original population. In reality, the sample standard deviation also varies and, 

therefore, it is possible to obtain a very low sample mean and, on the same lot, a very low sample 

standard deviation. This combination could result in an estimated distribution that lies entirely 

below the specification limit. 
 350

Figure 98. Spread of possible sample means for a normal distribution with 

10 PD below the lower specification limit and sample size = 3. 

Distribution of sample 

means for n = 3 

PD L = 10 PD U = 0

Skewness = 0 

Distribution of sample 

means for n = 3 

μ

Spec
 351

The plots in figure 99 illustrate the same population as in figure 98, but with a sample size of 

n = 10. The larger sample size indicates that the distribution of the sample means will have less 

spread than that for a sample size of n = 3. It is still possible to have an estimated PD value of 

greater than 50; however, as drawn with the constant standard deviation, it would not be possible 

to have a PD estimate as low as zero. Once again, remember that these simplified examples 

assume that the standard deviation does not vary and remains equal to the population’s standard 

deviation. They do, however, illustrate how natural sampling variability will lead to variability in 

the estimated PD (or PWL) values. 

The remaining figures in appendix F illustrate different PDL/PDU divisions and different 

population PD values for the cases of symmetrical distributions (skewness = 0) and for skewed 

distributions (skewness coefficient = 1.0). They can be interpreted in the same way as the 

discussions for figures 98 and 99. 
 352

Figure 99. Spread of possible sample means for a normal distribution with 

10 PD below the lower specification limit and sample size = 10. 

Distribution of sample 

means for n = 10 

PD L = 10 PD U = 0 

Skewness = 0 

Distribution of sample 

means for n = 10 

μ

Spec
 353

Figure 100. Spread of possible sample means for a normal distribution with 

5 PD outside each specification limit and sample size = 3. 

Distribution of sample 

means for n = 3 

Distribution of sample 

means for n = 3 

PD L = 5 PD U = 5

Skewness = 0 

μ

Spec Spec
 354

Figure 101. Spread of possible sample means for a normal distribution with 

5 PD outside each specification limit and sample size = 10. 

Distribution of sample 

means for n = 10 

Distribution of sample 

means for n = 10 

PD L = 5 PD U = 5 

Skewness = 0 

μ

Spec Spec
 355

Figure 102. Spread of possible sample means for a normal distribution with 

50 PD below the lower specification limit and sample size = 3. 

Distribution of sample 

means for n = 3 

PD L = 50 PD U = 0

Skewness = 0 

Distribution of sample 

means for n = 3 

μ = Spec
 356

Figure 103. Spread of possible sample means for a normal distribution with 

50 PD below the lower specification limit and sample size = 10. 

Distribution of sample 

means for n =10 

PD L = 50 PD U = 0 

Skewness = 0 

Distribution of sample 

means for n = 10 

μ = Spec
 357

Figure 104. Spread of possible sample means for a normal distribution with 

25 PD outside each specification limit and sample size = 3. 

Distribution of sample 

means for n = 3 

Distribution of sample 

means for n = 3 

PD L = 25 PD U = 25

Skewness = 0 

μ Spec Spec
 358

Figure 105. Spread of possible sample means for a normal distribution with 

25 PD outside each specification limit and sample size = 10. 

Distribution of sample 

means for n = 10 

Distribution of sample 

means for n = 10 

PD L = 25 PD U = 25 

μ Spec Spec

Skewness = 0 
 359

Figure 106. Spread of possible sample means for a distribution with skewness = 1.0, 

10 PD below the lower specification limit, and sample size = 3. 

Distribution of sample 

means for n = 3 

PD L = 10 

PD U = 0 

Skewness = 1.0 

Distribution of sample 

means for n = 3 

Spec μ
 360

Figure 107. Spread of possible sample means for a distribution with skewness = 1.0, 

