NUREG-1505, Rev. 1

A Nonparametric Statistical

Methodology for the Design

and Analysis of Final Status

Decommissioning Surveys

Interim Draft Report for Comment and Use

U.S. Nuclear Regulatory Commission

Office of Nuclear Regulatory Research

C.V. Gogolak, G.E. Powers, A.M. Huffert
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NUREG-1505, Rev. 1

A Nonparametric Statistical

Methodology for the Design

and Analysis of Final Status

Decommissioning Surveys

Interim Draft Report for Use and Comment

Manuscript Completed: June 1998

Date Published: June 1998

Prepared by

C.V. Gogolak*, G.E. Powers, A.M.Huffert

Division of Regulatory Applications

Office of Nuclear Regulatory Research

U.S. Nuclear Regulatory Commission

Washington, DC 20555-0001

*U.S. Department of Energy, Environmental Measurements Laboratory

201 Varick Street, 5th Floor

New York, NY 10014
)
ABSTRACT

This report describes a nonparametric statistical methodology for the design and analysis of final

status decommissioning surveys in support of the final rulemaking on Radiological Criteria for

License Termination published by the Nuclear Regulatory Commission in the Federal Register

on July 21, 1997. The techniques described are expected to be applicable to a broad range of

circumstances, but do not preclude the use of alternative methods as particular situations may

warrant. Nonparametric statistical methods for testing compliance with decommissioning criteria

are provided both for the case in which the radionuclides of concern occur in background and

also for the case in which they do not occur in background. The tests described are the Sign test,

the Wilcoxon Rank Sum test, and a Quantile test. These tests are performed in conjunction with

an Elevated Measurement Comparison to provide confidence that the radiological criteria

specified for license termination are met. The Data Quality Objectives process is used for the

planning of final site surveys. This includes methods for determining the number of samples

needed to obtain statistically valid comparisons with decommissioning criteria and the methods

for conducting the statistical tests with the resulting sample data.

°°° NUREG-1505

CONTENTS

ABSTRACT ................................. iii

FOREWORD ................................ xiii

ABBREVIATIONS...................................................... xiv

I INTRODUCTION..................................................... 1-1

1.1 Overview of NRC Site Decommissioning ............................. 1-1

1.2 Need for This Report ............................................ 1-2

1.3 Objective of This Report.......................................... 1-2

1.4 Structure of This Report.......................................... 1-3

2 OVERVIEW OF FINAL STATUS SURVEY DESIGN.......................... 2-1

2.1 Introduction ........................ ....... *2-1

2.2 Final Status Survey Design........................................ 2-1

2.2.1 Release Criteria.......................................... 2-2

2.2.2 Data Interpretation ........................................ 2-3

2.2.3 Survey Unit Classification.................................. 2-3

2.2.4 Final Status Survey Classification ............................ 2-5

2.2.5 Background ............................................ 2-6

2.2.6 Data Variability.......................................... 2-7

2.2.7 Reference Areas......................................... 2-7

2.2.8 Radionuclide-Specific Measurements.......................... 2-8

2.3 Statistical Concepts ............................................. 2-8

2.3.1 Null and Alternative Hypotheses ............................. 2-8

2.3.2 Decision Errors......................................... 2-10

2.4 Hypothesis Testing Example...................................... 2-li

2.4.1 Detection Limits........................................ 2-12

2.4.2 Final Status Surveys ..................................... 2-14

2.4.3 The Effect of Measurement Variability on the Decision Process ...... 2-16

2.5 Statistical Tests................................................ 2-17

2.5.1 Nonparametric Statistical Tests ............................. 2-18

2.5.2 Wilcoxon Rank Sum and Sign Tests ........................... 2-19

2.5.3 Mean and Median ....................................... 2-21

2.3.4 Estimating the Amount of Residual Radioactivity ................ 2-23

2.5.5 Quantile Test ...................................... 2-23

2.5.6 Elevated Measurements Comparison .......................... 2-24

2.5.7 Investigation Levels....................................... 2-25

3 PLANNING FINAL STATUS SURVEYS: DATA QUALITY OBJECTIVES .......... 3-1

3.1 Introduction..................................................... 3-1

3.2 State the Problem................................................. 3-2

3.3 Identify the Decision ..................... ...... 3-2

3.4 Identify Inputs to the Decision....................................... 3-3

3.5 Define the Study Boundaries ........................................ 3-5

3.5.1 Spatial Variability ......................................... 3-5

v NUJREG-1 505
CONTENTS

Page

3.5.2 Temporal Variability ......................................... 3-6

3.5.3 Reference Coordinates ....................................... 3-6

3.5.4 Sampling Grids ...... ...................................... 3-9

3.6 Develop a Decision Rule ............................................ 3-12

3.7 Specify Limits on Decision Errors .................................... 3-14

3.7.1 Type I and Type H Decision Errors for Statistical Tests ............ 3-14

3.7.2 Decision Errors for Elevated Areas ............................ 3-19

3.8 Optimize the Design ............................................... 3-21

3.8.1 Optimizing the Design for the Mean Concentration ............... 3-22

3.8.2 Optimizing the Design for Detecting Elevated Areas .............. 3-27

4 ANALYSIS OF FINAL STATUS SURVEY RESULTS:

DATA QUALITY ASSESSMENT ...................................... 4-1

4.1 Review the Data Quality Objectives (DQOs) and Sampling Design ........... 4-1

4.2 Conduct a Preliminary Data Review .................................... 4-2

4.2.1 Basic Statistical Quantities .................................... 4-3

4.2.2 Graphical Data Review ....................................... 4-7

4.3 Select the Statistical Test ............................................ 4-13

4.4 Verify the Assumptions of the Statistical Test ........................... 4-15

4.5 Draw Conclusions from the Data ...................................... 4-16

5 SIGN TEST: CONTAMINANT NOT PRESENT IN BACKGROUND ............... 5-1

5.1 Introduction ....................................................... 5-1

5.2 Applying the Sign Test: Scenario A .................................... 5-5

5.3 Applying the Sign Test: Scenario B ..................................... 5-7

5.4 Interpretation of Test Results ........................................... 5-8

5.4.1 If the Null Hypothesis Is Not Rejected ........................... 5-9

5.4.2 If the Null Hypothesis Is Rejected ............................. 5-10

6 WILCOXON RANK SUM TEST: CONTAMINANT PRESENT IN BACKGROUND ... 6-1

6.1 Introduction ....................................................... 6-1

6.2 Applying the WRS Test: Scenario A ................................... 6-5

6.3 Applying the WRS Test: Scenario B ................................... 6-9

6.4 Interpretation of Test Results .......................................... 6-13

6.4.1 If the Null Hypothesis Is Not Rejected .......................... 6-13

6.4.2 If the Null Hypothesis Is Rejected ............................. 6-13

7 QUANTILE TEST ......................................................... 7-1

7.1 Introduction ....................................................... 7-1

7.2 Applying the Quantile Test ........................................... 7-1

7.3 Calculation of aQ for the Quantile Test .................................. 7-3

7.4 Modified Example for the Quantile Test ................................ 7-5

8 ELEVATED MEASUREMENT COMPARISON ................................. 8-1

8.1 Introduction ........................................................ 8-1

8.2 Comparison against Individual Measurements ............................ 8-2

NUREG-1505 vi
CONTENTS

Page

8.3 Comparison Against Scanning Measurements............................. 8-4

8.4 Area Factors ................................................... 8-4

8.5 Example ................................................ 8-9

9 SAM PLE SIZE ................................... .......................... 9-1

9.1 Sample Size and Decision Errors ...................................... 9-1

9.2 Sample Size Calculation for the Sign Test Under Scenario A ................ 9-2

9.3 Sample Size Calculation for the Sign Test Under Scenario B .............. 9-7

9.4 Sample Size Calculation for the WRS Test Under Scenario A ............... 9-9

9.5 Sample Size Calculation for the WRS Test Under Scenario B .............. 9-14

10 POWER CALCULATIONS FOR THE STATISTICAL TESTS ................... 10-1

10.1 Statistical Power and the Probability of Survey Unit Release ............. 10-1

10.2 Power of the Sign Test Under Scenario A ............................. 10-1

10.3 Power of the Sign Test Under Scenario B ............................. 10-4

10.4 Power of the Wilcoxon Rank Sum Test Under Scenario A ................ 10-6

10.5 Power of the Wilcoxon Rank Sum Test Under Scenario B .............. 10-12

11 MULTIPLE RADIONUCLIDES ........................................... 11-1

11.1 Using the Unity Rule .............................................. 11-1

11.2 Radionuclide Concentrations With Fixed Ratios ........................ 11-1

11.3 Unrelated Radionuclide Concentrations ............................... 11-2

11.4 Example Application of WRS Test to Multiple Radionuclides .............. 11-3

12 M'ULTIPLE SURFACES .......................................... 12-1

\12.1 Choosing Survey Units .......... ........................... 12-1

12.2 Combining Dissimilar Areas into One Survey Unit ................. ..... 12-1

12.3 Using Paired Observations for Survey Units with Many Different Backgrounds 12-4

13 DEMONSTRATING INDISTINGUISHABILITY FROM BACKGROUND ......... 13-1

13.1 Determining Significant Background Variability ........................ 13-1

13.2 Determining if Reference Areas Have Significantly Different Background .... 13-2

13.3 Establishing the Concentration Level that Is Indistinguishable ............. 13-5

13.4 Using the Concentration Level that Is Indistinguishable in the WRS Test .... 13-9

13.5 Determining the Number of Reference Areas and the Number of Samples . 13-11

13.6 Determining when Demonstrating Indistinguishability Is Appropriate ........ 13-12

14 ALTERNATIVES AND MODIFICATIONS ............................. 14-1

14.1 Alternative Statistical Tests ................................... 14-1

14.2 Retesting ....................................................... 14-3

14.3 Composite Sampling ......... .............................. 14-3

15 GLOSSARY ............................................................. 15-1

16 BIBLIOGRAPHY ........................................................ 16-1

APPENDIX: STATISTICAL TABLES ......................................... A-1

vii NUREG-1505
CONTENTS

FIGURES

Page

Figure 2.1 Type I and Type II Errors in the Determination of a Detection Limit ......... 2-13

Figure 2.2 Type I and Type II Errors for Scenario A ................................ 2-14

Figure 2.3 Type I and Type II Errors for Scenario B ............................... 2-15

Figure 2.4 Differences in Concentration Compared to Measurement Variability .......... 2-17

Figure 2.5 Comparison of Normal and Lognormal Distributions ...................... 2-22

Figure 3.1 Sample Indoor Reference Coordinate System ............................. 3-7

Figure 3.2 Sample Outdoor Reference Coordinate System.......................... 3-8

Figure 3.3 Laying Out a Triangular Grid ..................................... 3-11

Figure 3.4 Completed Triangular Sampling Pattern ................................ 3-12

Figure 3.5 Example of Setting Acceptable Probabilities for Survey Unit Release ......... 3-16

Figure 3.6 Circular and Elliptical Areas Relative to the Sample Grid ................. 3-19

Figure 3.7 Probability that an Elliptical Area Is Not Sampled on a Triangular Grid 3-20

Figure 3.8 Probability that a Circular Area Is Not Sampled on a Systematic Grid ......... 3-21

Figure 3.9 Probability that a Survey Unit Is Released Using Sign Test Under Scenario A .. 3-26

Figure 3.10 Probability that a Survey Unit Is Released Using Sign Test Under Scenario B . 3-27

Figure 3.11 Probability that a Survey Unit Is Released Using WRS Test Under Scenario A 3-28

Figure 3.12 Probability that a Survey Unit Is Released Using WRS Test Under Scenario B . 3-29

Figure 4.1 Confidence Bands for the Ratio of the Range to the Standard Deviation ........ 4-7

Figure 4.2 Example of a Posting Plot ....................................... 4-9

Figure 4.3 Frequency Plots of Example Survey Unit Data ........................ 4-10

Figure 4.4 Ranked Data Plot for the Example Reference Area Data ................... 4-12

Figure 4.5 Ranked Data Plot for the Example Survey Unit Data ...................... 4-12

Figure 4.6 Example Quantile-Quantile Plot ................................... 4-14

Figure 5.1 Desired Probability That the Survey Unit Passes ........................... 5-2

Figure 5.2 Posting Plot of Survey Unit Data ....................................... 5-2

Figure 5.3 Histogram of Survey Unit Data ........................................ 5-4

Figure 5.4 Ranked Data Plot of Survey Unit Data .................................. 5-5

Figure 6.1 Desired Probability That the Survey Unit Passes ......................... 6-2

Figure 6.2 Posting Plot of Reference Area and Survey Unit Data ...................... 6-3

Figure 6.3 Histograms of Reference Area and Survey Unit Data ....................... 6-4

Figure 6.4 Quantile-Quantile Plot of Example Data ................................. 6-5

Figure 8.1 Square and Triangular Sampling Grids and Grid Areas ....................... 8-3

Figure 8.2 Example Outdoor Area Factors ........................................ 8-7

Figure 8.3 Example Indoor Area Factors ........................................... 8-8

Figure 8.4 Combined Ranked Data Plot of Reference Area and Survey Unit Measurements- .8-10

Figure 8.5 Posting Plot of Indoor Concrete Survey Unit ............................. 8-12

Figure 9.1 The Parameter p for the Sign Test Under Scenario A ....................... 9-4

Figure 9.2 Sample Size Multiplier and the Parameter p Versus A/c .................... 9-5

Figure 9.3 Dependence of Sample Size on p ................................... 9-6

NUREG-1505 viii
CONTENTS

FIGURES (cont'd.)

Page

Figure 9.4 The Parameter p for the Sign Test Under Scenario B ...................... 9-8

Figure 9.5 The Parameter P,. for the WRS Test Under Scenario A.....................9-10

Figure 9.6 Dependence of Sample Size on A/o for WRS Test ........................ 9-12

Figure 9.7 Comparison of Sample Sizes Required for the WRS Test and the Sign Test .... 9-13

Figure 9.8 Dependence of Sample Size on Pr ............ .......................... 9-14

Figure 9.9 The Parameter Pr for the WRS Test Under Scenario B .................... 9-15

Figure 10.1 Example Power Curves: Sign Test Scenario A .......................... 10-3

Figure 10.2 Example Power Curves: Sign Test Scenario B .......................... 10-7

Figure 10.3 Probability Example Survey Unit Passes: Sign Test Scenario B ............ 10-7

Figure 10.4 Example Power Curves: WRS Test Scenario A ........................ 10-12

Figure 10.5 Example Power Curves: WRS Test Scenario B ........................ 10-15

Figure 10.6 Probability Example Survey Unit Passes: WRS Test Scenario B ........... 10-16

Figdre 12.1 Example Survey Units: Case #1 ...................................... 12-2

Figure 12.2 Example Survey Units: Case #2 ...................................... 12-3

Figure 12.3 Example Survey Units: Case #3 ...................................... 12-3

Figure 12.4 Example Survey Units: Case #4 ...................................... 12-5

Figure 13.1 Impact of Background on Decision Errors .............................. 13-2

Figure 13.2 Example DQOs for the Probability That the Survey Unit Passes Versus the

Concentration Difference Between the Survey Unit and the Reference Area .. 13-10

TABLES

Table 2.1 Final Status Survey Design Classification ................................. 2-5

Table 2.2 Summary of Types of Decision Errors .................................. 2-11

Table 2.3 Summary of Statistical Tests .......................................... 2-18

Table 2.4 Summary of Investigation Levels ...................................... 2-25

Table 3.1 Summary of Types of Decision Errors . ................................. 3-15

Table 3.2 Number of Samples Required for the Sign Test............................ 3-22

Table 3.3 Number of Samples Required for the Wilcoxon Rank Sum Test ............... 3-23

Table 4.1 Example Final Status Survey Data ...................................... 4-4

Table 4.2 Basic Statistical Quantities Calculated for the Data in Table 4.1 ............... 4-5

Table 4.3 Ranks of the Example Data ........................................... 4-11

Table 4.4 Recommended Statistical Tests ........................................ 4-15

* Table 4.5 Methods for Checking the Assumptions of Statistical Tests.................. 4-16

Table 5.1 Summary Statistics for Example Data of Figure 5.2 ......................... 5-3

Table 5.2 Ranked Data for Example Figure 5.2 ............ ....................... 5-4

Table 5.3 Calculations for Sign Test in Scenario A ................................. 5-7

Table 5.4 Calculations for Sign Test in Scenario B .............. .................... 5-9

ix NUREG-1505
CONTENTS

TABLES (cont'd.)

Table 6.1 Summary Statistics for Example Data of Figure 6.2 ..................... 6-3

Table 6.2 Ranked Data for Example of Figure 6.2 .................................. 6-4

Table 6.3 WRS Test for Class 2 Interior Drywall Survey Unit ........................ 6-7

Table 6.4 Spreadsheet Formulas Used in Table 6.3 ................................. 6-8

Table 6.5 WRS Test Under Scenario B for Class 2 Interior Drywall Survey Unit ........ 6-11

Table 6.6 Spreadsheet Formulas Used in Table 6.5 ................................ 6-12

Table 7.1 Quantile Test Under Scenario B for Class 2 Interior Drywall Survey Unit ....... 7-2

Table 7.2 Example Calculation of aQ for the Quantile Test ........................... 7-4

Table 7.3 Spreadsheet Formulas Used in Table 7.2 ................................. 7-5

Table 7.4 Quantile Test Under Scenario B: Modified Example Survey Unit .............. 7-6

Table 8.1 Example Outdoor Area Factors ........................................ 8-6

Table 8.2 Example Indoor Area Factors .......................................... 8-8

Table 8.3 Data for Indoor Survey Unit and Reference Area ........................... 8-11

Table 9.1 Some Values of Z,, and Z1 _.p Used To Calculate the Sample Sizes ............ 9-2

Table 9.2 Some Values of (Z,.,, + Z1_p) 2 Used To Calculate Sample Sizes ............... 9-3

Table 9.3 Values of p for Use in Computing Sample Size for the Sign Test ..... %........ 9-5

Table 9.4 Values of P, for Use in Computing Sample Size for the WRS Test ........... 9-11

Table 10.1 Example Power Calculations: Sign Test Scenario A ....................... 10-3

Table 10.2 Example Power Calculations: Sign Test Scenario B ...................... 10-6

Table 10.3 Values ofp, and P2 for Computing the Mean and Variance of WMW .......... 10-10

Table 10.4 Example Prospective Power Calculation: WRS Test Scenario A ............ 10-11

Table 10.5 Example Prospective Power Calculation: WRS Test Scenario B ............ 10-15

Table 11.1 Example WRS Test for Two Radionuclides ............................. 11-4

Table 13.1 Critical Values, K,, for the Kruskal-Wallis Test ......................... 13-4

Table 13.2 Example Data for the Kruskal-Wallis Test .............................. 13-5

Table 13.3 Calculation of 02 for the Example Data ............................ 13-8

Table 13.4 Analysis of Variance for Example Data ................................ 13-9

Table 13.5 Power of the F-test When c2 = o2 .................................... 13-12

Table A.1 Cumulative Normal Distribution Function, 4)(z) ........................... A-1

Table A.2a Sample Sizes for the Sign Test ....................................... A-2

Table A.2b Sample Sizes for the Wilcoxon Rank Sum Test .......................... A-3

Table A.3 Critical Values for the Sign Test Statistic, S - ............................ A-4

Table A.4 Critical Values for the WRS Test .... ............................ A-6

Table A.5a 0.025 and 0.975 Percentiles of the Chi-Squared Distribution ............... A-9

Table A.5b 0.05 and 0.95 Percentiles of the Chi-Squared Distribution ................ A-10

Table A.6 1000 Random Numbers Uniformly Distributed Between Zero and One ........ A-11

Table A.7a Values of r and k for the Quantile Test When a Is Approximately 0.01 ...... A-13

Table A.7b Values of r and k for the Quantile Test When a Is Approximately 0.025 ..... A-14

NUREG-1505 X
CONTENTS

TABLES (cont'd.)

Page

Table A.7c Values of r and k for the Quantile Test When a Is Approximately 0.05 ...... A-15

Table A.7d Values of r and k for the Quantile Test When a Is Approximately 0. 10 ...... A-16

xi NUREG-1505

FOREWORD

The NRC has amended its regulations to establish residual radioactivity criteria for decommissioning of licensed nuclear facilities. As part of this initiative, the NRC staff has evaluated the

application of nonparametric statistical methods as an alternative to the parametric statistical

approach described in the U.S. Nuclear Regulatory Commission (NRC) draft report

NUREG/CR-5849, entitled, "Manual for Conducting Radiological Surveys in Support of License

Termination." The nonparametric statistical approach described in this report is expected to be

simpler and more cost-effective for the design and analysis of final status decommissioning

surveys when radiological criteria for decommissioning approach background radiation levels.

This report also shows the advantages of using the Data Quality Objectives process as it relates

to the planning and analysis of final site surveys. The application of the proposed DQO process

includes methods for determining the number of samples needed to obtain statistically valid

comparisons with decommissioning criteria and the methods for conducting the statistical tests

with the resulting sample data.