10 PD below the lower specification limit, and sample size = 10. 

Distribution of sample 

means for n = 10 

PD L = 10 

PD U = 0 

Skewness = 1.0 

Distribution of sample 

means for n = 10 

Spec μ
 361

Figure 108. Spread of possible sample means for a distribution with skewness = 1.0, 

 5 PD outside each specification limit, and sample size = 3. 

Distribution of sample 

means for n = 3 

PD L = 5 

PD U = 5 

Skewness = 1.0 

Distribution of sample 

means for n = 3 

μ

Spec Spec
 362

Figure 109. Spread of possible sample means for a distribution with skewness = 1.0, 

 5 PD outside each specification limit, and sample size = 10. 

Distribution of sample 

means for n = 10 

PD L = 5 

PD U = 5 

Skewness = 1.0 

Distribution of sample 

means for n = 10 

μ

Spec Spec
 363

Figure 110. Spread of possible sample means for a distribution with skewness = 1.0, 

 50 PD below the lower specification limit, and sample size = 3. 

Distribution of sample 

means for n = 3 

PD L = 50 

PD U = 0 

Skewness = 1.0 

Distribution of sample 

means for n = 3 

μ Spec
 364

Figure 111. Spread of possible sample means for a distribution with skewness = 1.0, 

 50 PD below the lower specification limit, and sample size = 10. 

Distribution of sample 

means for n = 10 

PD L = 50 

PD U = 0 

Skewness = 1.0 

Distribution of sample 

means for n = 10 

μ Spec
 365

Figure 112. Spread of possible sample means for a distribution with skewness = 1.0, 

 25 PD outside each specification limit, and sample size = 3. 

Distribution of sample 

means for n = 3 

PD L = 25 

PD U = 25 

Skewness = 1.0 

Distribution of sample 

means for n = 3 

Spec μ

Spec
 366

Figure 113. Spread of possible sample means for a distribution with skewness = 1.0, 

 25 PD outside each specification limit, and sample size = 10. 

Distribution of sample 

means for n = 10 

Distribution of sample 

means for n = 10 

μ

PD L = 25 

PDU = 25 

Skewness = 1.0 

Spec

Spec
 367

APPENDIX G: BIAS HISTOGRAMS FROM THE 

SIMULATION PROGRAM

The plots in appendix G illustrate the distributions of sample means for a number of different 

combinations of population PD, sample size, and skewness coefficients. These figures are an 

expansion of those shown and discussed in chapter 6. 

Figure 114 shows how sample size can affect the variability of the PD estimates. This figure 

shows the complete program output for a symmetrical population (i.e., skewness coefficient = 0) 

that actually has 10 PD and for which the sample size is n = 5. The results of the nine divisions 

for PDL/PDU are clearly indicated. The BIAS/SE results indicate the average bias and the 

standard error (measure of variability of the PD estimates), respectively. These terms are 

discussed and explained in detail in chapter 6. The bias values are relatively closely distributed 

about zero, indicating that the method for estimating PD (and hence PWL) is not biased for a 

symmetrical distribution. This is to be expected since the PD estimating method assumes a 

symmetrical normal distribution. 

Figure 115 shows portions of the program output for the same population, but with sample sizes 

of n = 3 and n = 10. Figures 116 and 117, and 118 and 119 show similar plots for populations 

with skewness coefficients = 1.0 and 2.0, respectively. The remaining figures illustrate other 

comparisons of the bias results. Figures 120 and 121 show the bias values histograms for 

sampling with the same sample size of n = 5 from symmetrical populations with different PD 

values (i.e., PD = 30 and PD = 50, respectively). Figures 122 and 123 show similar results for a 

sample size of n = 10. Figure 124 shows the bias values histograms for the same sample size of 

n = 5 and the same PD value = 50, but for populations with skewness coefficients = 0.0, 1.0, 2.0, 

and 3.0. Figures 125 through 127 show the complete program outputs for populations with a 

skewness coefficient = 3.0, a sample size of n = 5, and PD values = 10, 30, and 50. It is unlikely 

that populations with this degree of skewness will be found in practice. 
 368

Figure 114. Sample output screen for a population with PD = 10, 

skewness coefficient = 0.00, and sample size = 5. 
 369

Figure 115. Portions of output screens for PD = 10, skewness coefficient = 0.00, 

and sample sizes = 3, 5, and 10. 
 370

Figure 116. Sample output screen for a population with PD = 10, 

skewness coefficient = 1.00, and sample size = 5. 
 371

Figure 117. Portions of output screens for PD = 10, skewness coefficient = 1.00, 

and sample sizes = 3, 5, and 10.
 372

Figure 118. Sample output screen for a population with PD = 10, 

skewness coefficient = 2.00, and sample size = 5. 
 373

Figure 119. Portions of output screens for PD = 10, skewness coefficient = 2.00, 

and sample sizes = 3, 5, and 10.
 374

Figure 120. Sample output screen for a population with PD = 30, 

skewness coefficient = 0.00, and sample size = 5. 
 375

Figure 121. Sample output screen for a population with PD = 50, 

skewness coefficient = 0.00, and sample size = 5. 
 376

Figure 122. Sample output screen for a population with PD = 30, 

skewness coefficient = 0.00, and sample size = 10. 
 377

Figure 123. Sample output screen for a population with PD = 50, 

skewness coefficient = 0.00, and sample size = 10. 
 378

Figure 124. Portions of output screens for PD = 50, sample size = 5, 

and skewness coefficients = 0.00, 1.00, 2.00, and 3.00.
 379

Figure 125. Sample output screen for a population with PD = 10, 

skewness coefficient = 3.00, and sample size = 5. 
 380

Figure 126. Sample output screen for a population with PD = 30, 

skewness coefficient = 3.00, and sample size = 5. 
 381

Figure 127. Sample output screen for a population with PD = 50, 

skewness coefficient = 3.00, and sample size = 5. 
 
 383

APPENDIX H: DISTRIBUTION OF SAMPLE PWL ESTIMATES

GENERATING SAMPLES 

The plots in appendix H illustrate the distributions of sample PWL estimates for a number of 

different combinations of actual population PWL values. All of the plots are based on the same 

simulation results that came from simulating 1000 samples (sample size = 4) from a normal 

distribution with a mean of 0.00 and a standard deviation of 1.00. The sample means and 

standard deviations were then calculated for each of the 1000 samples. These values were then 

used to calculate sample PWL values for various actual population PWL values. 

Figure 128 shows the distribution of the sample means for the 1000 samples. Statistical theory 

says that this should be a normal distribution with a mean equal to that of the population that was 

sampled (i.e., 0.00). Furthermore, the theory also says that the standard deviation of these 1000 

sample means should be the population standard deviation (i.e., 1.00) divided by the square root 

of the sample size (i.e., 4 = 2 ). This means that the standard deviation of the sample means 

should be 0.50. 

The average for the 1000 simulated sample means was +0.01, while their standard deviation was 

0.484. These values are close to the theoretical ones and they are well within the range that might 

be expected from a simulation study. Figure 128 shows distribution histograms for the sample 

means and the sample standard deviations. As expected, the distribution of the sample means is 

approximately normal and is centered around the population mean of 0.00. Also, as expected, the 

distribution of the sample standard deviations is obviously skewed to the right. The average of 

the 1000 sample standard deviations is 0.941, which is close to the population value of 1.00. 

DISTRIBUTION OF SAMPLE PWL VALUES 

The mean and standard deviation for each sample were used to estimate the PWL for populations 

with actual PWL values = 90, 80, 70, 60, and 50. For each population, both one-sided 

specifications and two-sided specifications were evaluated. For the two-sided specifications, the 

population mean was set in the middle of the specification range so that there were equal PD 

values for the lower and upper specification limits. 