The initial draft of this report was published in August 1995. As a result of the comments

received, extensive revisions were made to include alternative scenarios for statistical hypothesis

testing. A number of new concepts have been introduced, and examples for some special cases

have been added. The results, approaches and methods described herein are provided for

information only and should not be considered a substitute for NRC requirements.

n W. Craig, Director

Division of Regulatory Applications

Office of Nuclear Regulatory Research

Xlll NUREG-1505
ABBREVIATIONS

ALARA

CFR

DCGL

DOE

DQA

DQO

EMC

EPA

LBGR

MCA

MDC

NIST

NRC

PC

PDL

PIC

QA

QC

TEDE

WRS

WSR

as low as is reasonably achievable

Code of Federal Regulations

Derived Concentration Guideline Level

U.S. Department of Energy

data quality assessment

data quality objective

elevated measurement comparison

U.S. Environmental Protection Agency

Lower Boundary of the Gray Region

multichannel analyzer

minimum detectable concentration

National Institute for Standards and Technology

U.S. Nuclear Regulatory Commission

personal computer

predicted dose level

pressurized ionization chamber

quality assurance

quality control

total effective dose equivalent

Wilcoxon Rank Sum Test

Wilcoxon Signed Ranks Test

NUREG-1505 xiv
1 INTRODUCTION

1.1 Overview of NRC Site Decommissioning

At sites and facilities licensed by the Nuclear Regulatory Commission (NRC), the formal

decommissioning process begins when a licensee decides to terminate licensed activities. The

majority of licenses terminated each year by NRC involve little or no site remediation and,

therefore, present no complex decommissioning problems owing to residual radioactivity.

However, license termination at a small number of sites is far more complex because

contamination may be spread into various areas within the facility and surrounding areas by the

movement of materials and equipment, by activation, and by the dispersion of air, water, or other

fluids through or along piping, equipment, walls, floors, and drains. Decontamination of such

areas is likely to be performed at nuclear power plants, non-power (research and test) reactors,

fuel fabrication plants, uranium hexafluoride production plants, and independent spent fuel

storage installations. A small number of universities, medical institutions, radioactive sourcp

manufacturers, and companies that use radioisotopes for industrial purposes may also contain

radioactive contamination that requires remediation.

NRC regulations in 10 CFR 30.36, 40.42, 50.82, 70.38, and 72.54 require licensees to remove

their facilities from service safely. As part of the decommissioning process, licensees are

required to demonstrate that residual radioactivity in facilities and environmental media has been

reduced to acceptable levels. Typically, licensees demonstrate compliance with radiological

criteria for license termination by conducting final status surveys of the site or facility and

reporting the survey results to NRC for evaluation. Where appropriate, the NRC staff conducts

confirmatory surveys to verify that lands and structures have been adequately remediated.

On July 21, 1997, the NRC amended the regulations in 10 CFR Part 20 to include explicit

radiological criteria for decommissioning (62 FR 139, pp. 39057- 39092). Subpart E of the

amended regulations contains dose-based radiological criteria for restricted and unrestricted

release, consisting of a total effective dose equivalent (TEDE) limit for residual radioactivity

above background. These regulations replace prior NRC guidance based on surface and volume

activity concentration limits for specific radionuclides.

To implement the dose criteria in the amended 10 CFR Part 20, final status surveys and

confirmatory surveys must be capable of detecting very low levels of residual radioactivity in the

presence of background at a variety of NRC-licensed facilities and sites. An essential component

of such surveys is a statistical methodology that is appropriate for radiological data at or near

background levels. This document presents such a methodology.

1.2 Need for This Report

Previously, the NRC staff used guidance for conducting final status radiological surveys that is

contained in draft report NUREG/CR-5849 (1992), entitled "Manual for Conducting

Radiological Surveys in Support of License Termination." This report contains an alternative

statistical approach for designing radiological surveys. The framework for the survey design is

1-1 NUREG-1505
INTRODUCTION

the Data Quality Objectives (DQO) process. The DQO process uses statistical hypothesis testing

rather than the construction of confidence intervals. This allows a balance to be reached between

the risk of possibly releasing an incompletely remediated site and the risk of possibly requiring

further remediation at an already adequately remediated site. One of the primary goals of the

DQO process is the determination of acceptable decision error rates for the hypothesis test, i.e.,

those that will reflect the relative importance of these risks at a specific site. The DQO process is

used to incorporate site-specific information and sound scientific judgment into the survey design

and data analysis so that the objective of safely releasing a site can be met while reducing the

number of unnecessarily arbitrary and conservative assumptions that are sometimes invoked in

the face of uncertainty.

Using the DQO framework, the amount and type of data to be collected are related to the

specific decision to be made rather than sampling at a fixed density. The number of samples of

measurements needed in a survey unit is determined by the acceptable decision error rates, the

magnitude of the release criterion relative to the overall variability of the data, and the sensitivity

of the scanning method used. The type and amount of scanning required depend primarily on the

classification of the survey unit. Three classes of survey units are used to direct the survey effort

at a level commensurate with the potential for residual radioactivity in excess of the release

criterion. Acceptable areas of elevated activity are determined by radionuclide-specific area

factors derived from an appropriate dose model..

The nonparametric statistical techniques described in this report do not require the data to be

normally or log-normally distributed and are, therefore, expected to be more appropriate for

determining the number of samples required for radiological surveys and analyzing data collected

at or near background levels. These tests perform almost as well as the parametric tests even

when the data are normally distributed, are less sensitive to outliers, and are better able to handle

data sets that include non-detects.

There are two possible approaches to demonstrating compliance with criteria that specify a dose

limit due to residual radioactivity distinguishable from background, depending on which of the

following questions is emphasized:

(1) Does the dose due to residual radioactivity exceed the limit?

(2) Is the residual radioactivity indistinguishable from background?

In the initial draft of NUREG-1505 (August 1995) the approach emphasized question 2. This

final report addresses both approaches.

1.3 Objective of This Report

This report describes a nonparametric statistical methodology that NRC licensees may consider

when evaluating methods for demonstrating compliance with the radiological criteria for license

termination in Subpart E of 10 CFR Part 20. The DQO process (EPA QA/G-4 and QA/G-9,

1994) is used as the framework for the planning of final site surveys. The statistical approach

described in this report is expected to be a resource-efficient solution for the design of final status

decommissioning surveys when radiological criteria for decommissioning approach background

NUREG-1505 1-2
INTRODUMTON

levels. The proposed process includes methods for determining the number of samples needed to

obtain statistically valid comparisons with decommissioning criteria and the methods for

conducting the statistical tests with the resulting sample data.

No single statistical formulation can adequately anticipate every contingency that will arise in

deciding whether a survey unit can be safely released. The DQO process should be used to

determine whether a proposed action will further the objective of safely releasing the site. The

decisions reached may not always be accompanied by a numerical procedure leading to that

decision. However, such decisions should always be accompanied by a description of which

actions were taken, and why. The DQO process provides a methodology for resolving the often

complex issues surrounding site remediation and decommissioning.

1.4 Structure of This Report

This report is divided into four major parts. The first part deals with general final status survey

design criteria, definitions, and data quality objectives (Chapters 2 and 3). The second part

describes preliminary data analysis and data quality assessment (Chapter 4). The third part

describes the use of the statistical tests recommended in this report (Chapters 5 through 8).

These first eight chapters contain all of the information required to design and conduct final

status surveys, and to analyze and interpret the results. Chapters 9 through 14 deal with

extensions of, and alternatives to, the statistical procedures that may be applicable in some

situations. Chapters 15 and 16 contain a Glossary and Bibliography, respectively. The appendix

contains the statistical tables needed to perform the analyses described in this report.

1-3 NUREG- 1505

2 OVERVIEW OF FINAL STATUS SURVEY DESIGN

2.1 Introduction

It is recognized that demonstrating that residual concentrations of radioactivity at a site are at

very low levels in the presence of background may be a complex task involving sophisticated

sampling, measurement, and statistical analysis techniques. The difficulty of the task can vary

substantially depending on a number of factors, including the radionuclides in question, the

background level for those and other radionuclides at the site, and the temporal and spatial

variations in background at or near the site. Sufficient radiological data must be collected to

characterize both the residual radioactivity at the site and the background radioactivity levels in

the vicinity of the site. The number of measurements required to accomplish this task will be

determined on a site-specific basis and will depend upon the nature of the facility, its size, the

selection of thestatistical tests used, and certain statistical parameter values that influence how

compliance with radiological criteria is determined.

2.2 Final Status Survey Design

Decommissioning is defined in 10 CFR 20.1003 as removing a facility or site safely from service,

and reducing residual radioactivity to a level that permits (1) release of the property for

unrestricted use and termination of the license; or (2) release of the property under restricted

conditions and termination of the license. A survey unit is a geographical area of specified size

and shape for which a separate decision will be made whether or not that area meets the release

criteria. This decision is made following afinal status survey of the survey unit. Thus, a survey

unit is an area for which a final status survey is designed, conducted, and results in a release

decision. The objective of this report is the design of efficient final status surveys. These surveys

should obtain all of the data required for making the decision, but avoid the collection and

analysis of superfluous samples.

Usually there are two conditions that would lead to the determination that a particular survey unit

requires further remediation before unrestricted release:

(1) If the average level of residual radioactivity within the survey unit exceeds the regulatory

limit, or

(2) If there are small areas within the survey unit with elevated residual radioactivity that

exceed the regulatory limit.

Sampling at discrete points within the survey unit is a simple method for determining if the first

of these conditions exists. The term sampling is used here in its statistical sense, namely

obtaining data from a subset of a population. Sampling in this sense Would include both direct in

situ measurements and the collection of physical samples for laboratory analysis.

On the other hand, sampling at discrete points within a survey unit is not a very efficient method

of determining if the second condition exists. Scanning is a much better method for detecting

isolated areas with elevated activity. However, scanning is generally not as sensitive as sampling.

2-1 2-1 NUREG- 1505
OVERVIEW

A major component of the survey designs discussed in this report is the efficient use of sampling

at distinct locations combined with scanning to accurately determine the final status of a survey

unit. The statistical procedures described in this report are used to establish the number of ;

samples taken at distinct locations needed to determine if the mean concentration in the survey

unit exceeds the regulatory limit, with a specified degree of precision. Thus, these statistical

procedures are as important in the planning and design of the final status survey as they are in the

analysis and interpretation of the resulting data.

2.2.1 Release Criteria

In the past, release criteria have often been expressed as activity concentration limits. The criteria

for license termination given in 10 CFR 20.1402 and 10 CFR 20.1403 are expressed in terms of

total effective dose equivalent (TEDE). This cannot be measured directly. Exposure pathway

modeling is used to calculate the estimated volume or surface area concentration of specific

radionuclides that could result in a TEDE equal to the release criterion. This concentration is

termed the derived concentration guideline level (DCGL). The units for the DCGL are the same

as the units for measurements performed to demonstrate compliance (e.g., Bq/kg, Bq/m', etc.).

This allows direct comparisons between the survey results and the DCGL.

A complete discussion of DCGLs is beyond the scope of this report. There is, however, one

aspect of exposure pathway modeling that bears directly on survey unit design. That is the

dependence of the TEDE on the assumed area of contamination used in the exposure pathway

model.

The two conditions of Section 2.2 that may cause a survey unit to fail the TEDE release criterion

may have very different corresponding DCGLs because of the different size of the areas of

residual radioactivity. Consequently, this report considers two distinct DCGLs:

(1) The DCGLw is derived assuming that residual radioactivity is uniformly distributed over

a wide area, i.e. the entire survey unit. This can often be the default DCGL provided by an

exposure pathway model.

(2) The DCGLEMc is derived assuming that residual radioactivity is concentrated in a much

smaller area, i.e., in only a small percentage of the entire survey unit.

The DCGLEMc can never be less than the DCGLw, but it may be significantly greater. The ratio of

the DCGLEMc to the DCGLw defines a radionuclide specific area factor, FA, such that the

DCGI-mMc = (FA) (DCGLw), when the residual radioactivity is confined to an area of size A.

Detailed procedures for developing these area factors are beyond the scope of this report.

However, in the simplest case, an area factor can be determined from the ratio of the result

obtained from an exposure pathway model using the entire survey unit area to the result obtained

assuming the residual radioactivity is confined to a smaller area. The value of the DCGL EMC that

is calculated for survey planning purposes is based on an area, A, determined by the spacing

between adjacent sampling locations.

NUREG-1505 2-2
OVERVIEW

2.2.2 Data Interpretation

The use of the two DCGLs discussed above differs when interpreting the results of the final

status survey data. The DCGLw is used to form a statistical hypothesis concerning the level of

residual radioactivity that may be uniformly distributed across the survey unit. A nonparametric

test is applied to the sampling data taken at distinct locations in the survey unit to determine

whether this level meets the release criterion.

The DCGLtmc, however, is used to trigger further investigation of a portion of the survey unit.

Any measurement from the survey unit is considered elevated if it exceeds the DCGLmc. This is

the elevated measurement comparison. The existence of an elevated measurement in a survey

unit indicates the possibility of an area of residual radioactivity that may cause the dose criteria to

be exceeded. The elevated measurement alone does not indicate that the survey unit fails to meet

the release criterion, only that it is a possibility that must be investigated further. The DCGLEMC

is based on the area factor used for the survey design. The area factor used in the survey design is

based on the area bounded by adjacent sampling points. The actual area of elevated activity could

be smaller. Thus, the area factor based on the actual area of contamination may be larger.

Further investigation will usually be necessary to determine the actual extent and concentration

level of a specific elevated area.

2.2.3 Survey Unit Classification

To maximize the efficiency of the final status surveys, it is clear that the greatest effort should be

expended on the areas that have the highest potential for contamination. Final status survey

designs depend fundamentally on the classification of survey units according to contamination

potential. The survey unit classification determines the final status survey design and the

procedures used to develop the design.

Areas that have no potential for residual contamination are classified as non-impacted areas.

These areas have no radiological impact from site operations and are typically identified early in

decommissioning. Areas with some potential for residual contamination are classified as

impacted areas.

Impacted areas are further divided into one of three classifications:

(1) Class 1 Areas: Areas containing locations where, prior to remediation, the concentrations

of residual radioactivity may have exceeded the DCGLW,

(2) Class 2 Areas: Areas containing no locations where, prior to remediation, the

concentrations of residual radioactivity may have exceeded the DCGL W

(3) Class 3 Areas: Areas with a low probability of containing any locations with residual

radioactivity.

Class 1 areas have the greatest potential for contamination and therefore receive the highest

degree of survey effort for the final status survey. Non-impacted areas do not receive any level of

survey coverage because they have no potential for residual contamination. Impacted areas for

2-3 NUREG-1505
OVERVIEW

which there is insufficient information to justify a lower classification should be classified as

Class 1.

Examples of Class 1 areas include: (1) site areas previously subjected to remedial actions,

(2) locations where leaks or spills are known to have occurred, (3) former burial or disposal sites, (4) waste storage sites, and (5) areas with contaminants in discrete solid pieces of material and

high specific activity.

Remediated areas are identified as Class I areas because the remediation process often results in

less than 100% removal of the contamination. The contamination that remains on the site after

remediation is often associated with relatively small areas with elevated levels of residual

radioactivity. This results in a non-uniform distribution of the radionuclide and a Class 1 - classification. If an area is expected to have levels of residual radioactivity below the DCGL w

and was remediated for purposes of ALARA, the remediated area might be classified as Class 2

for the final status survey.

Examples of areas that might be classified as Class 2 for the final status survey include:

(1) locations where radioactive materials were present in an unsealed form, (2) potentially contaminated transport routes, (3) areas downwind from stack release points, (4) upper walls and

ceilings of buildings or rooms subjected to airborne radioactivity, (5) areas handling low

concentrations of radioactive materials, and (6) areas on the perimeter of former contamination

control areas.

To justify changing the classification from Class 1 to Class 2, there should be measurement data

that provides a high'degree of confidence that no individual measurement would exceed the

DCGLw. Other justifications for reclassifying an area as Class 2 may be appropriate, based on

site-specific considerations.

Examples of areas that might be classified as Class 3 include buffer zones around Class 1 or

Class 2 areas, and areas with very low potential for residual contamination but insufficient

information to justify a non-impacted classification.

The number of distinct sampling locations needed to determine if a uniform level of residual

radioactivity within a survey unit exists does not depend on the survey unit size. However, the

sampling density within a survey unit should reflect the potential for small elevated areas of

residual radioactivity. Thus, the appropriate size for survey units formed within each of the three

area classifications differs. Survey units with a higher potential for residual radioactivity should

be smaller. Suggested maximum areas for survey units are:

Class 1 Structures ........ 100 m2 floor area

Class 1 Land areas ....... 2,000 m2

Class 2 Structures ........ 100 to 1,000 m2

Class 2 Land areas ....... 2,000 to 10,000 in2

Class 3 Structures ........ no limit

Class 3 Land areas ....... no limit

NUREG-1505 2-4
OVERVIEW

The area of the survey unit should also be consistent with that assumed in the exposure pathway

model used to calculate the DCGLw. Survey units with structure surface areas less than 10 m2or

land areas less than 100 m2 may have unnecessarily high sampling densities, and should be

avoided.

2.2.4 Final Status Survey Classification

Class I areas have the highest potential for containing small areas of elevated activity exceeding

the release criterion. Consequently, both the number of sampling locations and the extent of

scanning effort is the greatest. The final status survey is driven by the effort to provide

reasonable assurance that if any areas with concentrations in excess of the DCGL EMCexist that

then these areas will be found. Sampling is done on a systematic grid. The distance between

sampling locations is made small enough that any elevated area that might be missed by

sampling would be found by scanning. Scanning is performed over 100% of the survey unit. The

minimum detectable concentration (MDC) of the scanning method must be lower than the

DCGLEMC.

Class 2 areas may contain residual radioactivity, but the potential for elevated areas is very small.

Sampling is done on a systematic grid. The distance between samples is limited by limiting the

maximum size of the survey unit. Scanning is performed systematically over the survey unit.

Since Class 2 is an intermediate classification, scanning coverage may range from as little as

10% to nearly 100% of the survey unit, depending on whether the potential for an elevated area is

nearer that for a Class 1 area or for a Class 3 area.

Class 3 areas should contain little, if any, residual radioactivity. There should be virtually no

potential for elevated areas. Sampling is random across the survey unit, and the sample density

can be very low. Scanning is limited to those parts of the survey unit where it is deemed prudent,

based on the judgment of an experienced professional.

Table 2.1 summarizes the differences in the final status survey design for each of the three survey

unit classifications.

Table 2.1 Final Status Survey Design Classification

Class Sampling Scanning

1 Systematic 100% Coverage

2 Systematic 10- 100%

3 Random Judgmental

2-5 NUREG-1505
OVERVIEW

2.2.5 Background

The release criteria in 10 CFR Part 20.1402 and 1403 specify a dose limit (TEDE) due to residual

radioactivity that is distinguishable from background radiation. According to 10 CFR 20.1003,

background radiation means radiation from cosmic sources, naturally occurring radioactive

material, including radon (except as a decay product of source or special nuclear material), and

global fallout as it exists in the environment from the testing of nuclear explosive devices or

from nuclear accidents like Chemobyl which contribute to background radiation and are not

under the control of the licensee. Background radiation does not include radiation from source,

byproduct, or special nuclear materials regulated by the Commission. The term distinguishable

from background means that the detectable concentration of a radionuclide is statistically

different from the background concentration of that radionuclide in the vicinity of the site, or, in

the case of structures, in similar materials using adequate measurement technology, survey and

statistical techniques.

For the purposes of survey design, the method of accounting for background radiation will

depend not only on the radionuclides involved, but also on the type of measurements made. For

radionuclide specific measurements of radionuclides that do not appear in natural background, it

is clear that no adjustments for background are needed. In some cases, a sample-specific

background adjustment may be possible. For example, residual 23

1.1 activity may be

distinguishable from natural 238

U by the amount of 226

Ra present in a sample. In other cases, it

will not be possible to make such a distinction. In particular, such a distinction will not be

possible, even if the radionuclide does not appear in background, when gross activity or exposure

rate measurements are used.

For the elevated measurement comparison of individual sampling results, an adjustment for

background will not ordinarily be necessary, since the DCGLEMcis a multiple of the DCGLw. For

statistical testing of the results against the release criterion, however, one approach is used when

the measurements represent net residual radioactivity, but a different approach is necessary when

the measurements represent total radioactivity including background.

When a specific background can be established for individual samples, the results of the survey

unit measurements can be compared directly to the DCGL, since each is a measurement of the

residual radioactivity alone. Because only one set of measurements is involved in this

comparison, the statistical test is called a one-sample test.

S

When a specific background cannot be established for individual samples, the survey unit

measurements cannot be directly compared to the DCGL, since each is a measurement of the

total of any residual radioactivity plus the survey unit background. In this case, the measurements

in a survey unit must be compared to similar measurements in local reference areas that have

been matched to the survey unit in terms of geological, chemical, and biological attributes, but

which have not been affected by site operations. The distribution of the measurements in a

survey unit is compared to the distribution of background measurements in a reference areas.

Because two sets of measurements are used in making this comparison, the statistical test is

called a tvo-sample test.

NUREG-1505 2-6
OVERVIEW

2.2.6 Data Variability

The ease or difficulty with which compliance may be demonstrated depends primarily on the size

of the DCGLw relative to the amount of variability in the measurement data. This is commonly

known as the signal-to-noise ratio. As this ratio becomes smaller, more measurement data will be

needed to determine compliance with the release criterion, i.e. to extract the signal from the

noise.

The variability in the measurement data is a combination of the precision of the measurement

process, and the real spatial variability of the quantity being measured in the survey unit.