Figures 129 through 133 show the sample PWL distributions for both one-sided and two-sided 

populations with 90, 80, 70, 60, and 50 PWL, respectively. A trend regarding the shape of the 

distributions is apparent. For the population with its mean farthest from the specification limit(s) 

(i.e., the 90-PWL population), the distribution of the sample PWL values is skewed to the left. 

This is seen by the long tail to the left and the fact that about 60 percent of the distribution is to 

the right of the population PWL of 90. As the population mean moves closer to the specification 

limit(s) (i.e., moves from 80 to 70 to 60 to 50), skewness decreases until the distribution becomes 

symmetrical. 

The reason that the sample PWL distributions are skewed stems from the fact that the 

distribution of the sample standard deviations is skewed. This skewness has less impact as the 

mean of the population approaches the specification limit. And, for a one-side specification to 
 384

have 50 PWL, the population mean must be centered on the specification limit. Any population 

that is centered on the one specification limit will have 50 PWL regardless of the value of its 

standard deviation. So, in this instance, the distribution of the sample PWL values will be 

symmetrical about the population mean. For any sample mean that is below the population mean, 

the estimated PWL will be less than 50, while any sample mean above the population mean will 

have an estimated PWL greater than 50. This is shown in figures 128 and 133 where both have 

48.9 percent below and 51.1 percent above their population values. 

Figure 134 shows diagrams of the populations that were sampled. These populations can be 

compared with the distributions of the sample PWL values shown in figures 129 through 133. 

For the one-sided specifications, the distribution whose mean is farthest from the specification 

limit (i.e., 90 PWL) has more PWL values above the true value than below it. As the distance 

between the population mean and the specification limit decreases, the distributions become 

symmetrical about the true population value. 

For the two-sided specifications, once again, the population whose mean is farthest from the 

specification limits (i.e., 90 PWL) has more PWL values above the true value than below it. 

However, as the population means come closer to the specification limits (i.e., 60 PWL and 50 

PWL), the distributions have more values below the true value than above it. 

The two situations that were simulated (one-sided specifications and two-sided specifications, 

with the populations centered between the two limits) are the extremes with respect to how the 

PD for a population with a given PWL value can be distributed outside the two limits. There are 

obviously many other ways in which these PWL values could be located with respect to the 

specification limits. From these examples, it appears that the shape of the distribution of the 

sample PWL values will vary depending on how the PWL range of the population is located 

inside the limit or limits. 

However, regardless of the shape of the distribution of the sample PWL values that were 

simulated, the average of the 1000 sample PWL values all differed from the population PWL 

value by no more than about 0.5 percent. Although the EP values were all very close for each of 

the distributions, the different shapes indicate that there may be small differences in the 

percentage of time that a population PWL value is estimated to be too high and too low, 

depending on the PWL of the population and where the population is centered with respect to the 

specification limit or limits. 
 385

Figure 128a. Distribution of sample means for 1000 samples from a normal population 

with μ = 0.00, σ = 1.00.

Figure 128b. Distribution standard deviations for 1000 samples from a normal population 

with μ = 0.00, σ = 1.00. 

1 0 1 3

9

28

60

102

131

154

162

148

81

76

25

13

4 1 1

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Sample Means

Sample Means, n = 4

489 ≤ μ < 511

15

49

346

195

138

108

86

40

14 6 2 0 1

0.2 0.4 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

Sample Standard Deviations

Sample Standard Deviations, n = 4

605 ≤ σ < 395
 386

Figure 129a. Distribution of sample PWL values for 1000 samples from a normal 

population with PWL = 90, one-sided specifications. 

Figure 129b. Distribution of sample PWL values for 1000 samples from a normal 

population with PWL = 90, two-sided specifications. 