Variability can be reduced by using more precise measurement methods, but the spatial

variability remains. The mechanism by which spatial variability can be reduced is by choosing

survey units that are as homogeneous as possible with respect to the expected level of residual

radioactivity. This means that survey units should generally be formed from areas with similar

construction, use, contamination potential, and remediation history.

If the measurement data include a background contribution, the spatial variability of background

adds to the overall measurement variability. Thus, the survey units where such measurements

will be used should be as homogeneous as possible with respect to expected natural background

as well. Further information on natural background and its variability can be found in NUREG1501 (August 1994).

An additional source of variability is introduced when survey unit measurement data including

background are compared to measurements from a reference area.oAny systematic difference in

background level between the survey unit and the reference area will be indistinguishable from a

difference in residual radioactivity in the two areas. This situation is not unique to

decommissioning or the methodology of this report. It is always true when a background

adjustment must be made using data from a location other than the sampling location, e.g. using

control dosimeters at remote locations to account for background in monitoring dosimeters.

2.2.7 Reference Areas

A reference area (or background area) is a geographical area from which representative samples

of background will be selected for comparison with samples collected in specific survey units at

the remediated site. The reference area should have similar physical, chemical, radiological, and

biological characteristics to the site area being remediated, but should not have been

contaminated by site activities. The reference area is where background is measured and defined

for the purpose of decommissioning. To minimize systematic biases in the comparison, the same

sampling procedure, measurement techniques, and type of instrumentation should be used at both

the survey unit and the reference area. The distribution of background measurements in the

reference area should be the same as that which would be expected in the survey unit if that

survey unit had never been contaminated. It may be, necessary to select more than one reference

area for a specific site, if the site includes so much physical, chemical, radiological, or biological

variability that it cannot be represented by a single reference background area.

I 2-7 NUREG-1505
OVERVIEW

2.2.8 Radionuclide-Specific Measurements

As indicated in Section 2.2.5 and 2.2.6, if radionuclide-specific survey methods are used, and if

the radionuclide of interest does not appear in background, reference area measurements are not

needed, and one-sample statistical tests are used. If other survey methods are used, such as gross

activity or exposure rate measurements, then the individual contributions due to background and

any residual radioactivity will not be separately identifiable, suitable reference area

measurements will be needed, and two-sample statistical tests are used.

Even if the radionuclide of interest does appear in background, the variability of radionuclide

specific measurements will generally be smaller than those of gross activity measurements in the

same area. Depending on the level of residual activity that it is necessary to detect, many more

measurements may be required if gross activity or exposure rate measurements are used than if

radionuclide-specific measurements are made. At very low levels, it may be difficult or

impossible to distinguish the residual radioactivity contribution unless radionuclide-specific

methods are used. However, it may be economical in some circumstances to perform a larger

number of simpler, less expensive measurements. One of the primary advantages of the Data

Quality Objectives process, is that alternative measurement strategies can be compared at the

planning stage. Exploring the statistical design of the final status survey in advance, the most

efficient method for the problem can be chosen.

2.3 Statistical Concepts

This section introduces some of the statistical concepts and terminology used in hypothesis

testing. A use of statistical hypothesis testing that is familiar in the radiation protection

measurements field is the calculation of lower limits of detection. The methodology of this report

can be viewed as an application of these same concepts to a survey unit rather than to a

laboratory measurement. This analogy is pursued further in Section 2.6.

2.3.1 Null and Alternative Hypotheses

The decisions necessary to determine compliance with the criteria for license termination are

formulated into precise statistical statements called hypotheses. The truth of these hypotheses can

be tested with the survey unit data. The state that is presumed to exist in reality is expressed as

the null hypothesis (denoted by H0). For a given null hypothesis, there is a specified alternative

hypothesis (denoted as H.), which is an expression of what is believed to be the state of reality if

the null hypothesis is not true.

For the purposes, of this report, the important decision is whether or not a site meets the

applicable license termination and release criteria. This decision will be supported by the

individual decisions on whether each survey unit meets the applicable release criteria. In this

report, two different scenarios, designated Scenario A and Scenario B, are considered.

In Scenario A, the null hypothesis is:

H0: The survey unit does not meet the release criterion

yersus the alternative

Ha: The survey unit meets the release criterion.

NUREG-1505 2-8
OVERVIEW

In Scenario B, the null hypothesis is:

H0: The survey unit meets the release criterion.

versus the alternative

Ha: The survey unit does not meet the release criterion.

As indicated in Section 2.2.1, the release criterion is specified in terms of a dose, which is

converted via pathway modeling to a residual radioactivity concentration limit, the DCGLW, If

the concentration of residual radioactivity that is distinguishable from background in the survey

unit exceeds the DCGLw , the survey unit does not meet the release criterion.

When choosing the scenario to use, it is important to note that the null hypothesis cannot be

proved, i.e. accepted as true. The null hypothesis is either rejected or not rejected. The data are

either consistent with the null hypothesis, or they are not. It is stated this way because there are

two circumstances leading to the decision not to reject the null hypothesis:

(1) the null hypothesis is true.

(2) the null hypothesis is false, but the data did not provide enough evidence to show it.

The burden of proof is on the alternative. Thus, in Scenario A, the survey unit will not be

released until proven clean. In Scenario B, the survey unit will be released unless it is shown to

be contaminated above background. Rejecting the null hypothesis has different implications for

survey unit release in the two scenarios. For this reason, a survey unit will be said to pass the

final status survey if it is concluded that it may be released. Otherwise it will be said to fail. In

Scenario A, the emphasis is on the dose limit. In Scenario B, the emphasis is on

indistinguishability from background. In Scenario A, the survey unit is assumed to fail unless the

data show it may be released. In Scenario B, the survey unit is assumed to pass unless the data

show that further remediation is necessary.

In Scenario A, the measured average concentration in the survey unit must be significantly less

than the DCGLw in order to pass. In Scenario B, the measured average concentration in the

survey unit must be significantly greater than background in order to fail. In Scenario A,

increasing the number of measurements in a survey unit increases the probability that an

adequately remediated survey unit will pass. In Scenario B, increasing the number of

measurements in a survey unit increases the probability that an inadequately remediated survey

unit will fail.

Which scenario should be used? Because of insufficient evidence, the null hypothesis may not be

rejected even when it is false. Thus, the null hypothesis should be the one that is the easiest to

live with even if it is false. The alternative should be the hypothesis that carries the severest

consequences if it falsely chosen. To make the proper choice of scenario, the possible types of

decision errors and the probability of making them should be examined. This is the subject of the

next section.

In most cases, when the DCGLw is fairly large compared to the measurement variability,

Scenario A should be chosen. This is because even contamination below the DCGLw should be

measurable. Requiring additional remediation when it is not strictly necessary may still have

some benefit in the form of reduced radiation exposure. Releasing a survey unit that really should

2-9 NUREG-1505
OVERVIEW

be remediated further is a less tolerable mistake. It is anticipated that Scenario A will be simpler

to implement for most licensees.

When the DCGLw is small compared to measurement and/or background variability, Scenario B

should be chosen. This is because contamination below the DCGLw will be difficult to measure.

Requiring additional remediation when it is not necessary, may essentially require remediation of

background. This is an impossible task. Releasing a survey unit that has residual radioactivity

within the range of background variations is a less severe consequence in this case. It is fairly

straightforward to specify what is meant for a survey unit to meet the release criterion, but a

survey unit may be distinguishable from background either because it is uniformly contaminated

or because it contains spotty areas of residual radioactivity. For this reason, the data analysis for

Scenario B involves two statistical tests performed in tandem.

In this report, the two scenarios are developed in parallel. Within the limits imposed by the

magnitude of the data variability relative to the DCGLw, essentially the same information about

the survey unit should be obtained, and the same conclusion regarding compliance should be

reached using either scenario. The difference is in the emphasis.

2.3.2 Decision Errors

Errors can be made when making site remediation decisions. The use of statistical methods

allows for controlling the probability of making decision errors. When designing a statistical

test, acceptable error rates for incorrectly determining that a site meets or does not meet the

applicable decommissioning criteria must be specified. In determining these error rates,

consideration should be given to the number of sample data points that are necessary to achieve

them. Lower error rates require more measurements, but result in statistical tests of greater power

and higher levels of confidence in the decisions. In setting error rates, it is important to balance

the consequences of making a decision error against the cost of achieving greater certainty.

There are two types of decision errors that can be made when performing the statistical tests

described in this report. The first type of decision error, called a Type I error, occurs when the

null hypothesis is rejected when it is actually true. A Type I error is sometimes called a "false

positive." The probability of a Type I error is usually denoted by at. The Type I error rate is

often referred to as the significance level or size of the test.

The second type of decision error, called a Type II error, occurs when the null hypothesis is not

rejected when it is actually false. A Type II error is sometimes called a "false negative." The

probability of a Type II error is usually denoted by P3. The power of a statistical test is defined as

the probability of rejecting the null hypotheses when it is false. It is numerically equal to 1-3,

where P3 is the Type II error rate.

The setting of acceptable error rates is a crucial step in the planning process. Specific

considerations for establishing these error rates are discussed in Chapter 3. Table 2.2 summarizes

the types of decision errors that can be made for the specific hypotheses of Scenario A and

Scenario B.

NUREG-1505 2-10
OVERVIEW

Table 2.2 Summary of Types of Decision Errors

Scenario A True Condition of Survey Unit

Decision Based on Survey Does Not Meet Meets

Release Criterion Release Criterion

Does Not Meet Release Survey unit fails Survey unit fails

Criterion Correct Decision Type II Error

_ _ _ _ _(Probability = 1-c:) (Probability = 13)

Meets Release Criterion Survey unit passes Survey unit passes

Type I Error Correct Decision

(Probability = c) (Power = 1-13)

Scenario B True Condition of Survey Unit

Decision Based on Survey Meets Does Not Meet

Release Criterion Release Criterion

Meets Release Criterion Survey unit passes Survey unit passes

Correct Decision Type II Error

(Probability = 1-a) (Probability = P)

Does Not Meet Release Survey unit fails Survey unit fails

Criterion Type I Error Correct Decision

(Probability = a) (Power = 1-13)

2.4 Hypothesis Testing Example

The following example illustrates the use of the concepts discussed above as currently used in the

determination of detection limits for radioactivity measurements. The analogy is most direct for

Scenario B.") The calculation of detection limits, which is generally familiar to radiation

protection professionals, also involves hypothesis testing (HPSR/ EPA 520/1-80-012, 1980;

NUREG/CR-4007, 1984; Currie, 1968). In this situation, there is a measurement error, often

taken to be the Poisson counting error, a, equal to the square root of the number of counts. There

is a background counting rate, and any additional radioactivity in a sample must be

distinguishable above that. Generally it is assumed that the number of counts is sufficiently large

so that a normal approximation to the Poisson distribution of counts is appropriate.

) For Scenario A, the analogy would have to be restructured for the problem of deciding whether a given

sample, assumed to contain added radioactivity exceeding L D actually contained a smaller amount. In essence, the

null and alternative hypotheses would be reversed.

2-11 NUREG-1505
OVERVIEW

2.4.1 Detection Limits

For the calculation of detection limits, the hypotheses are:

Null Hypothesis:

H0: The sample contains no radioactivity above background.

and

Alternative Hypothesis:

Ha: The sample contains added radioactivity.

The count obtained from the sample measurement is the test statistic, and it has a different

probability distribhtion under the null and alternative hypothesis (see Figure 2.1). If a sample that

contains no radioactivity above background is declared to contain radioactivity above

background, a Type I error is made. Conversely, if a sample that contains radioactivity above

background is declared to contain no radioactivity above background, a Type II error is made.

The Type I error rate, a, depends on the variability of background, i.e., it is controlled by

requiring that the net counts exceed a certain multiple of the measurement standard deviation.

Under the null hypothesis, namely when there is no radioactivity above background, the net

counts have mean B - B = 0.

The standard deviation of the net count is

oB-B + B = o2+2 = Vr2 (2-1)

where B is the background count, and a = vR-is its standard deviation, since for a Poisson

distribution the standaid deviation is the square root of the mean. Unless the mean number of

counts is very low, a normal distribution with the same mean and standard deviation can be used

to approximate the Poisson distribution of the background counts. This determines the critical

level, Lc. If a net count above the critical detection level is obtained, the null hypothesis is

rejected. That is, the decision is made that the sample being measured contains radioactivity

above background.

LC = zx_- oBB = z, _, r2a (2-2)

z_,, is the 1-a percentile of a standard normal distribution, e.g. if a = 0.05, then 1 - ia = 0.95 and

z,_-, = 1.645. Note that the distribution of background counts (lefthand curve in Figure 2.1) is

used for this calculation.

The Type II error rate, P3, depends on the variability of the added radioactivity and is controlled

by requiring that the net counts exceed a certain number of standard deviations above the critical

level.

LD = Lc+ z20p o = + Zl_. aL za ry/'2 + ZlP [L+ 202 (2-3)

NUREG-1505 2-12
OVERVIEW

since

aLD = V(LD+B)+ = ýL+2o2

B = Background

Distribution Lc = Critical level

of ofintrcuntsn net countsPrbbltofTpIerr Lt = Detection limit

under null = Probability of Type I error

hypothesis 13 = Probability of Type II error

Distribution of

net counts

under alternative

* hypothesis

Lc LD

Figure 2.1 Type I and Type II Errors in the Determination of a Detection Limit

The distribution of counts under the alternative hypothesis (right hand curve in Figure 2.1) is

used to derive Equation 2-3. If the probability of a Type II error is set the same as the probability

of a Type I error, then zl.,, = Zj.p = k. Solving Equation 2-3 for LD, the count detection limit is

found to be

LD = k 2 + 2k V/2 a = k2 + 2 Lc (2-4)

The power, 1- P3, is the probability that the measurement will indicate the presence of additional

radioactivity in the sample, when the sample actually contains additional activity in the amount

necessary to produce an average of LD counts above background during the measurement.

2-13 NUREG-1505
OVERVIEW

2.4.2 Final Status Surveys

The statistical procedures described in this report for final status surveys have many similarities

to the detection limit calculation. Corresponding to Figure 2.1, the relationship between the

hypothesis, decision error rates and measurement distributions in Scenario A and Scenario B are

shown in Figures 2.2 and 2.3, respectively.

Scenario A LBGR = Lower Boundary of Gray Region

C = Critical Value

DCGL = Release Criterion

Distribution a = Probability of Type I error

of measurement p = Probability of Type II error

under

alternative

hypothesis

Distribution

of measurements

under

null hypothesis

LBGR C DCGL

Figure 2.2 Type I and Type II Errors for Scenario A

Some other points of similarity are:

(1) The null hypothesis is:

H0 : The sample contains no radioactivity above background.

becomes either

Ho: The survey unit does not meet the release criterion (Scenario A).

or

H0: The survey unit meets the release criterion (Scenario B).

NUREG-1505 2-14
OVERVIEW

(2) The alternative hypothesis is:

Ha: The sample contains added radioactivity above the detection limit.

becomes either

H.: The survey unit meets the release criterion (Scenario A).

or

H.: The survey unit does not meet the release criterion (Scenario B).

Sc

Di

mL

un

nu

,enario B LBGR Lower Boundary of Gray Region

C Critical Value

DCGL = Release Criterion

stribution of = Probability of Type I error

easurements p = Probability of Type II error

ider

9Il hypothesis Distribution of

measurements

under alternative

hypothesis

/ t I

LBGR C DCGL"

Figure 2.3 Type I and Type II Errors for Scenario B

(3) The Type I error rate is computed using the distribution of counts assuming the null

hypothesis is true. Similarly, the Type I error rates for the tests described in this report will be

calculated using the distribution of the measurements under the null hypothesis.

(4) The Type II error rate is computed using the distribution of counts assuming the alternative

hypothesis is true. Similarly, the Type II error rates for the tests described in this report will

2-15 NUREG-1505
OVERVIEW

be calculated using the distribution of the measurements under the alternative hypothesis.

This also gives the power of the tests.

(5) The variability of the count obtained from the sample, a, plays a crucial role in determining

the value of the detection limit. Similarly, the variability of the radioactivity measurements in

the reference areas and survey units plays a crucial role in how well the tests described in this

report will perform.

(6) Corresponding to the detection limit, Lo, is a critical level of counts, Lc. Any sample

producing more than the critical level of counts is assumed to contain additional

radioactivity. Thus, the decision whether or not to reject the null hypothesis is based on

comparing the counts actually obtained from the sample to the critical detection level.

Similarly, the decision whether or not to reject the null hypothesis for a survey unit is based

on the critical level of a test statistic which is computed from the measurement data. Note

that while Lc and LD are expressed in counts, there is a corresponding concentration level in

the sample being measured that will, on average, give rise to that number of counts.

(7) The critical level of counts, LC is calculated so that the decision to reject the null hypothesis is

made with probability ac when the true concentration in the sample being measured is zero.

The critical value of the final status survey test statistic is calculated so that the decision to

reject the null hypothesis in Scenario B is made with probability a when the true

concentration is equal to a certain value called the LBGR (Lower Boundary of the Gray

Region). The LBGR is a concentration value between zero and the DCGLw at which

probability of the survey unit incorrectly failing the final status survey is specified. The

LBGR is discussed further in Section 3.7.

(8) The detection limit can usually be made lower by counting for a longer time, thereby reducing

the relative measurement error, at additional cost. Similarly, the ability of the tests described

in this report to distinguish smaller amounts of residual radioactivity from background more

accurately can be improved by taking a greater number of samples, at additional cost.

(9) U~sually, a detection limit is calculated given the Type I and Type II error rates and the

background variability. However, if a certain detection limit is pre-specified instead, the

procedure given above shows how to relate it to the Type I and Type II error rates, and the

measurement variability- Similarly, the procedures of this report will show the

interrelationship of the decommissioning criteria (dose above background), the Type I and

Type II error rates, and the measurement variability.

2.4.3 The Effect of Measurement Variability on the Decisionmaking Process

Figure 2.4 further illustrates the affect of the measurement standard deviation on the decision

process. Shown are three hypothetical measurement distributions, with true mean concentration

equal to zero, one, and three times the measurement standard deviation, a. Assume for simplicity

there is no background to subtract. Then the critical level, Lc = z1_ = 1.645owhen a = 0.05.

Thus, there is a 5% chance of a positive result when the true concentration is actually zero. If the

true concentration is 3a, the probability of a positive result is very high since most of the

distribution lies above Lc (91% using the normal distribution table with z = 3-1.645 = 1.355).

NUREG-1505 2-16
OVERVIEW

However, if the true concentration is 1 a, then there is less than a 50% chance of a positive result

(26% using the normal distribution table with z = 1-1.645 = -0.645). If a true mean

concentration of C = l must be measured, then the uncertainty must be reduced by taking more

measurements. If nine measurements are averaged, then the standard deviation of the mean, o*,

falls by a factor of three (one over the square root of the number of measurements). In the "new

standard deviation units" C = 1 o = 30*. Thus, a difference of 1 a can be distinguished with nine

measurements as easily as a difference of 3u can be distinguished with one measurement.

3'o

0.

LOL

-3 -2 -1 0 1 2 3 4 5 6

Concentration (in units of measurement standard deviation)

Figure 2.4 Differences in Concentration Compared to Measurement Variability

2.5 Statistical Tests

There are two important uses of the statistical tests described in this report. The first is in the

analysis of the final status survey data to demonstrate compliance with the release criterion.

However, the second, and perhaps more important use, is in the design of the final status survey.

In some cases it may be clear from the data, Without any formal analysis, whether or not a survey

unit meets the decommissioning criteria. Provided that an adequate number of measurements are

made (either in situ or from samples), Table 2.3 can be used to determine whether or not a formal

statistical test is necessary.

2-17 NUREG-1505
OVERVIEW

It is usually not obvious, a priori, what number of samples is necessary in order to show whether

or not a survey unit meets the decommissioning criteria with acceptable rates of error. The DQO

process described in Chapter 3 provides a general method for designing surveys so that accurate

remediation decisions can be made cost effectively. Simple estimates of the number of samples

required for the statistical tests may be made using the mathematical relationships between the

error rates, residual radioactivity levels, and measurement variability.

Table 2.3 Summary of Statistical Tests

Radionuclide not in background and radionuclide-specific measurements made

Survey Result Conclusion

All measurements below DCGLw Survey unit meets release criterion

Average above DCGLw Survey unit does not meet release criterion'

Otherwise (some measurements above DCGLw, Conduct Sign test and elevated

but average below DCGLw) measurement comparison

Radionuclide in background or non-radionuclide-specific measurements made

Survey Result Conclusion

Difference between maximum survey unit Survey unit meets release criterion

measurement and minimum reference area

measurements is below DCGLw

Difference of survey unit average and reference Survey unit does not meet release criterion

area average is above DCGLw

Otherwise (Maximum difference above DCGLw, Conduct WRS test and elevated

but average difference below DCGLw) measurement comparison

-2.5.1 Nonparametric Statistical Tests

Many statistical tests can be used for determining whether or not a survey unit meets the release

criteria. Any one test may perform better or worse than others, depending on the hypotheses to be

tested, i.e., the decision that is to be made and the alternative, and how well the assumptions of

the test fit the situation.

The basic distinction between parametric and nonparametric statistical techniques is that

parametric techniques use specific assumptions about the probability distributions of the

measurement data. The most commonly made assumption is that the data fit a normal.

distribution. Such is the case when the Student's t statistic is used. Additional data and statistical

tests would generally be necessary in order to show that the assumption of normality is justified

(EPA QA/G-9), 1995.