0 0 0 0 0 0 1 0 1 1 6 9

25

47

67 73 84 85 75

526

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sample PWL Values

Actual PWL = 90, n = 4

One–Sided Spec

399 ≤ 90 PWL < 601

0 0 0 0 0 0 0 0 2 3 8 15 30

48

67 74 62

80 74

537

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sample PWL Values

Actual PWL = 90, n = 4

Two–Sided Spec

389 ≤ 90 PWL < 611
 387

Figure 130a. Distribution of sample PWL values for 1000 samples from a 

normal population with PWL = 90, one-sided specifications.

Figure 130b. Distribution of sample PWL values for 1000 samples from a normal 

 population with PWL = 80, two-sided specifications. 

0 0 0 1 2 1 1 2 10 15

40

59

75

98 94 86

77 71

59

309

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sample PWL Values

Actual PWL = 80, n = 4

One–Sided Spec

484 ≤ 80 PWL < 516

0 0 0 0 0 0 0 8 12 18

48

62

87

76

92 89 88

64

49

307

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sample PWL Values

Actual PWL = 80, n = 4

Two–Sided Spec

492 ≤ 80 PWL < 508
 388

Figure 131a. Distribution of sample PWL values for 1000 sample from a normal 

population with PWL = 80, one-sided specifications.

Figure 131b. Distribution of sample PWL values for 1000 samples from a normal 

population with PWL = 80, two-sided specifications. 

5

0 1 2 2 4

13

22

37

48

78

117

84

97

89

80

61 60

32

168

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sample PWL Values

Actual PWL = 70, n = 4

One–Sided Spec

510 ≤ 70 PWL < 490

0 0 0 0 0 1

15

21

57

77

87

95

89 93 93

55 57

50 49

161

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sample PWL Values

Actual PWL = 70, n = 4

Two–Sided Spec

535 ≤ 70 PWL < 465
 389

 Figure 132a. Distribution of sample PWL values for 1000 samples from 

a normal population with PWL = 60, one-sided specifications. 

Figure 132b. Distribution of sample PWL values for 1000 samples from a normal 

population with PWL = 60, two-sided specifications. 

13

1

5 6

13

19

34

50

68

94

104

95

106

93

70

59

38

31

23

78

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sample PWL Values

Actual PWL = 60, n = 4

One–Sided Spec

502 ≤ 60 PWL < 498

0 0 0 0

5

23

40

75

105

121

95

114

76

61 60

38

51

22

29

85

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sample PWL Values

Actual PWL = 60, n = 4

Two–Sided Spec

578 ≤ 60 PWL < 422
 390

Figure 133a. Distribution of sample PWL values for 1000 samples from a normal population with PWL = 50, onesided specifications.

Figure 133b. Distribution of sample PWL values for 1000 samples from a normal 

population with PWL = 50, two-sided specifications. 

30

7

15

23

29

40

73

66

92

114

103

115

79

57

43

32

22

17

13

30

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sample PWL Values

Actual PWL = 50, n = 4

One–Sided Spec

489 ≤ 50 PWL < 511

0 1 3 4

24

85

117

101

121

113

97

89

66

37

29 29

34

11

7

32

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Sample PWL Values

Actual PWL = 50, n = 4

Two–Sided Spec

569 ≤ 50 PWL < 431
 391

Figure 134. Illustration of the populations for which the distributions of sample PWL 

values are shown in figures 129 through 133. 

90 PWL

80 PWL

70 PWL

60 PWL

50 PWL
 
 393

REFERENCES 

1. Burati, J.L., et al. Optimal Procedures for Quality Assurance Specifications, Publication No. 

FHWA-RD-02-095, Federal Highway Administration, Washington, DC, April 2003. 

2. AASHTO Implementation Manual for Quality Assurance, AASHTO Quality Construction 

Task Force, American Association of State Highway and Transportation Officials, February 

1996. 

3. AASHTO Quality Assurance Guide Specification, AASHTO Quality Construction Task 

Force, American Association of State Highway and Transportation Officials, February 1996. 

4. “Glossary of Highway Quality Assurance Terms,” Transportation Research Circular No. 

E-C037, Transportation Research Board, Washington, DC, April 2002. 
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