NUREG-1505 2-18
OVERVIEW

Nonparametric techniques (sometimes referred to as distribution-free statistical methods) can be

used'without assuming a particular underlying distribution. Thus, nonparametric techniques are

often more appropriate in situations when the probability distribution of the data is either

unknown or is some continuous distribution other than the normal distribution. That a statistical

approach is nonparametric or distribution free does not imply that it is free of any and all

assumptions about the data distribution. Most nonparametric procedures require that measured

values be independent and identically distributed. Some nonparametric procedures assume that

the underlying probability distribution is symmetric. However, these requirements are usually

less restrictive than the assumption that the data follow a particular symmetric distribution, such

as the normal distribution.

Parametric methods rely on the assumptions about the data distribution to infer how large the

difference between two measurements is expected to be. These methods are better only if the

assumptions are true. Many nonparametric techniques are based on ranking the measurement

data. The data are ordered from smallest to largest, and assigned numbers (ranks) 1, 2, 3,...

accordingly. The analysis is then performed on the ranks rather than on the original measurement values. The advantage of this approach is that the probability that one measurement is

larger (i.e. ranked higher) than another can be computed exactly by combinatorial (enumeration

and counting) methods without reference to a specific probability distribution.

If the underlying distribution is known, a parametric test can make use of that additional

information. If the underlying distribution is different than that assumed, however, the results can

be unpredictable. The nonparametric methods described in this report have been found to

perform nearly as well as the corresponding parametric tests, even when the conditions necessary

for applying the parametric tests are fulfilled. There is often relatively little to be gained in

efficiency from using a specific parametric procedure, but potentially much to be lost. Thus, it

may be considered prudent to use nonparametric methods in all cases.

For survey measurements at or near background, there may be some measurement data which are

at or below instrumental detection limits2 These data-are sometimes reported as "less than" or

"non-detects". Such data are not easily treated using parametric methods. It is recommended that

the actual numerical results of measurements always be reported, even if these are negative or

below calculated detection limits. However, if it is necessary to analyze data which include nonnumerical results, nonparametric procedures based on ranks can still be used in many cases.

This is an additional advantage to the use of these methods.

2.5.2 Wilcoxon Rank Sum and Sign Tests

The Wilcoxon Rank Sum (WRS) test and Sign test are used in this report to detect a uniform

shift in the mean of a distribution of measurements. Without assuming symmetry in the

measurement distribution, these tests are technically for the median. However, computer

simulations have shown that these tests generally produce the correct decisions more often when

the assumption of symmetry is violated than the commonly used Student's t-test, which assumes

normality in addition to symmetry. Nevertheless, extremes of asymmetry are guarded against by

conducting the elevated measurement comparison in addition to the WRS and Sign tests. This

issue is discussed further in the next section.

2-19 NUREG-1505
OVERVIEW

The WRS test is a two-sample test that compares the distribution of a set of measurements in a

survey unit to that of a set of measurements in a reference area. The Sign test is a one-sample test

that compares the distribution of a set of measurements in a survey unit to a fixed value, namely

the derived concentration limit for a specific radionuclide.

The WRS test, also known as the Mann-Whitney test, is performed for Scenario A by first adding

the value of the DCGL to each of the reference area measurements, and then listing the combined

set of survey unit and adjusted reference area measurements in increasing nrimerical order from

smallest to largest. The next step is to replace the measurement values by their ranks, i.e., their

position number in the ordered list. Thus, the ranks are simply integer values from 1 through N,

where N is the total number of combined measurements. The rank 1 is assigned to the smallest

value, 2 to the second smallest observation, etc. Then, the sum of the ranks of the survey unit

and the sum of the ranks of the adjusted reference area measurements is computed. Because the

sum of the combined ranks is a fixed constant equal to N(N+ 1)/2, the sum of the adjusted

reference area measurement ranks is equal to N(N+1)/2 minus the sum of the ranks of the survey

unit measurements.

If the level of residual radioactivity in the survey unit is exactly at the DCGL, then any given

rank is equally likely to belong to either an adjusted reference area measurement or a survey unit

measurement. If the survey unit has residual radioactivity less than the DCGL, the survey unit

site ranks will tend to be smaller than the adjusted reference area ranks. The larger the average of

the ranks of the adjusted reference area measurements relative to the ranks of the survey unit

measurements, the smaller the probability that it is by chance, and the greater the evidence that

the residual radioactivity in the survey unit is actually below the DCGL. If the sum of the ranks

of the adjusted reference area measurements exceeds a calculated critical value, the decision is

made to reject the null hypothesis and to conclude that the survey unit actually meets the release

criterion. In some cases, the result will be obvious without any computations. If, for example, all

of the survey unit measurements are less than the smallest of the adjusted reference area

measurements, then the sum of the ranks of the adjusted reference area measurements is at its

maximum possible value, and the null hypothesis will always be rejected.

For Scenario B, the WRS is performed by first subtracting the value of the LBGR from each of

the survey unit measurements, and then listing the combined set of adjusted survey unit and

reference area measurements in increasing numerical order from smallest to largest and finding

their ranks. As above, the sum of the ranks of the survey unit measurements and the sum of the

ranks of the unadjusted reference area measurements is computed. Again, the sum of the

reference area measurement ranks plus the sum of the ranks of the adjusted survey unit

measurements is equal to N(N+l)/2. If the level of residual radioactivity in the survey unit is

exactly at the LBGR, then any given rank is equally likely to belong to either a reference area

measurement or a survey unit measurement. Thus, there is no reason to believe that the average

of the survey unit ranks will differ greatly from the average of the reference area ranks. With

residual radioactivity at the LBGR, the probability that the average of the survey unit ranks will

be larger than the average of the reference area ranks is 50% by random chance. However, the

larger the average of the survey unit ranks, the smaller the probability that it is by chance, and the

greater the evidence that the survey unit is contaminated. If the sum of the survey unit ranks

exceeds a calculated critical value, one can decide that the evidence shows that the residual

radioactivity in the survey unit exceeds the LBGR.

NUREG-1505 2-20
OVERVIEW

The one-sample Sign test is performed for Scenario A by first subtracting each survey unit

measurement from the derived concentration limit. Then the number of positive differences is

counted. Large numbers of positive differences are evidence that the survey unit measurements

do not exceed the derived concentration guideline.

The Sign test is performed for Scenario B by first subtracting the LBGR from each survey unit

measurement. Then the number of positive differences is counted. Large numbers of positive

differences are evidence that the survey unit measurements exceed the LBGR.

The Sign test uses no assumptions about the shape of the data distribution. An alternative test,

the Wilcoxon Signed Rank (WSR) test, assumes that the measurement distribution is symmetric.

When this assumption is valid, the WSR test generally has greater power than the Sign test. The

WSR and other alternative statistical tests are discussed in Section 14.1.

2.5.3 Mean and Median

The WRS and Sign tests are designed to determine whether or not a degree of residual

radioactivity remains uniformly throughout the survey unit. Since these methods are based on

ranks, the results are generally expressed in terms of the median. When the underlying

measurement distribution is symmetric, the mean is equal to the median. The assumption of

symmetry is less restrictive than that of normality, since the normal distribution is itself

symmetric. If, however, the measurement distribution is skewed to the right, the average will

generally be greater than the median. In severe cases, it may happen that the average exceeds the

DCGLw while the median does not. This is why the average is used to screen the data set before

any statistical test is performed (see Table 2.3).

Figure 2.5 illustrates the potential differences between the median and the mean. The normal and

lognormal distributions are commonly used examples of symmetric and skewed distributions,

respectively. In this figure, the probability densities all have arithmetic mean equal to one. The

coefficient of variation (arithmetic standard deviation divided by the mean) varies from 0.1 to

1.0. For values of the coefficient of variation larger than about 0.3, the lognormal begins to

diverge significantly from the normal. When the coefficient of variation is 1.0, the difference

between the median and the mean is large.

When the underlying data distribution is highly skewed, it is often because there are a few high

measurements. Since the elevated measurement comparison is used to detect such measurements,

the difference between using the median and the mean as a measure for the degree to which

uniform residual radioactivity remains in a survey unit tends to diminish in importance. This is

especially true in Scenario A, where the null hypothesis is that the survey unit does not meet the

release criterion.

In Scenario B, the null hypothesis is that the survey unit meets the release criterion. If the

measurement distribution is highly skewed, there may a substantial portion of the survey unit

with residual radioactivity higher than the DCGLw, but perhaps not in excess of the DCGLEMC.

In such cases, the median may be below the LBGR, while the mean is above it. The Quantile test,

discussed in Section 2.5.5, can be used to detect when remediation activities have failed in only a

few areas within a survey unit. Conducting the Quantile test in tandem with the WRS has been

2-21 NUREG-1505
OVERVIEW

found to improve the accuracy of the tests (EPA 230-R-94-004, 1994).

In some cases, data from an asymmetric distribution can be transformed so that the transformed

data have a more symmetric distribution. The analysis is then performed using the transformed

data. A common example is that the logarithms of lognormally distributed data have a normal

distribution. However, such transformations introduce additional complications. The mean of the

transformed data is not generally equal to the transform of the mean of the original data. For

example, the mean of the logs transforms back to the geometric mean, which is the median of a

lognormal data set. The computations necessary for testing the average of lognormal data can be

complex (see Chapter 14). The behavior of this lognormal test when the assumption of

lognormality is violated is not known.

4

Lognormal (1.0, O. 1)-ý)

Median = 0.995

3 ------- Normal (1.0, 0.1)

2 Lognormal (1.0, 0.3). '

Median =0.96

1L Lonra (1.0 1.0 Normal (1.0, 0.3)

\\ Normal (1.0, 1.0)

...... ......... ................ :.....

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Concentration

Figure 2.5 Comparison of Normal and Lognormal Distributions

(Arithmetic mean and arithmetic standard deviation shown in parentheses)

The EPA (EPA QA/G-4, 1994) has compared the use of the mean and the median as the

parameter of interest whose true value the decision maker would like know and that the data

will estimate. Some of the positive and negative attributes of each are listed below:

NUREG-1505 2-22
OVERVIEW

MEAN

Positive Attributes

* Useful when action level is based on long-term, average health effects

* Useful when the population is uniform with relatively small spread.

" Generally requires fewer samples than other parameters.

Negative Attributes

* Not a very representative measure of central tendency for highly skewed populations.

* Not useful when the population contains a large proportion of values that are less than

measurement detection limits.

MEDIAN

Positive Attributes

" Useful when action level is based on long-term, average health effects

* More representative measure of central tendency than the mean for skewed populations.

" Useful when the population contains a large number of values that are less than measurement

detection limits.

* Relies on few statistical assumptions.

Negative Attributes

" Will not protect against the effect of extreme values.

* Not a very representative measure of central tendency for highly skewed populations.

2.5.4 Estimating the Amount of Residual Radioactivity

The result of the statistical test is the decision whether or not to reject the null hypothesis.

Following the statistical hypothesis tests, it may also be necessary to estimate the amount of

residual radioactivity in the survey unit so that dose calculations can be made. It is usually best to

use the mean (average) residual radioactivity for this purpose (EPA PB92-963373, 1992). If the

data distribution is symmetric, the mean is equal to the median. If, however, the data distribution

is skewed, the mean may be greater than the median.

2.5.5 iQuantile Test

In contrast to locations with concentrations above the DCGLEMO more moderate departures from

uniformity in residual radioactivity concentrations may also exist within a survey unit. One

portion of a survey unit may have virtually no residual radioactivity, while another portion does

contain some residual radioactivity. There may be several portions of one type or another in a

survey unit, resulting in a patchy contamination pattern. The Quantile test is designed to detect

this type of residual radioactivity. The Quantile test is only needed in Scenario B.

Like the WRS test, the Quantile test (EPA 230-R-94-004, 1994; Johnson et al., 1987) is a twosample test. It is also performed by first subtracting the value of the LBGR from each of the

survey unit measurements, and then listing the combined set of adjusted survey unit and

reference area measurements from smallest to largest. However, only the largest measurements

in the list are examined. The number of measurements that will be considered in the Quantile

test is denoted by r. A count is made of the number of measurements among the largest r

measurements that are from the survey unit. This number is denoted by k. If there is no

contamination, measurements from the reference area and from the survey unit would be

2-23 NUREG-1505
OVERVIEW

expected to appear among the r largest measurements roughly in proportion to the number of

measurements made in each of them. If patchy residual contamination exists, then the r largest

measurements of the combined data sets (reference area and survey unit) are more likely to come

from the survey unit. When there are m background measurements and n survey site measurements, then k should be about r times nI(m+n). If the number of measurements from the survey

unit among the largest r is too much larger than this, then there is evidence that the survey unit

contains patchy areas of residual radioactivity in excess of the LBGR.

While it is possible to perform a one-sample version of the Quantile test, it will seldom be

necessary in practice. With no interfering background, patchy areas of contamination in excess of

the LBGR will be apparent in simple posting plots and histograms (see Chapter 4) of the survey

unit data.

2.5.6 Elevated Measurements Comparison

An elevated measurement comparison is performed by comparing each measurement from the

survey unit to the DCGLEMC. If the survey unit is being compared to a reference area, the net

survey unit measurement is first obtained by subtracting the mean of the reference area

measurements. A net survey unit measurement that equals or exceeds the DCGLEMCis an

indication that a survey unit may contain residual radioactivity in excess of the release criterion.

This type of measurement comparison is sometimes called a "hot spot test." The latter term may

be misleading because it is-not a formal statistical test, but a simple comparison of measured

values against a limit. In addition, there is no commonly accepted definition of what constitutes

a hot spot in either area or magnitude of residual radioactivity. Yet, this term may imply some

degree of radiological hazard. In this report, the term "area of elevated residual radioactivity" is

used to describe a limited area of residual activity that may cause the decommissioning dose

criteria to be exceeded. It is only these areas that might be considered hot spots. For planning

purposes, the potential extent of an "area of elevated residual radioactivity" is based on the

distance between sampling points in the survey sampling grid.

In addition to direct measurements or samples at discrete locations, parts of each survey unit will

also be scanned. For the quantitative measurements obtained at discrete locations, performing the

EMC is a straightforward comparison of two numerical values. Some sophisticated scanning

instrumentation is also capable providing quantitative results with a quality approaching those

from direct measurements or samples. Other scanning measurements, however, may be more

qualitative. In that case, action levels should be established for the scanning procedure so that

areas with concentrations that may exceed the DCGLEMc are marked for a quantitative

measurement.

A single unusually large measurement may occur by chance. The elevated measurement

comparison flags these measurements for further study. When a measurement is flagged, it

should first be determined that it is not due to sampling or analysis error. Such a determination

may include re-sampling the area at which the measurement was originally taken.

If an nieasurement exceeding the is DCGLEMC is confirmed, then the size of the area of elevated

residual radioactivity, A', and the average concentration within it, CA,, is determined. This will

NUREG-1505 2-24
OVERVIEW

generally involve taking further measurements in the vicinity of the elevated measurement to

adequately delimit its extent. Using the area factor FA, for the area A', CA, should not exceed the

product (FA,)(DCGLIv) in order for the survey unit to meet the release criterion.

2.5.7 Investigation Levels

In contrast to an elevated measurement, a measurement may be found that exceeds the

concentration level expected from the survey unit's classification. Investigation levels

are established for each class of survey unit to guard against the possible mis-classification of

survey units. If a measurement exceeds the investigation level, additional investigation is

required to determine if the final status survey for the survey unit was adequate to determine

compliance with the release criteria.

For example, in a Class 1 survey unit, measurements above the DCGLw may not be unusual or

unexpected. In Class 2 areas, however, neither measurements above the DCGLw nor elevated

areas are expected. Thus in these areas, any measurement at a discrete location exceeding the

DCGLw should be flagged for further investigation. Unless the scanning sensitivity is such that

an action level can be specified for areas with concentrations potentially exceeding the DCGLw,

any positive indication of residual radioactivity during the scan could warrant further

investigation.

Because there is a low expectation of any residual radioactivity in a Class 3 area, it may be

prudent to investigate any measurement exceeding even a fraction of the DCGL. What level

that should be will depend on the site, the radionuclides of concern, and the measurement and

scanning methods chosen. This level should be set using the DQO Process during the survey

design phase of the Data Life Cycle. In some cases it may be prudent to follow this procedure for

Class 2 survey units as well.

Suggested investigation levels that might be appropriate for each class of survey unit and type of

measurement are shown in Table 2.4.

Table 2.4 Summary of Investigation Levels

Survey Unit Flag Direct Measurement or Sample Scanned Area Marked When

Classification I I Action Levels Indicate:

Class I > DCGLEMc. >DCGLEMC

Class 2 > DCGLw >DCGLw

Class 3 > fraction of DCGLw >MDC

In the last two sections, we have considered elevated measurements that require investigation for

compliance with the dose criteria, and investigation levels that flag potential survey unit misclassifications. In addition to these, are the QA/QC procedures that should be in place in any

2-25 NUREG-1505
OVERVIEW

measurement program. Gross errors of analysis can often be spotted by the use of simple

preliminary data analysis techniques, such as posting plots and histograms. This is sometimes

called exploratory data analysis. These techniques also form a part of the Data Quality

Assessment process (EPA QAIG-9, 1995). Some of these methods are discussed in Chapter 4.

NUREG-1505. 2-26
3 PLANNING FINAL STATUS SURVEYS:

DATA QUALITY OBJECTIVES

An essential consideration in designing survey plans for site decommissioning is that the

radiological data that are collected and analyzed are sufficient and of adequate quality for

decision-making purposes. It is imperative that the type and quality of radiological data that

will be needed to support license termination be considered early in the decommissioning

process.

Before commencement of survey work, it is essential that a survey plan be developed that is

based on the data needed for decision making and the level of quality needed to support the

decision. Such a plan should specify what samples need to be obtained, how and where they will

be collected and analyzed, what quality assurance procedures will be used, the method of

comparing site areas to reference areas, and what level of decision errors will be considered

acceptable. These decisions become paramount for determining compliance with

decommissioning criteria that are near background levels.

3.1 Introduction

The Data Quality Objectives (DQO) process is a series of planning steps based on the scientific

method that is designed to ensure that the type, quantity, and quality of environmental data used

in decision making are appropriate for the intended application (EPA QA/G-4, 1994). DQOs are

qualitative and quantitative statements that

" clarify the study objective

* define the most appropriate data to collect

" determine the most appropriate conditions for collecting the data and

* specify acceptable levels of decision errors that will be used as the basis for establishing

the quantity and quality of data needed to support the decision.

The DQO process comprises the following steps:

(1) State the problem, i.e., the objective of the sampling effort.

(2) Identify the decision, i.e., the decision to be made that requires new data

(3) Identify inputs to the decision, i.e., the data that are needed and how they will be used to

support the decision.

(4) Define the study boundaries, i.e., the spatial and temporal aspects of the environmental

media that the data represent.

(5) Develop a decision rule, i.e., an "if. .. then" statement that defines the conditions for choice

among alternative actions.

(6) Specify limits on decision errors.

(7) Optimize the design for obtaining data, i.e., the most time- and resource-effective

sampling and analysis plan.

The DQO process is iterative, so that any and all of the specifications may change as new

information is obtained during the course of site remediation, up until the final status survey is

3-1 3-1 NUREG- 1505
PLANNING

actually performed.

It is important to specify the type and quality of radiological data that will be needed for final

status surveys early in the decommissioning process. This process entails early specification of

sample collection and analysis procedures, the determination of DCGLs, the classification of

survey units, the method of comparing survey units to reference areas, the null and alternative

hypotheses, Type I and Type II error rates, and quality assurance procedures.

In the following sections, each of the seven steps in the DQO process is discussed as it pertains to

the planning, design, and performance of the final status survey.

3.2 State the Problem

For most NRC licensees, the objective of the decommissioning process is to remove their

facilities safely from service and reduce residual radioactivity to a level that permits release of

the property and termination of the license. The data that will be needed to support this objective

will demonstrate that any residual radioactivity remaining on the site results in a dose that does

not exceed the release criterion. This objective will be met by performing a final status survey in

individual survey units. For each survey unit, a separate decision will be made on the attainment

of the release criterion.

The final status survey occurs near the end of the decommissioning process, following historical

site assessment, scoping, characterization, and remediation. These earlier steps in the

decommissioning process provide crucial information for the design of the final status surveys.

Thisinformation includes the identification of potential residual radioactive materials, the

general locations and extent of residual radioactivity, and estimates of the concentration levels

and its variability. Some of this information may be part of the licensee's decommissioning plan.

3.3 Identify the Decision

For the final status survey, the essential decision is whether the decommissioning criteria have

been met in individual survey units. The decommissioning criterion is expressed in terms of a

total effective dose equivalent (TEDE) limit above background due to residual radioactivity. The

decision will be based on radiological data collected in a survey designed for this purpose.

Procedures for the design of the final status survey and for the statistical analysis of the results

are the primary focus of this report.

An essential part of identifying the decision, is a knowledge of the applicable residual

radioactivity concentration limits. These are the Derived Concentration Guideline Levels

(DCGLs) discussed in Section 2.2.1. NRC has developed models to provide generic dose

conversion factors for residual radioactivity that can be applied within a hierarchy of modeling

approaches. The models provide a mechanism for translating the residual radioactivity at a site

into TEDE using the site-specific source term and varying levels of related site information. The

provisions of 10 CFR Part 20, Subpart E require that a licensee consider the entire applicable

source term and all credible dose pathways when determining whether any residual radioactivity

meets the decommissioning criteria.

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Since the dosimetric models are used to define the TEDE release criterion in terms of a DCGL,

careful consideration should be to the assumptions made in those models. Screening models are

generally the easiest to use. These models are constructed to cover a wide range of possible

conditions, and so are also generally the most conservative. Their use may result in very low

DCGLs. Using site specific parameters in such a model can reduce this conservatism

considerably, but will require some justification. A balance should be sought between the

complexity of site specific modeling and the potential cost in remediation and surveys of using

DCGLs that are overly conservative.

For sites which contain residual radioactivity distinguishable from background from more than

one radionuclide, there are two methods that can be used. If the concentrations of the

radionuclides within a survey unit are related, one radionuclide can be used as a surrogate for the

others using a modified DCGLW. Otherwise, the TEDE due to the mixture of radionuclides is

compared to the release criterion by applying a mixture rule. This is done by determining the

ratio between the concentration of each radionuclide in the mixture and the DCGLwfor that

radionuclide alone. The sum of the ratios for all radionuclides in the mixture should not exceed

one. The case of multiple radionuclides is discussed further in Chapter 11.

3.4 Identify Inputs to the Decision

Although the final status survey is performed near the end of the decommissioning process, it is

possible to produce a more efficient survey design if the requirements of this survey are

identified early in the decommissioning planning. By knowing in advance the type, quantity, and

quality of data that are needed in the final status survey, information obtained from earlier

decommissioning surveys may be used to support the final status survey.

Previous steps in the DQO process have identified the critical radionuclides, and established their

corresponding concentration or surface activity limits (DCGLs) for various post-remediation land

use scenarios. In subsequent steps, acceptable limits on decision errors, and the number of

measurements necessary to meet them, will be established. To accomplish this, an estimate of the

expected variability of the measurement data will be needed. Information from scoping,

characterization, and remediation control surveys can be very useful for estimating the mean and

standard deviation expected for residual radioactivity in a survey unit and for background

radioactivity in one or more reference areas. In the absence of such data, experience and

scientific judgment can be used to estimate the expected measurement variability or a separate

scoping survey may be conducted. The effort required for an adequate estimate of the expected

measurement variability will depend on its magnitude relative to the DCGLs. The smaller the

value of the DCGL relative to the expected measurement standard deviation, the more important

it will be to have an accurate estimate of that standard deviation. Thus, surveys performed earlier

in the decommissioning process can provide valuable information for designing the final status

survey. As more information comes available, both the measurement and statistical methods that

will be needed to meet release criteria can .be refined.

The selection and proper use of appropriate instruments and techniques will be critical factors in

assuring that the survey accurately determines the radiological status of the site. In this report,

three basic types of measurements are considered:

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(1) scanning

(2) direct field measurements

(3) laboratory analysis of samples.

Scanning is the process by which the surveyor moves a portable radiation detection instrument

over a surface (i.e., ground, wall, floor, equipment) to detect the presence of radiation. A scan is

performed to locate radiation anomalies that might indicate elevated areas of residual activity that

will require further investigation or action. If scan survey results exceed a scanning action level

determined on the basis of the potential contaminant and the detector and survey parameters, the

location is noted for further action (direct measurement or sampling).

Direct field measurements are those made at a fixed location using portable instruments (e.g.,

survey meter, pressurized ionization chamber (PIC), in situ spectrometer). The result of a direct

measurement, as opposed to a scan, is a quantitative measure of the radioactivity present at the

location measured.

Taking samples, with subsequent analyses conducted in a laboratory, will be required for certain

radionuclides and radiations that cannot be adequately detected using direct measurements. For

some nuclides or environmental media, this may be the only realistic technique to employ.

The analysis techniques used may be radionuclide specific or for total radioactivity.

The survey designs with which these measurements are made fall into two categories:

(1) authoritative (judgment) sampling

(2) probability sampling

Authoritative or judgment sampling occurs when measurements are made or samples are

collected at locations where anomalous radiation levels are observed or suspected. The term

"biased sampling" is sometimes used to indicate that the sample locations are not chosen on a

random or systematic basis. Biased radiological measurements and samples also may be taken to

further define the areal extent of potential contamination and to determine maximum radiation

levels within an area.

When data quality objectives involve statistical estimation or hypothesis testing, some form of

probability sampling is required. The type of probability sampling recommended for use in final

status surveys is either simple random sampling (for Class 3) or systematic sampling on a

systematic grid with a random start (for Class 1 and Class 2).

Of the three measurement types, only the results of direct measurements and sampling are used in

conducting the nonparametric statistical tests. All three types of measurement result are subject

to an elevated measurement comparison against an upper limit value.

The type of instrumentation or sampling and analysis methodologies or both used for final status

surveys will influence the number of samples or direct measurements, or both, that are required

for the appropriate statistical analysis of the data. As a rule, the less precise the measurement, the

greater the number of measurements that will be required for the statistical tests to achieve the

NUREG-1505 3-4
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desired level of uncertainty. The selection of survey instruments may involve a cost analysis of

whether it is better to use a more precise (and more expensive) measurement method with

correspondingly fewer measurements, or to use a less precise (and perhaps less costly) method

that would require the collection of more measurements. The information necessary to calculate

the required number of samples, given the expected variability of the data, is discussed in

discussed in Section 3.7.

Similar considerations are involved in the choice of making radionuclide-specific measurements

versus total alpha, beta, or gamma activity or total exposure rate measurements or both. If total

(gross) methods are used, the results will include the variability of natural background. This

additional variability will not only require more measurements to overcome but will also

necessitate comparison with a reference area using the two-sample Wilcoxon Rank Sum test of

Chapter 6 rather than the one-sample Sign test of Chapter 5.

If the radionuclide of concern appears as part of background, there is no alternative to a survey

unit comparison to a reference area; however, the measurement precision will still affect the

number of samples required. Radionuclide-specific methods should be considered in this case as

well, since the variability of the total activity present will be greater than that due to any

particular radionuclide or series alone.

Instrumentation can be selected using guidelines that compare its performance capabilities to the

applicable decommissioning criteria. Consideration should be given to the characteristics of the

type of detector, in particular, the minimum detectable concentration (MDC) for the radionuclide

under investigation. The simplest of devices, survey meters, may be appropriate for hand

scanning of building surfaces for certain nuclides at certain activity levels. Fixed-place detectors

at grid points can be used in other situations. In some situations, the sensitivity needed at background levels will require that measurements be nuclide specific, thereby requiring spectrometric

techniques. Consideration should also be given to newer technologies as they are developed.

3.5 Define the Study Boundaries

Defining the spatial and temporal boundaries will help ensure that the samples taken in the

survey are representative of the survey unit for which the decommissioning decision will be

made. Spatial boundaries describe what measurements or samples should be taken and in what

areas. Temporal boundaries describe when the measurements or samples should be taken, and

any time constraints on the data collection and analysis. Uniformity over a given area should be

checked wherever possible. This can be done by inspecting the site and knowing its history from

data collected earlier in the decommissioning process, or by scanning measurements. The

selection of measurement and sampling points must ensure that the sample is representative of

the site category under investigation.

3.5.1 Spatial Variability

As has been discussed in Sections 2.2.6 and 2.2.7, some estimate of the variability of the data is

needed for a good survey design. The smaller the variability within each reference area or survey

unit, the smaller the number of samples that will be needed to achieve the specified Type I and

Type II error rates for the test. Thus, it is advantageous to identify survey units that are relatively

3-5 NUREG-1505
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homogeneous in radiological character. Reference areas and survey units should be as similar as

possible with regard to their background characteristics.

Considering the variability in collected data that is expected in any environmental sampling

program, accurate interpretation of the results is essential. The choice of individual survey units

and any reference areas to which they are to be compared is especially significant. In the analysis

of the data, any systematic difference in the measurements from a survey unit and a referenece

area is assumed to be due to residual radioactivity. The choice of a reference area is a spatial

extrapolation of the background radionuclide concentrations there to the survey unit. It would be

obviously inappropriate to compare uranium concentrations in soils collected from two sites of

different geology, such as a sandy beach area and an inland region with heavy clay soil. In the

case of the fallout radionuclide 'a"Cs, concentrations in surface soils could only be extrapolated

to other local plots of land that have received the same deposition (rainfall) and have the same

history (for example, plowed agricultural land, forest, or undisturbed lawn). For instance, the

presence of 137Cs in soil, and the observation that it is not atthe same level from place to place,

does not necessarily indicate a local facility contribution. Such variations may have resulted

from disturbance to the site through either natural or human action, which led to removal or

addition of material containing fallout from atmospheric nuclear weapons tests, as well as

differences in the spatial distribution of the original deposition.

In some situations involving radionuclides that appear as part of natural background, the

screening level DCGLs may be small compared to the spatial variations among even nearby and

closely matched reference areas and survey units. In such cases, an effort should be made to

reduce exposure pathway modeling conservatism by using site specific parameters and realistic

occupancy scenarios. In particularly difficult cases it may be necessary to explore alternative

statistical methods for establishing whether residual radioactivity in a survey unit is

distinguishable from background. An example of such an analysis is given in Chapter 13.

3.5.2 Temporal Variability

Temporal variability will contribute to the overall uncertainty of comparisons of survey units and

reference areas, although generally to a lesser extent than spatial variability. However, it is best

to avoid temporal variability to whatever extent possible. This might be accomplished by

collecting data from areas to be compared over as short a time interval as possible, and avoiding

circumstances known to cause short-term background variations. There may be reasons why

samples cannot be taken in certain places or at certain times. These constraints should be

identified so that they can be accounted for in the planning process.

3.5.3 Reference Coordinates

As'part of this step in the DQO process, a site diagram should be prepared showing each

potential survey unit and the reference area to which it will be compared. For each unit, the types

of samples that will be taken, the analyses needed, and a schedule for sampling and analysis

should be listed.

Reference coordinate systems are established at the site to facilitate selection of

measurement/sampling locations and provide a mechanism for referencing a measurement to a

NUREG-1505 3-6
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specific location so that the same survey point can be relocated.

A survey reference coordinate system consists of a grid of intersecting lines, referenced to a fixed

site location or benchmark. Typically, the lines are arranged in a perpendicular pattern, dividing

the survey location into squares or blocks of equal area. Reference coordinate system patterns on

horizontal surfaces may be identified numerically on one axis and alphabetically on the other axis

or-in distances in different compass directions from the grid origin. Interior walls are treated as

extensions of the floor along the horizontal plane. An example of a structure interior reference

coordinate system using letters and numbers-in shown in Figure 3.1. An example land area

reference coordinate system using distance from an origin along north-south and east-west lines

is shown in Figure 3.2.

14

13

12

:WALL:

I - U mi Gm. - . -

11

10

9

8

7

6

5

4

W ALL-------- ......

------ -- - -

-- - -- ,-,- - -

--- ,AL, -

--------------------------

-------------------------

-------------------------

VLOORICEIL

-------------

---------------------------------------

WALL

. .--. . .-. . . - ..- -- -..-- 

--- --- ---- ---

3

2 WALL:

2

A B C D E F G I J K L

FEET 0 , 6

0 2

METERS

Figure 3.1 Sample Indoor Reference Coordinate System.

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E Scale In meters 10 20

nrr

90. .

20 ......

10K80

0 10 20 30 40 50 so 70 s 90

Figure 3.2 Sample Outdoor Reference Coordinate System

One procedure for constructing a reference coordinate system is to d raw a map of the area to be

sampled and a rectangle that encloses it. Define a coordinate system for locating points (X,Y)

within the rectangle, e.g., the number of meters east, X, and the number of meters north, Y, from

the southwest corner (0,0) of the rectangle. The northeast corner will then have coordinates

(Xr, Yma,). Note that the local coordinate system need not line up with the principal compass

points. It may be convenient to align one of the axes with a site boundary or other local feature.

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For the example in Figure 3.2 the coordinate system has been laid out in the north-south and eastwest directions. There are 9 ten-meter east-west coordinates, and 11 ten-meter north-south

coordinates. The total area is 9,900 M2, of which approximately 9,000 in2 is the affected area

within the fence line. The soil area to be surveyed is about 4,500 n2

. The remainder of the area

is covered by buildings, walkways, etc., which will bepart of other survey units.

3.5.4 Sampling Grids

Sampling locations in Class 1 and Class 2 survey units are laid out on random start systematic

grids. The essential procedure for determining where samples should be taken in either reference

areas or survey units is the same. On a site map, a reference coordinate system is laid out as in

Section 3.5.3, with enough detail to locate positions with an error that will be small compared to

the distance between samples. A square or equilateral triangular systematic sampling grid

pattern is superimposed on the coordinate system. The length, L, of a side of either the square or

the triangle used to generate the pattern, is the distance between sampling locations. This

distance is determined by the total number of samples or measurements to be taken. This number,

n, is calculated to satisfy the requirements of the statistical tests and is discussed in Section 3.8.1.

The length (or spacing), L, of the systematic pattern is given by:

L= W[ A for a triangular grid, 0.866 n

and

L = - for a square grid Fn

where A is the area of the survey unit.

After L is determined, a random coordinate location is identified as the starting location for the

survey pattern. Beginning at the random starting coordinate, a row of points is identified,

parallel to the X axis, at intervals of L.

For a square grid, the second and subsequent rows of points are spaced at intervals of length L

along the Y axis.

For a triangular grid, the second row of points is located parallel to the first row, but at a Y-axis

distance of 0.866L from the first row. The survey points along the second row are located

midway (on the X-axis) between the points on the first row. This process is then repeated to fill

out the triangular pattern across the survey unit.

If points are identified that fall outside the survey unit or at locations which cannot be surveyed,

additional points are determined, using the same random process as was used to determine the

starting point, until the desired total number of points is identified.

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An example based on Figure 3.2 is shown in Figure 3.3. The procedure used for a laying out the

triangular sampling grid for the soil area survey unit is as follows:

(1) Locate a random starting point by drawing two random numbers from a uniform distribution

on the interval [0,1]. Random numbers can be generated using the random number function

of a spreadsheet or a scientific calculator. Table A.6 contains 1000 random numbers

generated using a spreadsheet, and similar tables can be found in many statistics texts.

Choose any starting point in the table, and then take numbers consecutively either across

rows or down columns. For example, in Table A.6, starting at row 23 in column 2 and

working down, the two numbers 0.93062 and 0.029842 are found. Scale the first number by

the length of the east-west coordinate axis to get 83.76 = (90)(0.93062). Round the

coordinates to the nearest values that can be easily measured in the field (e.g., nearest meter).

This gives 84 meters to the nearest meter. Similarly scale the second number by the length of

the north-south coordinate axis to get 3.28 = (110)(0.029842) or 3 meters to the nearest

meter. This gives (84,3) as the starting coordinate for the sampling grid. Since this does not

fall within the area to be sampled (it falls on an area of asphalt), the next two random

numbers (0.863244,0.921291) are taken, giving (78, 101). Continue until a point that falls

within the sampling area is obtained. In this case (78, 101) does fall in the area to be

sampled. The points are shown on Figure 3.3..

(2) Compute the spacing, L, of the sampling locations on the triangular grid using the number of

sampling locations required, n, rounded down to the nearest meter. Rounding down helps

assure that the requisite number of sampling points are identified on the sampling grid.

L A 4500 17.5 meters = 17 meters

0.866n = (0.866)(17)

Note that the area, A, is the net area of the survey unit, i.e. with buildings and paved areas that are

not part of the soil area survey unit subtracted.

(3) From the starting location, lay out a row of sampling points parallel to the X-axis and

distance L apart, as is shown in Figure 3.3.

(4) To start additional rows, locate the midpoint between two adjacent sampling locations on the

sample row and mark a spot at a distance

0.866 66A _ (0.866)(4500) - 15.14 meters =15 meters

0.66n (17)

perpendicular to the row. Again, this number should be rounded down if necessary. This is

the starting location for the new row. This is also shown in Figure 3.3.

(5) Continue until all grid points within the sampling area have been located. Ignore any

sampling locations that fall outside the area to be sampled. The completed sampling grid is

shown in Figure 3.4.

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I

I

Figure 3.3 Laying Out a Triangular Grid

Using this procedure, the number of sampling points on the triangular grid within the sampling

area may differ from the desired number, n, depending on the shape of the area. In this example,

because of the irregular shape of the region caused by its wrapping around the building, 20

sampling points are found on the grid. If the number of points is greater than the desired number,

use all the points.

If the number of points is less than the desired number, the additional points required may be

identified using the same procedure that was used to determine the grid starting point. These will

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be at individual random locations within the sampling area, and should be used regardless of

where they occur relative to the grid. If more than a few random sample locations are needed, it

is preferable to lower the grid spacing, L, and redraw the sampling grid.

-. - - ~ _ _ I(78,1,0j)

.. 1..... 4 ................... **..~ 1..'*~

-- ----------- - _ --

.-. 177//, 0~Y

Figure 3.4 Completed Triangular Sampling Pattern

3.6 Develop a Decision Rule

A decision rule relates the concentration of residual radioactivity in the survey unit to the release

criterion so that decisions can be made based on the results of the final status survey.

The decision rule proposed in this report for the final status survey consists of a statistical test

and an elevated measurement comparison. The specific recommended statistical tests were

discussed in Section 2.4, and the elevated measurement comparison was discussed in Section 2.5.

Alternative statistical tests may be appropriate in specific circumstances, providing the

assumptions of those tests are verified.

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In some cases, it will not be necessary to formally conduct the statistical tests. If the

radionuclide is not in background and radionuclide-specific measurements made, the survey unit

meets the release criterion if all of the measurements are below the Derived Concentration

Guideline Level for the mean residual radioactivity (DCGLw) defined in Section 2.2.1. On the

other hand, if the average of the measurements is above the DCGLw, the survey unit does not

meet the release criterion. It is only when some measurements are above the DCGLW but the

average is below the DCGLw, that the Sign test and the elevated measurement comparison need

to be used.

If the radionuclide appears in background or if non-radionuclide-specific measurements made,

the survey unit always meets the release criterion when the difference between the maximum

survey unit measurement and the minimum reference area measurement is below DCGL w. If the

difference between the survey unit and reference area averages is above DCGL w the survey unit

fails to meet the release criterion. When the maximum difference is above the DCGL w but the

average difference below DCGLw, conduct the WRS test and elevated measurement comparison.

Recall that the DCGLw is the concentration level corresponding to the release criterion when the

residual radioactivity is spread throughout the survey unit rather than is smaller elevated areas.

The Lower Boundary of the Gray Region (LBGR) is the concentration level below which further

remediation is not reasonably achievable. The null and alternative hypotheses for the statistical

tests that were discussed in Section 2.3.1 can be restated in terms of the DCGLwand LBGR as

follows.

Scenario A

Null Hypothesis:

H0: The mean concentration of residual radioactivity above background in the survey unit

exceeds the DCGLw.

versus

Alternative Hypothesis:

Ha: The mean concentration of residual radioactivity above background in the survey unit does

not exceed the LBGR.

Scenario B

Null Hypothesis:

H.: The mean concentration of residual radioactivity in the survey unit is indistinguishable from

background up to a level specified by the LBGR.

versus

Alternative Hypothesis:

Ha: The mean concentration of residual radioactivity in the survey unit distinguishable from

background is in excess of the DCGLw.

These hypotheses are stated in terms of the mean concentration, which is the parameter of

interest. As discussed in Section 2.5, nonparametric tests are used to test these hypotheses, using

the elevated measurement comparison to correct potential inaccuracies when the measurement

distribution is very skewed.

The preceding paragraphs have indicated some decision rules for final status surveys. However,

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there has been as yet no statement about how confident one can be that the decision is correct.

While a formal statistical test may, in retrospect, not be needed to analyze the data, the survey

must always be designed as if it were. Otherwise there is no basis for deciding the number of

measurements to be taken, or with what precision. Nor will there be any basis for confidence in

the resulting decision. This is the subject of the next Section. The hypothesis testing framework

allows a estimate to be made of the Type I and Type II error probabilities. In addition, it is

possible to similarly calculate the probability that the null hypothesis will be rejected (i.e., the

power of the test) at any specific residual radioactivity concentration level, given some

assumptions concerning the distribution of the residual radioactivity. In Section 3.7 on specifying

the limits on decision errors, it will be seen that this allows the ALARA concept to be explicity

incorporated into th6 decision-making process.

A different sort of decision rule is required for the elevated measurement comparison. As

indicated in Section 2.2.1, the value of the DCGLEMcis based on a specific size area of elevated

residual radioactivity. The area used during the survey planning to determine the DCGLEMCis

based on the distance between the sampling locations on a systematic grid. The actual extent of

an elevated area cannot be determined from a single measurement. When a measurement exceeds

the DCGLEMc, further investigation is required to determine both the size of the elevated area and

its average concentration of residual radioactivity. Only then can it be determined if the TEDE

due to the elevated area exceeds the release criterion. Thus, the decision rule for the EMC is a

two stage process. In the first stage, areas are flagged as potentially elevated at specified

investigation levels. In the second stage, the actual average concentration over the actual extent

of elevated area is compared to the release criterion. The level at which measurements should be

flagged depends on the survey unit classification, as discussed in Section 2.5.7.

3.7 Specify Limits on Decision Errors

A statistical decision error occurs when the null hypothesis is rejected when it is true (Type I) , or

not rejected when it is false (Type II). The source of the uncertainty leading to such errors is the

measurement variability, a, discussed in Section 2.6. While the possibility of a decision error

can never be totally eliminated, it can be controlled. Limits on decision errors are set to establish

performance goals for the survey design. The two types of decision errors are classified as Type I

and Type II decision errors, and were summarized in Table 2.2 for Scenario A and Scenario B.

The probability of making a Type I decision error, orthe level of significance, is called alpha (c).

The probability of making a Type II decision error is called beta (P3). The power of a test (1-13) is

the probability of rejecting the null hypothesis when it is false.

This step in the DQO process is crucial.. It is at this point that the limits on the decision errors

rates are developed in order to establish appropriate goals for limiting uncertainty in the data.

This is done by establishing the goals for the Type I error rate and the Type II error rate. The

procedure for doing this follows.

3.7.1 Type I and Type II Decision Errors for Statistical Tests

A Type I error is made when the null hypothesis is rejected when it is true. A Type II error is

made when the null hypothesis is not rejected when it is false. Thus the Type I and Type II errors

have different meanings about meeting the release criteria, depending on whether Scenario A or

NUREG-1505 3-14
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Scenario B is being considered. In an effort to avoid confusion, it will be convenient to say that a

survey unit passes the statistical test if it is deemed to meet the release criterion as a result of that

test. Otherwise, the survey unit will be said to fail the statistical test. The error rates can then be

expressed as the probability that a survey unit passes when it should fail or fails when it should

pass. This is summarized in Table 3.1. In Scenario A, the probability that a survey unit passes

when it should fail is a and the probability it fails when it should pass is 13. In Scenario B, the

probability that a survey unit passes when it should fail is 13 and the probability it fails when it

should pass is cc. In Scenario A and B, the roles of a and 13 are reversed because the null and

alternative hypotheses are reversed.

Table 3.1 Summary of Types of Decision Errors

If Survey Unit passes ...when the True Condition of Survey Unit is...

the statistical test...

That It Does Not Meet That It Meets

Release Criterion Release Criterion

under Scenario A Type I Error Correct Decision

(Probability = a) (Power = 1-13)

under Scenario B Type II Error Correct Decision

(Probability = 13) (Probability = 1-a)

If Survey Unit fails the ...when the True Condition of Survey Unit is...

statistical test...

That It Does Not Meet That It Meets

Release Criterion Release Criterion

under Scenario A Correct Decision Type II Error

(Probability = 1 -a) (Probability = 13)

under Scenario B Correct Decision Type I Error

(Power = 1-13) (Probability = a)

Acceptable error rates can be established for either scenario by determining the desired

probability for passing the survey unit as a function of the concentration of residual radioactivity

actually remaining in the survey unit. Using the statistical tests recommended in this report, the

probability that a survey unit passes decreases as the residual radiation concentration increases.

At concentrations above background near the DCGLw, this probability should be low in order to

be adequately protective of public health. The probability that the survey unit passes should be

high whenever the concentrations are near-background in order to avoid unnecessary remediation

costs. Somewhere in the range between residual radioactivity concentrations of zero and the

DCGLw there is often is a concentration level such that remediation below this level is not

considered to be reasonably achievable, considered unreasonable. The concentration range

between this lower level and the DCGLw, defines a'gray region of residual radioactivity

concentrations in which the consequences of decision errors are relatively minor. The

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specification of a gray region is important because variability in the data may be such that a

decision may be "too close to call" when the true but unknown value of the residual radioactivity

concentration is very near the DCGLw. The Lower Boundary of the Gray Region, the LBGR, is,

by definition, the concentration value at which the acceptable probability of failing a survey unit

when it should pass (13 in Scenario A, a in Scenario B) is specified.

Specifying acceptable error rates is a means for bringing considerations of practicality directly

into the decision making process. The probability limits assigned to points above and below the

gray region should reflect the risks involved in making decision errors. These probabilities can

then be converted to acceptable Type I and Type II error rates, a and P3, using Table 3.1.

Figure 3.5 illustrates this process. For example, suppose it is considered that remediation to

concentrations below one-half the DCGLw cannot be achieved with reasonable effort. The

desired probability that the survey unit passes should then be set at a high value when the true

residual radioactivity concentrations are at or below that level. This is the LBGR for this

example. When the true concentration is at the DCGLw, a small probability for passing the

survey unit is desired. The line segments connecting the LBGR and DCGLw points with

concentration values both higher and lower reflect the fact that the probability that the survey unit

passes decreases with increasing concentration.

U)

Cu)

C,)

.0

0~

100

80

60

40

20

0

4' ' 4

44 4 ~

4' 4t 4

1~ 4,

0 LBGR DCGL 1.5 DCGL

True concentration in Survey Unit

Figure 3.5 Example of Setting Acceptable Probabilities for Survey Unit Release

There is a relationship between a and 13 that is used in developing a survey design. When a fixed

number of concentration measurements are made, increasing a generally decreases 13, and

decreasing a generally decreases 13. Increasing the number of measurements allows either or both

error rates to be decreased. Once the LBGR and DCGLw are specified, the number of

NUREG-1505 3-16
PLANNING

measurements needed to meet the desired values of ix and P3 from the statistical test can be

estimated using the estimated variance of the measured concentration distribution. This is

discussed further in Section 3.8. The technical details are given in Chapter 9.

Constructing a curve such as that in Figure 3.5 is equivalent to specifying the desired power

curve for the statistical test that'is used to analyze the final status survey data. The desired power

curve for the statistical test is selected during the DQO process by specifying the desired values

for a and P3 at the lower (LBGR) and upper (DCGLw) boundary of the gray region. By definition,

the power of a test at any specific concentration is the probability that the null hypothesis is

rejected when that is the true concentration in the survey unit. In conducting a statistical test, the

value of the test statistic is calculated, and compared to a critical value. The critical value

depends only on a and the number of measurements, n. The actual power may larger or smaller

than the desired power at any specific value of the assumed true concentration. Thus, Figure 3.5

is the desired power for Scenario A, but it is one minus the power for Scenario B. More

information about the power of the tests, how it is calculated, and a procedure for comparing the

desired power to the actual power is given in Chapter 10.

The critical value of the test statistic depends, explicitly, only on a and the number of

measurements, n. One consequence of this, is that the Type I error rate has traditionally been

given precedence in experimental designs. Often, a is set at an arbitrarily low value, without

regard to the impact on increasing P3. EPA (QA/G-9, 1995) recommends that a more balance

approach be used, that the errors rates, a and P3, be considered simultaneously, and that several

different sets of values be examined before finalizing the survey design. This is part of the

optimization process discussed in Section 3.8.

Another consequence of the explicit dependence of the critical value on a, is the practice of

calculating p-values, or levels of significance. Recall that a is the probability that the null

hypothesis is rejected when it is actually true. No data are needed to calculate this probability.

However, once the data are obtained, one may calculate the probability that a data set as extreme

as that observed would occur when the null hypothesis is true. Unless this calculated "p-value" is

greater than a, the null hypothesis is rejected. Difficulties can arise when a p-value very close to

a is calculated. There is a tendency, in-this case, to believe that one has "just missed" the desired

result. There is also a temptation to believe that if more data are taken, the p-value will fall on the

"correct" side of a. Unfortunately, unless the survey is designed specifically to be performed in

two stages, the p-value calculated following the second stage of data collection will no longer be

the correct one for the original null hypothesis. Some specific ways to construct two-stage tests

are referenced in Chapter 14.

The value of a is fixed during the DQO process so that the critical value of the test statistic will

be an objective standard of comparison for the measured data. This is necessary so that a clear

line between pass and fail is drawn, despite the measurement uncertainty. In setting the value of

a, it can be useful to consider what level of discomfort will be felt about the decision if the pvalue that is observed should fall a little bit to either side of it. The time to adjust a and P3 is

during the DQO process, not after the data are taken.

As stated earlier, the acceptable probabilities for survey unit release, and the corresponding

values of a and 13, that are set as goals during in the DQO process should reflect the risk involved

3-17 NUREG-1505
PLANNING

in making a decision error. The following are important considerations for this process:

In radiation protection practice, the public health risk is modeled as a linear function of dose

(BEIR, 1990). Therefore, a 10% change in dose results in a 10% change in risk. This

situation is quite different from one in which there is a threshold. In the latter case, the risk

associated with a decision error can be quite high, and a low decision error rate is desirable.

When the risk is linear, higher error rates might be considered adequately protective at the

boundaries of the gray region, especially since these errors are known to decrease as the

concentration increases. One should consider, as part of the DQO process, the magnitude,

significance, and potential consequences of decision errors at all concentration values. This

is the purpose of the power curve.

* The DCGL itself has uncertainty associated with it. Since dose cannot be measured directly,

dose pathway models are used. Many assumptions are made in converting doses to derived

concentrations. To be adequately protective of public health, many models, especially

screening models, are generally designed to guard against underestimating the dose that may

be delivered by a given concentration of residual radioactivity. That is, the model

assumptions tend to be such that the true dose delivered by residual radioactivity in the

survey unit is very likely to be lower than that predicted by the model. Unfortunately, it is

difficult to quantfy this. Nonetheless, it is probably safe to say that most models are

conservative. This is an additional consideration that could support the use of higher

acceptable error rates in some situations. The assumptions made in any model used to

predict DCGLs for a site can be examined to determine if the use of site specific parameters

result in large changes in the DCGLs, or whether a site-specific model should be developed

rather than designing a survey around DCGLs that may be too conservative.

* The economic risk of requiring additional remediation when a survey unit already meets the

release criterion may be highly non-linear. The costs will depend on whether the survey unit

has already had remediation work performed on it, and the type of residual radioactivity

present. There may be a threshold below which the remediation cost rises very rapidly. If

so, a high probability of release is appropriate at that threshold value. This is primarily an

issue for survey units that have a substantial likelihood of falling at or above the gray region

for residual radioactivity. For survey units that are very lightly contaminated, or have been

so thoroughly remediated that any residual radioactivity is expected to be far below the

DCGL, smaller release probabilities may be tolerated, especially if final status survey

sampling costs are a concern. Again, it is important to examine the probability that the

survey unit passes over the entire range of possible residual radioactivity concentrations,

below as well as above the gray region.

* Lower decision error rates may be possible if alternative sampling and analysis techniques

can be used that result in higher precision. The same might be achieved with moderate

increases in sample sizes. These alternatives should be explored before accepting higher

design error rates. However, in some circumstances, such as high background variations,

lack of a radionuclide specific technique, and/or radionuclides whose concentrations are

very difficult and expensive to measure, error rates that are lower than the uncertainties in

the dose estimates may be neither cost effective nor necessary for adequate radiation

protection.'

NUREG-1505 3-18
PLANNING

3.7.2 Decision Errors for Elevated Areas

When the concern is to test the average residual radioactivity concentration, the actual surface

area of the survey unit is immaterial except insofar as it should be consistent with that assumed

in the dose pathway model used. It is only the distribution of the measured concentrations in the

survey unit, its mean and its variance that are important. When the concern is finding isolated

areas of elevated activity, the size of the survey unit must be explicitly taken into account. This is

because the probability of discovering an elevated area depends on the sampling density, i.e., the

distance between sampling locations.

From Section 3.5.4, the length (or spacing), L, of the systematic pattern is given by:

L= A for a triangular grid

0.866 n

and

L = F for a square grid

where A is the area of the survey unit.

A computer code for determining the probability that an elliptically shaped elevated area would

be missed by a systematic sampling grid was developed by Singer (1972). An elliptical area can

be described by the length A, of its semi-major axis and its shape (the ratio of major and minor

axis lengths), S. For a circle the length is simply the radius and the shape, S, is one. Figure 3.6

shows an example of a circular (S = 1.0) area with radius L/2 and an elliptical (S = 0.2) area with

semi-major axis length L, compared to both square and triangular sampling grids with spacing L.

Figure 3.6 Circular and Elliptical Areas Relative to the Sample Grid

3-19 NUREG-1505
PLANNING

Singer's computer code, ELIPGRID, has been improved and modified for use on personal

computers by Davidson (see ORNL/TM-12774). This code, ELIPGRID-PC, was used to generate

the data for Figure 3.7.

0)

4a)

.0

0)

100%

90%

80%

70%

60%

50%

40%

30%

20%

10%

0%

0 0.2L

Semi-Major

0.4L 0.6L 0.8L

of Elevated Area

1.OL

Axis Length

(In units of the sample grid spa~cing, L)

Figure 3.7 Probability an Elliptical Area Is Not Sampled on a Triangular Grid

This figure shows the probability that an elevated area of a given size and shape is not detected

using a triangular sampling grid with spacing L. Note that the area of an ellipse of length A is

nSA2, so that for a given value of, an ellipse of length A with shape S = 0.2 has only one-fifth the

area of a circle (S = 1.0) of the same length, i.e., radius. That is one reason, in addition to the area

becoming longer and thinner, that the probabilities increase as S decreases.

The black and white squares in Figure 3.7 correspond the circle (S = 1.0, A= 0.5L) and the ellipse

(S = 0.2, A= L) shown in Figure 3.6. The probability is less than 10% that the circle would go

undetected. The probability is about 40 percent that the ellipse would go undetected, even though

its area (0.628 L2) is only slightly smaller than the area of the circle (0.785 L2).

The circles in Figure 3.7 correspond to elevated areas equal to the triangular grid area, 0.866L2.

The probability of missing them is rather low unless the shape parameter is also very low.

NUREG-1505 3-20
PLANNING

In Figure 3.8, the probabilities of missing a circular elevated area with triangular and square

systematic grids are compared. The square grid is only slightly less efficient than the triangular

grid. It can be concluded that, in most cases, an elevated area of the same size as, or larger than,

that defined by the sampling grid is likely to be discovered during the final status survey.

0.9 3⁄4

* 0.8 . . Square

0 \J .. . ." Triangular

4 0.7

to

00.0.5

0.4

0

*~0.2

r~0.1

0

0 0.1L 0.2L 0.3L O.4L 0.5L 0.6L 0.7L

Radius of Circular Area (in units of grid spacing, L)

Figure 3.8 Probability a Circular Area Is Not Sampled on a Systematic Grid

3.8 Optimize the Design

The DQO process need be neither static nor sequential. Some of the activities involved may be

taking place concurrently, and may be visited more than once. At any stage in the process, new

information may be available that should then be incorporated into planning the final status

surveys.

Optimization of the final status survey involves examining all of the factors that effect the

decision errors and sample sizes so that costs and potential risks are balanced. The primary

factors to be considered in optimizing the design for determining the mean concentration are the

DCGLw and the measurement standard deviation. The estimate of the measurement standard

deviation should include both the uncertainty in measurement process and any anticipated spatial

and temporal concentration variations. The delineation and classification of survey units and

reference areas can affect the spatial variability. Scan sensitivity is a primary consideration in

optimizing the design to ensure no elevated areas remain in a survey unit. The Area Factor and

3-21 NUREG-1505
PLANNING

the scan MDC are the important parameters which can impact survey costs and uncertainty.

3.8.1 Optimizing the Design for the Mean Concentration

There are relationships between the measurement uncertainty, a, the width of the gray region, A,

the desired decision error limits (a and P3) and the number of measurements needed to meet

those limits. This is illustrated in Table 3.2 for the case when no reference area is needed (onesample test). Table 3.3 is used when the survey unit is compared to a reference area (two-sample

test), and lists the number of samples to be taken in each. The method used to generate these

tables is discussed in Chapter 9.

Table 3.2 Number of Samples, N, Required in Survey Unit to Meet Error Rates ac, and , With Relative Shift A/a, When Using the Sign Test

o=0.1cc =0.05 a = 0.10 a = 0.25

A/ay 0.01 0.05 0.10 0.25 0.01 0.05 0.10 0.25 0.01 0.05~ 0.10 0.25 0.01 0.05 0.10 0.25

0.1 4095 2984 2463 1704 2984 2048 1620 1018 2463 1620 1244 725 1704 1018 725 345

0.2 1035 754 623 431 754 518 410 258 623 410 315 184 431 258. 184 88

0.3 468 341 282 195 341 234 185 117 282 185 143 83 195 117 83 40

0.4 270 197 162 113 197 136 107 68 162 107 82 48 113 68 48 23

0.5 178 130 107 75 130 89 71 45 107 71 54 33 75 45 33 16

0.6 129 94 77 54 94 65 52 33 77 52 40 23 54 33 23 11

0.7 99 72 59 41 72 50 40 26 59 40 30 18 41 26 18 9

0.8 80 58 48 34 58 4032 21 48 32 24 1534 2115 8

0.9 66 48 40 28 48 34 27 17 40 27 21 12 28 17 12 6

1.0 57 41 34 24 41 29 23 15 34 23 18 11 24 15 11 5

1.1 50 36 3021 36 2621 1430 2116 1021 1410 5

1.2 45 33 27 20 33 23 18 12 27 18 15 9 20 12 9 5

1.3 41 30 26 17 30 21 17 11 26 17 14 8 17 11 8 4

1.4 38 28 23 16 28 20 16 10 23 16 12 8 16 10 8 4

1.5 35 27 22 15 27 18 15 10 22 15 11 8 15 10 8 4

1.6 34 24 21 15 24 17 14 9 21 14 11 6 15 9 6 4

1.7 33 24 20 14 24 17 14 9 20 14 10 6 14 9 6 4

1.8 32 23 20 14 23 16 12 9 20 12 10 6 14 9 6 4

1.9 30 22 18 14 22 16 12 9 18 12. 10 6 14 9 6 4

2.0 29 22 18 12 22 15 12 8 18 12 10 6 12 8 6 3

3.0]2720 1712120 14.11 8117 11 9 5112 8 5 3I

NUREG- 1505 323-22
PLANNING

Table 3.3 Number of Samples, N12, Required in Both Reference Area and Survey Unit to

Meet Error Rates a and P3 With Relative Shift A/o, When Using the Wilcoxon

Rank Sum Test

a = 0.01 a = 0.05 cc = 0.10 a = 0.25

A/a 0.01 0.05 0.10 0.25 0.01 0.05 0.10 0.25 0.01 0.05 0.10 0.25 0.01 0.05 0.10 0.25

0.1

0.2

0.3

0.4

0.5

5452 3972 3278 2268

1370 998 824 570

614 448 370 256

350 255 211 146

227 .166 137 95

3972 2726 2157 1355

998' 685 542 341

3278 2157 1655 964

824 542 416 243

2268 1355 964 45c

570 341 243 11(

448 307 243 1531 370 243 187 1091 256 153 109

255 175 139 871 211 139 106

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.25

2.5

2.75

3.0

3.5

4.0

161

121

95

77

64

55

48

43

38

35

32

30

28

26

25

22

21

20

19

18

18

117

88

69

56

47

40

35

31

28

25

23

22

20

19

18

16

15

15

14

13

13

97

73

57

47

39

33

29

26

23

21

19

18

17

16

15

14

13

12

12

11

11

67

51

40

32

27

23

20

18

16

15

14

13

12

11

11

10

9

9

8

8

8

166 .114

117 81

88 61

69 48

56 39

47 32

40 28

35 24

31 22

28 19

25 18

23 16

22 15

20 14

19 13

18 13

16 11

15 11

15 10

14 10

13 9

13 9

90 57

64 40

48 30

38 24

31 20

26 16

22 14

19 12

17 11

15 10

14 9

13 8

12 8

11 7,

11 7

10 7

9 6

9 6

8 5

8 5

8 5

7 5

137

97

73

57

47

39

33

29

26

23

21

19

18

17

16

15

14

13

12

12

11

11

90 69

64 49

48 37

38 29

31 24

26 20

22 17

19 -15

17 13

15 12

14 11

13 10

12 9

11 9

11 8

10 8

9 7

9 7

8 6

8 6

8 6

7 6

62

41

29

22

17

14

12

10

9

8

7

7

6

6

5

5

5

4

4

4

4

4

4

146

95

67

51

40

32

27

23

20

18

16

15

14

13

12

11

11

10

9

9

8

8

8

87

57

40

30

24

20

16

14

12

11

10

9

8

8

7

7

7

6

6

5

5

5

5

62

41

29

22

17

14

12

10

9

8

7

7

6

6

5

5

5

4

4

4

4

4

4

The width of the gray region, A, is a parameter that is central to the nonparametric tests discussed

in this report. It is also referred to as the shift. In this report, the gray region is always bounded

from above by the DCGLw corresponding to the release criterion. The Lower Boundary of the

3-23 NUREG-1505
PLANNING

Gray Region (LBGR) is selected during the DQO process along With the target values for a and

3, as discussed in Section 3.7.1. The width of the gray region, or shift, A, is equal to (DCGL -

LBGR). The absolute size of the shift is actually of less importance than the relative shift A/a,

where a is an estimate of the standard deviation of the measured values in the survey unit. The

estimated standard deviation, a, includes both the real spatial variability in the quantity being

measured, and the precision of the chosen measurement method. The relative shift, A/o, is an,

expression of the resolution of the measurements in units of measurement uncertainty.

Expressed in this way, it is easy to see that relative shifts of less than one standard deviation, A/a

< 1, will be difficult to detect. On the other hand, relative shifts of more than three standard

deviations, A/a > 3, are generally easy to detect.

It is evident from Tables 3.2 and 3.3, that the number of measurements that will be required to

achieve given error rates (a and P3) depends entirely on the value of A/a. Note also that the

number of measurements required is symmetric in a and 3. For example, if A/a =1, cc = 0.05 and

P3 = 0.10, then, from Table 3.1, the number of samples needed for the Sign test is 23. For the

same value of A/a, but with error rates reversed (i.e., a = 0.10 and 13 = 0.05), the number of.

samples needed for the Sign test is again 23. Thus, these tables may be used to plan the number

of measurements needed, regardless of whether Scenario A or Scenario B is used. It is only when

the statistical test is actually performed on the measurement results that the distinction between

a and P3 becomes important.

For fixed values of a and 13, small values of A/a result in large numbers of samples. It is

desirable to design for A/a > 1 whenever possible. There are two obvious ways to increase A/a.

The first is to increase the width of the gray region by making LBGR small. The disadvantage is

that the acceptable probability of the survey unit passing will be specified at this smaller LBGR.

Thus, a survey unit will generally have to be lower in residual radioactivity to have a high

probability of being judged to meet the release criterion. The second way to increase A/c is to,

make a smaller. One way to make a small is by having survey units that are relatively

homogeneous in the amount of measured radioactivity. This is an important consideration in

selecting survey units that have both relatively uniform levels of residual radioactivity and also

have relatively uniform background radiation levels. Measurements performed during scoping,

characterization, and remedial action support surveys can be useful for determining an estimate

of a for the final status survey planning.

Another way to make a small is by using more precise measurement methods. The more precise

methods might be more expensive, but this may be compensated for by the decrease in the

number of required measurements. One example would be in using a radionuclide specific

method rather than gross radioactivity measurements for residual radioactivity that does not

appear in background. This would eliminate the variability in background from a, and would

also eliminate the need for reference area measurements. On the other hand, the costs associated

with performing additional measurements with an inexpensive measurement system may be less

than the costs associated with fewer measurements of higher precision.

The effect of changing the width of the gray region and/or changing the measurement variability

on the estimated number of measurements (and cost) can be investigated using Table 3.1 and 3.2.

Generally, the design goal should be to achieve A/a values between one and three. ' The number

of samples needed rises dramatically when A/a is smaller than one. Conversely, little is usually

NUREG-1505 3-24
PLANNING

gained by making A/c larger than about three. If A/c is greater than three or four, one can take

advantage of the measurement precision available by making the width of the gray region

smaller. It is even more important, however, that overly optimistic estimates for a be avoided.

The consequence of taking fewer samples than are needed given the actual measurement

variations will be increased error rates, leading to either unnecessary remediations (Scenario A)

or improper survey unit release (Scenario B).

On the other hand, a smaller number of samples may still result in acceptable error rates. When

A/a is small, and the number of samples is large, a modest increase in the acceptable error rates

may result significant reduction in the number of samples required. Given the other uncertainties

involved, the cost savings may justify larger acceptable error rates. The advantage of the

optimization step of the DQO process is that several alternatives can be explored on paper before

time and resources are committed.

One consideration in setting the error rates are the health risks associated with releasing a survey

unit that might actually contain residual radioactivity in excess of the DCGL. If a survey unit did

exceed the DCGL, the first question that arises is "How much above the DCGL is the residual

radioactivity likely to be?" Figures 3.9 through 3.12 can be used to estimate this.

These figures show the probability of the survey unit passing the statistical tests as a function of

the true concentration of residual radioactivity in the survey unit. Figures 3.9 and 3.10 are for the

one-sample Sign test, under Scenario A and B, respectively. Figures 3.11 and 3.12 are for the

two-sample WRS test, under Scenario A and B, respectively. In these figures, the black-colored

curves are those for ac = 0.01, the white-colored curves are those for a = 0.10, and the graycolored Curves are those for a = 0.25. For each value of a, survey unit sample sizes of 10, 15, 20,

30, 50 and 100 are shown. Note that in Scenario A, a is the probability that the survey unit passes

when the concentration is equal to the DCGLw. In Scenario B, a is the probability that the survey

unit passes when the concentration is equal to the LBGR.

For example, if the DCGLw is 1.0, the LBGR is 0.5, a is 1.0, a = 0.05 and 3 = 0.05, then A/a =

0.5 and Table 3.2 indicates that 89 samples would be required. If cc = 0.1 and P3= 0.1, then only

54 samples are required. How likely is it that a survey unit with residual radioactivity 50% higher

than the DCGLw would pass? A concentration 50% higher than the DCGLw is 1.5, which is the

same as the DCGLw + 0.5a. For the Sign test in Scenario A, Figure 3.9 (second white curve

from the left) shows that the probability of the survey unit passing is near zero for a

concentration of 1.5 when a = 0.1 and the sample size is 50. While a survey unit with residual

radioactivity equal to the DCGLw might have a 10% chance of being released, a survey unit at

the DCGLw + 0.5a has almost no chance of being released. On the other hand, a survey unit with

a residual radioactivity that is at 50% of the DCGLw, i.e., 0.5, is at the DCGLW - 0.5a, and has

a 90 % chance of being released. If the'sample size were nearer 100, the leftmost white curve

shows that this probability would increase to about 99%. Thus, if the cost of remediation below a

concentration of 0.5 was very high, the larger sample size might be chosen, but with the objective

of achieving a = 0.10 and P = 0.01.

A similar result is obtained for Scenario B, where a concentration of 1.5 =LBGR + 1.0o.

Figure 3.10 (second white curve from the right) shows that with a = 0.1 and a sample size of

50, a survey unit with a concentration level of LBGR + 1.0Oa has less than a 1% chance of being

3-25 NUREG-1505
PLANNINd

released. Setting a = 0.01 does not appear to appreciably increase this probability (second black

curve from the right).

Scenario A

C

n

>b

N

0

0.600

0.40

0.30

03-Bla ck: a=0.01

Dashed: a =0.10

0 -Gray: 0 a 0 .25N

0.20

DCGL-2.0a DCGL-1.5o DCGL-A.No DCOL-O.5o DCGL DCGL+0.5a

Residual Radioactivity Concentration

Figure 3.9 Probability a Survey Unit Is Released Using the Sign Test Under Scenario A

Using these figures, the probability that the survey unit passes over the entire range of possible

residual radioactivity values, can be compared to the DQOs as expressed, for example, in

Figure 3.5. In this way, the sample design can be optimized, taking into account the risks and

costs associated with a decision error. The construction of the curves is discussed further in

Chapter 10.

NUREG-1505 3-26
PLANNING

Scenario B

1.fn n. I I

~zcj

U

I

I

U

I

A.

I

(0

.0

U

.0

0

1~ A.

0.70,V

~0.60-

~0.50

0.40

0.30-

Black: a 0.01

Dashed: a 0.10

Gray: a 0.25

0.20Z___

0.10

U.1WU

LBGR-0.5a LBGR LBGR+0.5a LBGR+ ai.O n LBGR+1.5a LBGR+2.Oa

Residual Radioactivity Concentration

Figure 3.10 Probability a Survey Unit Is Released Using the Sign Test Under Scenario B

3.8.2 Optimizing the Design for Detecting Elevated Areas i

As discussed in Section 2.2, one objective of the final status survey is to provide reasonable

assurance that there are no small areas of elevated residual radioactivity left within the survey

unit that might cause the release criterion to be exceeded. However, it is inefficient to treat all

survey units equally in this regard. During the process of survey unit classification, Class 1

survey units are identified as those with the potential for such elevated areas. Measurements and

3 -27 NUREG-1505
PLANNING

sampling on a systematic grid, in conjunction with scanning, are used to assure that any small

areas of elevated radioactivity that might remain within a Class 1 survey units will not produce a

dose in excess of the release criterion. To accomplish this, an additional step in the survey design

optimization is required.

Scenario A

C

N

0

L.

IL

1.000

0.90 r

0.80-

I0.70 _ ____ _________

10

*0.60'

0.50-

0.40-

0.30- X - Black: a 0.01

Dashed: a =0.10

0.20 -________

U.1U

.UXu

DCGL-2.0o DCGL-t.5a RduL-.aoa DCGL-O.5o DCGL DCGL4O.5a

Residual Radioactivity Concentration

Figure 3.11 Probability a Survey Unit Is Released Using WRS Test Under Scenario A

The number of samples, N, to be taken on a random start systematic grid in a survey unit of area

A, determines the spacing, L, between the samples (see Section 3.4.5). Corresponding to this

spacing is the grid area delimited by neighboring sampling locations. This grid area is 0.866 L2

for a triangular grid and L' for a square grid.

NUREG-1505 3-28
PLANNING

Scenario B I nn I I

~ZZN

0

1.

4L

0.801

100--

0.70 -

0.60,

0.50o

0.40

0.30 Black: a 0.01

Dashed: a = 0.10

Gray: a 0.25 0.201

0.10.

f f i 

LBGR-0.5a LBGR LBGR+0.5o LBGR.1.Oa LBGR1.5aS LBGR+2.Oo

Residual Radioactivity Concentration

Figure 3.12 Probability a Survey Unit Is Released Using WRS Test Under Scenario B

A given concentration of residual radioactivity spread over a smaller area will, in general, result

in a smaller dose. Thus, the DCGLEMc used for the elevated measurement comparison is usually

larger than the DCGLw used for the statistical test. The amount of residual radioactivity that

would have to exist within the grid area between sampling locations in order to exceed the

guideline dose is a multiple, FA, of the residual radioactivity derived concentration guideline

level, DCGLw. Values for the area factor, FA, can be determined by comparing the dose

conversion factor (DCF) obtained from the results of a pathway analysis under the scenario that a

unit activity concentration of a given radionuclide is distributed uniformly across the survey unit

3 -29 NUREG-1505
PLANNING

to the DCF obtained when a unit concentration of that radionuclide is confined to the smaller grid

area. For some radionuclides, especially those that deliver dose primarily via internal pathways,

the dose is approximately proportional to inventory, and so the ratio of the DCGLEmc to the

DCGLw is nearly proportional to the ratio of the survey unit area to the grid area. However, this

may not be the case for radionuclides that deliver a significant portion of the dose via external

exposure. The exact relationship between the DCGLEmc and the DCGLw is generally a

complicated function of the dose modeling pathways.

The scanning procedure used for the survey unit should have a minimum detectable

concentration (MDC) less than the DCGLEMc. The DCGLEMc depends on the grid area which in

turn depends on the spacing of the samples. Once a scanning technique is selected, the actual

MDC can be compared to the DCGLEMC. If the actual scan MDC is less than the DCGLEMC, the

survey design is complete. When the scanning method is sensitive enough to detect residual

radioactivity concentrations at the DCGLEMC, the combination of sampling and scanning will be

sufficient to provide reasonable assurance that release criterion is met by any residual

radioactivity remaining in the survey unit. Any area smaller than the grid area would require a

residual radioactivity concentration within it larger than DCGLEMc = (FA)(DCGLw) in order to

exceed the release criterion. Recall from the discussion in Section 3.7.2 that any area larger than

the grid area is likely to be hit by a sampling location on the systematic grid at least once.

If the scanning MDC is greater than the DCGLEMO then the survey design must be modified. A

larger DCGLEMc is generally obtained by decreasing the sampling grid area, i.e., adding more

sampling locations to the grid. The number of additional sampling locations can be found by

determining the area factor necessary to raise the DCGLEMC to a level detectable by scanning:

FA = (Scan MDC) / (DCGLw).

The sampling grid area, AuMc, that corresponds to this area factor can be used to determine the

sample size for the survey unit. Dividing the survey unit area, As, by the revised sampling grid

area AuDc yields the required sample size, nut* = As IAuDc.

Thus, for Class 1 Survey Units, the number of samples may be driven more by the need to detect

small areas of elevated activity than by the requirements of the statistical tests. This in turn will

depend primarily on the sensitivity of available scanning instrumentation, the size of the area of

elevated activity, and the dose model. For many radionuclides, scanning instrumentation is

readily available that is sensitive enough to detect residual radioactivity concentrations at the

DCGLEmc derived for the sampling grid of direct measurements used in the statistical tests.

Where instrumentation of sufficient sensitivity is not available, the number of samples in the

survey unit can be increased until the area between sampling points is small enough (and the

resulting area factor is large enough) that DCGLEMC can be detected by scanning. For some

radionuclides (e.g., 311), the scanning sensitivity is so low that this process would never

terminate-i.e., the number of samples required could increase without limit. Thus, an important

part of the DQO process is to determine the smallest size of an area of elevated activity that it is

important to detect, Amn, and an acceptable level of risk, RPA , that it may go undetected. Figures

3.7 and 3.8 can be used for this purpose. The ELIPGRID-PC computer code (ORNL/TM-12774,

1994) can also be used to calculate these risks.

NUREG-1505 3-30
PLANNING

In this part of the DQO process, the concern is less with areas of elevated activity that are found

than with providing adequate assurance that negative scanning results truly demonstrate the

absence of such areas. In selecting acceptable values for A ,m and RA, maximum use of

information from the historical site assessment and all surveys prior to the final status survey

should be used to determine what sort of areas of elevated activity could possibly exist, their

potential size and shape, and how likely they are to exist.

3-31 NUREG-1505

4 ANALYSIS OF FINAL STATUS SURVEY RESULTS:

DATA QUALITY ASSESSMENT

Data Quality Assessment (DQA) is the scientific and statistical evaluation of data to determine if

the data are of the right type, quality, and quantity to support their intended use (EPA QA/G-9,

1995). There are five steps in the DQA process:

(1) Review the Data Quality Objectives (DQOs).and sampling design.

(2) Conduct a preliminary data review.

(3) Select the statistical test.

(4) Verify the assumptions of the statistical test.

(5) Draw conclusions from the data.

4.1 Review the Data Quality Objectives (DQOs) and Sampling Design

During survey design, acceptable error rates are specified for the statistical tests; and the desired

probability that a survey unit will pass the release criterion is charted against the amount of

residual radioactivity that actually may be present in order to test the efficacy of a proposed

design. During the interpretation of survey results, it is important to determine that the objectives

of the design have been met. The first and most straightforward way to check this is to ascertain

that the number of usable measurements meet the requirement of the statistical tests as outlined

in Section 3.8.1. The sample standard deviation, s, should also be compared to the estimate of the

measurement variability, a, that was used to determine the number of samples required. The

consequence of there being too few measurements, or of there being higher than expected data

variability, is that the Type II error rate P3 will be larger than planned, and the power of the test to

detect departures from the null hypothesis, 1 - 3, is reduced. In Scenario A this means that a

survey unit that meets the release criterion has a higher probability of being incorrectly deemed

not to meet it. In Scenario B this means that a survey unit that does not meet the release criterion

has a higher probability of being incorrectly deemed to meet it. After examining the number of

usable measurements and their variability, the retrospective power of the test can be determined

using the methods of Chapter 10. This is not usually necessary when the null hypothesis is

rejected since the Type I error rate, co, is fixed when the statistical test is performed using the

actual number of usable measurements.

Since the occurrence of missing or unusable data can impact the Type II error rates, a reasonable

allowance for such occurrences should be built into the planning process by adding more

measurements to the sample sizes listed in Tables 3.2 and 3.3.

The power of the statistical tests will'also be reduced if data variability is greater than that

assumed during the survey planning. The number of measurements required to meet the

acceptable error specified during the planning process will not be sufficient if a was

underestimated. As mentioned in Section 2.2.6, the overall data variability may be considered to

consist of two more or less independent components, the component due to uncertainty in the

measurement process, measn and the component due to spacial variability in the concentrations

4-1 NUREG-1505
ANALYSIS

being measured, ospatial, Spatial variability was discussed in Section 3.5.1. The overall variability

is approximately a = 0mes2 + Ospatial2. If either standard deviation is one-third or less of the

other, there is not much point in trying to reduce it further. If the smaller contributor were

eliminated entirely, at most a would be reduced by a factor of 0/910-0.95, 1 i.e., only about a

5% gain. Efforts should be directed at reducing the dominant component of the data variability.

The quality of data is critical to the successful execution of a survey. Even if the measurement

uncertainty is dominated by the spatial variability, poorly calibrated instruments could lead to

either improperly labeling an area as still contaminated or releasing it when, in fact, it is above

the guidelines. For this reason, calibrations must be performed regularly with traceable

standards; the inherent precision of the survey instrument must be evaluated to determine if it

meets the needs of the survey plan. Energy responses of instruments must be known so that

appropriate applications are made to different radiation fields. Replicate, reference, and blank

measurements are also an integral part of the survey methodology. Comparisons of field

measurement results to those of laboratory sample analyses forms an important quality control

check.

Bounds on measurement uncertainties should be established in the planning process and regularly

assessed throughout the measurement program. Uncertainties in the measurements add to the

variance in distribution of data sets and should be taken into consideration when selecting

parameters for the statistical tests and in the interpretation of results of these tests. Failure to

adequately consider the effect of measurement errors could result in the added expense of

additional measurements. In the worst case, inadequate control of the Type II statistical errors as

determined from a retrospective power calculation, could invalidate the final survey results and

require a re-survey. For this reason, it is better to plan the surveys cautiously:

" It is better to overestimate the potential data variability than to underestimate it.

* It is better to take too many samples than too few.

* It is better to overestimate minimum detectable concefitrations (MDCs) than to underestimate

them.

Further information on quality assurance for environmental data may be found in EPA QA/R-5

(1994), EPA QA/G-5 (1996), and ANSI/ASQC (1994)

4.2 Conduct a Preliminary Data Review

To learn about the structure of the data-identifying patterns, relationships, or potential

anomalies-one can review quality assurance (QA) and quality control (QC) reports, prepare

graphs of the data, and calculate basic statistical quantities.

Radiological survey data are usually obtained in units that have no intrinsic meaning relative to

DCGLs, such as the number of counts per unit time. For comparison of survey data to DCGLs,

NUREG-1505 4-2
ANALYSIS

the survey data from field and laboratory measurements should be converted to DCGL units.

4.2.1 Basic Statistical Quantities

Basic statistical quantities that should be calculated for the sample data set are the

" mean

" standard deviation

* median

The average of the data can be compared to the reference area average and the DCGL wto get a

preliminary indication of the survey unit status. Where remediation is inadequate, this

comparison may readily reveal that a survey unit contains excess residual radioactivity-even

before applying statistical tests. For example, if the average of the data exceeds the DCGL wand

the radionuclide of interest does not appear in background, then it is obvious that the survey unit

does not meet the release criterion. On the other hand, if every measurement in the survey unit is

below the DCGLw, the survey unit will always pass the Sign test.

The value of the sample standard deviation is especially important. If too large compared to that

assumed during the survey design, this may indicate an insufficient number of samples were

collected to achieve the desired test power.

The median is the middle value of the data set when the number of data points is odd, and is the

average of the two middle values when the number of data points is even. Thus 50% of the data

points are above the median, and 50% are below the median. Large differences between the

mean and the median would be an early indication of skewness in the data. This would also be

evident in a histogram of the data.

Table 4.1 lists an example of concentration data taken in a reference area and survey unit. For

this example, the quantity and units of measurement have been left arbitrary. Basic statistical

quantities can be calculated simply by using one of the many widely available personal computer

programs that perform data analysis. Table 4.2 shows the result of a "descriptive statistics"

command applied to the data of Table 4.1 using a spreadsheet program. In addition to the mean,

median and standard deviation, this table lists several other useful parameters such as the

minimum, maximum, mode, range, skewness and kurtosis.

For the example survey unit, the mean is 1.15 and the median is 1.05. The sample standard

deviation is 0.46. The difference between the median and the mean, divided by the sample

standard deviation is sometimes used as a simple measure of skewness. In this case,

(1.15 -1.05)/0.46 = 0.22.

The coefficient of skewness is the average cubed difference from the mean divided by the

standard deviation cubed. The sample estimate of skewness is g1 = M3 /m2 , where

n n n

m 3 = ,(xi-x) 3

In , and m2 = • (x,-x) 2/n. m= (xj-x) 2

In.

i=1 i=1 i=1

4-3 NUREG-1505
ANALYSIS

For a sample from a normal distribution, g, is approximately normal with mean zero and

standard deviation ý61n The sample skewness for the survey unit data is 0.96. This is nearly

four times VI-619 0 0.26, indicating that there is some positive skewness in this data set.

Table 4.1 Example Final Status Survey Data

___- Reference Area - Survey Unit- -_ _

Point Data Point Data Point Data Point Data Point Data Point Data

No. Value No. Value No. Value No. Value No. Value No. Value

1 1.1 31 1.9 61 0.8 91 1.2 121 0.7 151 0.8

2 1.3 32 1 62 1.1 92 1.4 122 1.9 152 1.1

3 0.7 33 0.7 63 0.6 93 0.8 123 1.3 153 1.2

4 0.7 34 1.9 64 0.8 94 0.6 124 1.4 154 0.7

5 1.6 35 1.0 65 1.2 ___ 95 1.4 125 0.5 155 1.4

6 1.0 36 0.6 66 0.8 96 2.9 126 1.0 156 1.6

7 1.1 37 1.5 67 1.0 97 0.9 127 1.3 157 0.4

8 0.7 .38 1.1 68 0.9 98 0.8 128 0.6 158 0.6

9 0.9 39 0.9 69 1.5 ___ 99 0.8 129 1.3 159 1.6

10 1.8 40 0.9 70 0.8 ___100 1.6 130 1.5 160 0.7

11 0.9 41 0.8 71 1.2 ___101 1.6 131 1.4 161 1.0

12 0.7 42 1.1 72 1.1 ___102 1.2 132 1.3 162 1.0

13 1.1 43 0.9 73 0.6 __ 103 1.2 133 0.8 163 1.8

14 1.1 44 1.2 74 1.0 ___104 2.5 134 1.5 164 1.3

15 0.9 45 1.2 75 0.9 __ 105 1.9 135 0.8 165 1.5

16 1.5 46 1.0 76 1.0 ___106 1.9 136 1.1 166 0.8

17 1.0 47 1.3 77 0.6 ___107 0.9 137 1.1 167 1.5

18 0.8 48 0.9 78 0.9 ___108 0.9 138 1.0 168 0.9

19. 0.6 49 0.8 79 1.0 ___109 0.8 139 1.1 169 0.9

20 1.1 50 1.7 80 0.8 ___110 1.0 140 1.6 170 0.8

21 0.7 51 0.7 81 0.6 Ill_11 1.7 141 1.5 171 1.5

22 0.6 52 1.0 82 1.2 ___112 1.5 142 0.8 172 1.0

23 0.9 53 0.8 83 1.2 ___113 2.1 143 0.7 173 0.7

24 0.6 54 1.0 184 1.3 ___114 2.0 144 0.6 174 1.1

25 1.5 55 0.5 85 1.0 115 1.7 145 0.9 175 1.4

26 0.9 56 1.1 86 0.9 ___116 0.7 146 0.8 176 1.0

27 1.5 57 1.1 87 0.7 ___117 1.0 147 0.5 177 1.2

28 0.8 58 0.9 188 10.9 1__ .118 1.0 148 0.6 178 0.5

"29 1.1 59 0.9 189 1.4 ___119 1.5 149 0.8 179 0.5

3 1.2 60 0.6 190 1 __ 120 1 150 10.8 180 1.7

NUREG- 150544 4-4
ANALYSIS

Table 4.2 Basic Statistical Quantities Calculated for the Data in Table 4.1

Reference Survey Unit

Mean 1.00 Mean 1.15

Standard 0.03 Standard 0.05

Error Error

Median 1.00 Median 1.05

Mode 0.90 Mode 0.80

Standard 0.30 Standard 0.46

Deviation Deviation

Sample 0.09 Sample 0.22

Variance Variance

Kurtosis 0.93 Kurtosis 1.44

Skewness 0.95 Skewness 0.96

Range 1.4 Range 2.5

Minimum 0.5 Minimum 0.4

Maximum 1.9 Maximum 2.9

Count 90 Count 90

The kurtosis is the average fourth power of the difference from the mean divided by the variance

squared. It is a measure of how "flat" the distribution is relative to normally distributed data. The

sample estimate of kurtosis is b2 = m4 /M2 , where

n mn4 = •_ X-~~ , and

i=1

in2 ~(Xi_ )2 In =(n_1)S2In.

For a sample from a normal distribution, b2 has mean three. The sample coefficient of kurtosis is

92 = b2 - 3. For very large samples from a normal distribution, g2 has mean zero and standard

deviation ý2-44n (Snedecor and Cochran, 1980). The sample coefficient of kurtosis for the

survey unit data is 1.44, and .Thus, the kurtosis appears to be significantly greater than zero. It is

an indicator of how well the sample variance, S2, estimates the true variance, o2, of the

measurement data. The variance of the sample estimate of the variance is

Var(s 2) = 2j4 [ + n-1 92 ]

The variance of s is given approximately by

4-5 NUREG-1505
ANALYSIS

Var(s) = Var(s 2) -L Var(s 2 ) 2 C14 [l+n-1 ] - s3 [Ln-1 ]1+ 4s 4s(n-1) 2•"'n g 2(n-l) 2n

where the propagation of error formula for the variance of the square root has been used (Taylor,

1990). For the example survey unit data with s = 0.36, 92= 1.44, and n = 90, we have

Var(s) = (6.46)3 [1+ i-1.448 (0.0973/178)(1.712) =0.000936. 2(89) 18

The standard deviation of s = V = •0.000936 = 0.0306. Thus, one can estimate that,

very roughly, s = 0.46 ± 0.03.

An approximate 1- a confidence interval for a2 may be obtained from

2_-,(x/f2) < <(n [0 -1)s2]/ _

I+ g21n X" +92/n , X 

where Xn I (a) is the 100ath percentile of the chi-squared distribution with n-1 degrees of

freedom (Box, 1953). Percentiles of the chi-squared distribution are given in Table A.5. With

s2 = 0.22,92 =1.44, n = 90, X89(0.975) 117, and X89(0.025) = 64.8, we find that with 95%

confidence [ (•(0.22)' 117 < o< < 1(89)(022) 64.8 or 0.1647<a2<0.2974,

which implies that 0.406 < 0 < 0.545. This is not too different from the cruder estimate

s = 0.46 ± (2) (0.03) = 0.46 ± 0.06, using a 2o interval about the mean.

Examining the minimum, maximum, and range of the data may provide additional useful

information. The minimum of the example survey unit data is 0.4 and the maximum is 2.9, so

the range is 2.9 - 0.4 = 2.5. This is 5.4 times the standard deviation of 0.46. Figure 4.1 shows

that is well within the expected spread of values for this ratio, which is sometimes called the

studentized range. These intervals were calculated for normally distributed data.

Absolute upper and lower bounds for the studentized range have been found by Thomson (1955).

These bounds are fairly wide, but are useful in checking for errors in calculation. The upper

bound is j2(Qnl) . The lower bound is 2j(n-•)/n , when n is even, and 2VIn/(n+ 1) , when n

is odd.

NUREG-1505 4-6
ANALYSIS

8

0 7-

a~6

L.0

/~0 00 oo' 10 5 •,

(U0

2

.10 100 1000

Sample Size

Figure 4.1 Confidence Bands for the Ratio of the Range to the Standard Deviation

Dotted: 90% Dashed: 95% Solid: 99%

Transformations are sometimes used bring data closer to a normal distribution, and decrease any

dependence of the variance on the mean. A rule of thumb sometimes used is that if the ratio of

the data maximum to the data minimum is less than 20, no data transformation is necessary to

stabilize the variance of the data (EPA 600/4-90/013, 1990). For the example survey unit data,

this ratio is 2.9/0.4 = 7.25.

Many of the "diagnostic checks" on the basic statistical quantities discussed in this section are

based on comparing the values computed for the sample data distribution to those that would be

expected if the data were normally distributed. When viewed as tests of normality, they are

generally not very powerful, and are not suggested here for that purpose. As noted earlier, the

nonparametric statistical tests used in this report do not assume the data are normally distributed.

Rather, these checks are being used as exploratory techniques to alert the data analyst of any

unusual features in the data.

4.2.2 Graphical Data Review

At a minimum, the graphical data review should consist of a posting plot and a histogram. Rank

or Quantile plots are also useful diagnostic tools, particularly in the two-sample case, to compare

the survey unit and reference area.

4-7 NUREG-1505
ANALYSIS

A posting plot, which is simply a map of the survey unit with the data values entered at the

measurement locations, will reveal potential anomalies in the data, especially possible patches of

elevated residual radioactivity. Even in a reference area, a posting plot can reveal spatial trends in

background data that might affect the results of the two-sample statistical tests.

The survey unit data in Table 4.1 were taken on a square systematic grid in a rectangular survey

unit. A simple posting plot is shown in the upper half of Figure 4.2. It is often useful to add some

color coding of data values to aid in identifying patterns. In the lower half of Figure 4.2, darker

shading was used for larger data values. The small slightly elevated region near 40 East and 20

North stands out more clearly when the shading is added.

If the posting plot reveals systematic spatial trends in the survey unit, the cause would need to be

investigated. In some cases, such trends could be due to residual radioactivity, but may also be

due to an inhomogeneous survey unit background. Other diagnostic tools for examining spatial

data trends may be found in EPA Report QA/G-9 (1996). Geostatistical tools may also be useful

in some cases (EPA 230/02-89-042, 1989a).

A frequency plot (or a histogram) is a useful tool for examining the general shape of a data

distribution. This plot is a bar chart of the number of data points within a certain range of values.

The frequency plot will reveal any obvious departures from symmetry, such as skewness or

bimodality (two peaks), in the data distributions for the survey unit or reference area. Skewness

or other asymmetry can impact the accuracy of the statistical tests. A data transformation (e.g.,

taking the logs of the data) can sometimes be used to make the distribution more symmetric. The

statistical tests could then be performed on the transformed data. The interpretation of the results,

however, can be more complex, since the quantity being tested is also transformed. For example,

the mean of log-transformed data is the log of the geometric mean of the data, not the log of the

arithmetic mean of the data.

The presence of two peaks in the survey unit frequency plot may indicate the existence of

isolated areas of residual radioactivity. In some cases it may be possible to determine an

appropriate background for the survey unit using this information. The interpretation of the data

for this purpose will generally be highly dependent on site-specific considerations and should

only be pursued after consultation with the responsible regulatory agency.

The presence of two peaks in the reference area frequency plot may indicate a mixture of

background concentration distributions due to different soil types, construction materials, etc.

The greater variability in the data due to the presence of such a mixture will reduce the power of

the statistical tests to detect an adequately remediated survey unit. These situations should be

avoided whenever possible by carefully matching the reference areas to the survey units, and

choosing survey units with homogeneous backgrounds.

A major concern in constructing a histogram or frequency plot is the bin width, i.e. the range of

concentration values over which the data are grouped and counted. If the bin width is too narrow,

there will be too much spurious detail in the plot. If the bin width is too wide, too much detail is

lost. A useful rule of thumb is to calculate the bin Width by rounding down the quantity 3.5sn" 3

,

where n is the number of data points, and s is the sample standard deviation (Scott, 1979). An

example is shown in Figure 4.3 using the example survey unit data. In this example,

NUREG-1505 4-8
ANALYSIS

3.5sn'"' = 3.5(0.46)(90)"'n = 3.5(0.32)(0.22) = 0.354, which was rounded down to 0.3. The

resulting histogram is shown as Figure 4.3a. For comparison, histograms constructed using bin

widths of 0.2 (Figure 4.3b) and 0.1 (Figure 4.3c) are also shown.

Figure 4.2 Example of a Posting Plot

4-9 NUREG-1505
ANALYSIS

15-

10 I (c)

I I I I 5

0 III -[ 141 40 CO 4- N V eq CO CM C! -0 40 Cc

o; C C; V: V: VZ ~-r C4 C4 C4

20-

10-

(" ?-* W- t-C V)0 V- 0

C) 0 40 0 V - W. V- %- W 4 N4 4 N4

(a)

201

IU

5

0 IL" -1. ý, . JMIL0

r.-

40 I- V N N

Figure 4.3 Frequency Plots of Example Survey Unit Data

with bin width 0.1 (c-top) and 0.2 (b-middle) and 0.3 (a-bottom)

A ranked data plot indicates the amount of data falling within a given range of values. The first

step in constructing a ranked data plot is to sort the data in increasing order. The data are then

assigned the number corresponding to their position in the list. The ranking of the example data

from Table 4.1 is shown in Table 4.3.

NUREG-1505 4-10
ANALYSIS

Table 4.3 Ranks of the Example Data

F - Reference Area ______I___I___-__ Survey Unit

Rank Data Rank Data Rank Data Rank Data Rank Data Rank Data

1 0.5 31 0.9 61 1.1 __ 1 0.4 31 0.9 61 1.4

2 0.6 32 0.9 62 1.1 2 0.5 32 0.9 62, 1.4

3 0.6 33 0.9 63 1.1 __ 3 0.5 33 0.9 63 1.4

4 0.6 34 0.9 64 1. 4 0.5 34 0.9 64 1.4

5 0.6 35 0.9 65 1.1 5 0.5 35 0.9 65 1.4

6 0.6 36 0.9 66 1.1 6 0.6 36 1.0 66 1.4

7 0.6 37 0.9 67 1.1 7 0.6 37 1.0 67 1.5

8 0.6 38 0.9 68 1.1 8 0.6 38 1.0 68 1.5

9 0.6 39 0.9 69 1.1 9 0.6 39 1.0 69 1.5

10 0.6 40 0.9 '70 1.2 10 0.6 40 1.0 70 1.5

11 0.7 41 0.9 71 1.2 11 0.7 41 1.0 71 1.5

12 0.7 42 0.9 72 1.2 12 0.7 42 1.0 72 1.5

13 0.7 43 0.9 73 1.2 13 0.7 43 1.0 73 1.5

14 0.7 44 0.9 74 1.2 14 0.7 44 1.0 74 1.5

15 0.7 45 1.0 75 1.2 15 0.7 45 1.0 75 1.6

16 0.7 46 1.0 76 1.2 16 0.7 46 1.1 76 1.6

17 0.7 47 1.0 77 1.3 17 0.8 47 1.1 77 1.6

18 0.7 48 1.0 78 1.3 18 0.8 48 1.1 78 1.6

19 0.8 49 1.0 79 1.3 19 0.8 49 1.1 79 1.6

20 0.8 50 .1.0 80 1.4 20 0.8 50 1.1 80 1.7

21 0.8 51 1.0 81 1.5 21 0.8 51 1.2 81 1.7

22 0.8 52 1.0 82 1.5 __ 22 0.8 52 1.2 82 1.7

23 0.8 53 1.0 83 1.5 23 0.8 53 1.2 83 1.8

24 0.8 54 1.0 84 1.5 24 0.8 54 1.2 84 1.9

25 0.8 55 1.0 85 1.5 25 0.8 55 1.2 85 1.9

26 0.8 56 1.0 86 1.6 26 0.8 56 -1.3 86 1.9

27 0.8 57 1.0 87 1.7 27 0.8 57 1.3 87 2.0

28 0.8 58 1.1 88 1.8 28 0.8 58 1.3 88 2.1

29 0.9 59 1.1 89 1.9 29 0.8 59 11.3 189 2.5

30 0.9 60 1.1 90 1.9 _ _L_ 3 . 0 13 9 .

The ranked data plots for this data are shown in Figure 4.4 and Figure 4.5. A small amount of

data in a range will result in a large slope. A large amount of data in a range of values will result

in a flatter slope. A sharp rise near the bottom or the top is an indication of asymmetry. In Figure

4.4, the is an indication of some slight asymmetry in the reference area data. There is stronger

evidence of asymmetry in the survey unit data in Figure 4.5.

4-11 4-11 NUREG-1505
ANALYSIS

Figure 4.4 Ranked Data Plot for the Example Reference Area Data

0 10 20 30 40 50 60

Ranks

70 80 90

Figure 4.5 Ranked Data Plot for the Example Survey Unit Data

NUREG-1505 4-12
ANALYSIS

A Quantile plot is similar to a ranked data plot. It is constructed by ranking the data from

smallest to largest, and simply plotting the data against the quantity: (rank-0.5)/(number of data

points) rather than against the ranks. In this way, the percentage of data in various concentration

ranges is easily found.

A useful aid to interpreting a ranked data or quantile plot is the addition of boxes containing the

middle 50% and middle 75% of the data. These are shown as the dashed lines in Figure 4.4. The

50% box has its upper right comer at the 75th percentile and its lower left comer at the 25th

percentile. These points are also called the quartiles. For the example survey unit data, these are

0.8 and 1.5, respectively, as indicated by the inner dashed box. They bracket the middle half of

the data values. The 75% box has its upper right comer at the 87.5th percentile and its lower left

comer at the 12.5th percentile. A sharp increase within the 50% box can indicate two or more

modes in the data. Outside the 75% box, sharp increases can indicate outliers. The median (50th

percentile) is indicated by the heavy solid line at the value 1.0, and can be used as an aid to

judging the symmetry of the data distribution.

A Quantile-Quantile plot is valuable because it provides a direct visual comparison of the two

data sets. If the two data distributions differ only in location (e.g., mean) or scale (e.g., standard

deviation), the points will lie on a straight line. If the two data distributions being compared are

identical, all of the plotted points will lie on the line Y = X. Any deviations from this would point

to possible differences in these distributions. A Quantile-Quantile plot can be constructed to

compare the distribution of the survey unit data with the distribution of the reference area data. If

the number of data points is the same in both sets, the construction of the Quantile-Quantile plot

is straightforward. This has already been done for the example data in Table 4.3. Simply plot

each pair of measurements matched with the same rank, i.e. the survey unit measurement, Y, with

rank R is plotted against the reference area measurement, X, with rank R. If the number of data

points in the survey unit and reference area are not equal, the construction of the Quantile -

Quantile plot will involve some numerical adjustments of the ranks. This and other useful

techniques for exploratory data analysis are discussed in EPA QA/G-9 (1996).

The Quantile-Quantile plot for the example data is shown in Figure 4.6. The middle data point

plots the median of the survey unit data against the median of the reference area data. That this

point lies above the line Y = X, shows that the median of Y is larger than the median of X.

Indeed, the most of the points lie above the line Y = X in the region of the plot beyond a

concentration value of about one. This is a sensitive indication that the distribution of the survey

unit data is shifted toward values higher than the reference area distribution. As with the quantile

plot, the addition of boxes containing the middle 50% and middle 75% of the data can be a useful

aid to interpreting a quantile-quantile plot.

4.3 Select the Statistical Test

An overview of the statistical considerations important for final status surveys appears in Section

2.3, 3.7, and 3.8. The statistical tests recommended in this report for final status surveys are

discussed in Section 2.4. The detailed instructions for applying these tests, with examples, appear

in Chapters 5, 6, and 7."

4-13 NUREG-1505
ANALYSIS

I16 ........... •

0*

0 I I *

* I a

I.I

00.5 1 1.5 2 2.53

Reference Area

Figure 4.6 Example Quantile-Quantile Plot

The nonparametric statistical tests in this report are described as one-sample (Sign) and twosample (WRS, Quantile) tests. Their application will depend upon the specific fadionuclides

under consideration, the concentration or surface activity limits for these radionuclides, and the

comparison to background levels in the surrounding environment. Application of these

techniques will also depend upon whether a gross dose or count rate survey is employed instead

of spectrometric measurements for individual nuclides.

The one-sample tests are appropriate when there is no need to compare the survey unit with a

reference area. The one-sample statistical test (Sign test) described in Chapter 5 can be used if

the contaminant is not present in background and radionuclide-specific measurements are made.

The one-sample test may also be used if the contaminant is present at such a small fraction of the

DCGLw value as to be considered insignificant. In this case no provision for background

concentrations of the radionuclide is made. Thus, the total concentration of the radionuclide is

compared to the release criterion. This option should only be.used if it is expected that ignoring

the background concentration will not sighificantly affect the decision on whether or not the

survey unit meets the release criterion. The advantage of ignoring a small background

contribution is that no reference area is needed. This can simplify the finhl status survey

considerably.

NUREG-1505 4-14
ANALYSIS

The two-sample WRS test (discussed in Chapter 6) should be used when the radionuclide of

concern appears in background or if measurements are used that are not radionuclide specific.

The two-sample Quantile test discussed in Chapter 7 is used only when the null hypothesis of

Scenario B is chosen.

The statistical tests recommended in this report are listed in Table 4.4. In every case, these tests

are supplemented by the elevated measurements comparison (cf. Sections 2.6, 3.7.2, 3.8.2 and

Chapter 8). Other statistical tests may be used provided that the data are consistent with the

assumptions underlying their use, as discussed in the next section. The nonparametric tests

generally involve fewer assumptions than their parametric equivalents. For example, the

Student's t test may be used if the data distribution is consistent with the assumption of

normality.. If the data do not exhibit a normal distribution, the nonparametric tests will generally

produce smaller decision error rates.

Table 4.4 Recommended Statistical Tests

Scenario Reference Area Test

A Yes Wilcoxon Rank Sum

A No Sign

B Yes Wilcoxon Rank Sum, Quantile

B No Sign

4.4 Verify the Assumptions of the Statistical Test

An evaluation should be made to determine that the data are consistent with the underlying

assumptions of the statistical testing procedures used. Certain departures from these

assumptions may be acceptable when given the actual data and other information about the study.

Much of the information gained in the preliminary data review (Section 4.2) is directly applicable

to verifying the assumptions of the statistical tests, and is a major reason for emphasizing their

,use.

A statistical test is called robust if it relatively insensitive to departures from its underlying

assumptions. The nonparametric procedures described in this report were chosen because they

are robust for the problem of testing the value of mean concentrations of residual radioactivity in

a survey unit. In cases where the data distributions are extremely skewed, these tests may not

detect limited areas with concentration much higher than the, average in the survey unit. This is

one reason for supplementing these tests with the elevated measurement comparison.

The nonparametric tests of Chapters 5, 6 and 7 assume that the data from the reference area or

survey unit consist of independent samples from each distribution. The WRS test assumes that

the reference area and survey unit data distributions are the same except for a possible shift in the

mean.

4-15 NUREG-1505
ANALYSIS

Spatial dependencies that potentially affect the assumptions can be assessed using the posting

plots. More sophisticated tools for determining the extent of spatial dependencies are also

available (e.g., EPA QA/G-9, 1996). These methods tend to be complex and are best used with

guidance from a professional statistician.

Asymmetry in the data can be diagnosed with a histogram or a ranked data plot. Data

transformations can sometimes be used to minimize the effects of asymmetry.

One of the primary advantages of the nonparametric tests used in this report is that they involve

fewer assumptions about the data than their parametric counterparts. If parametric tests are used,

(e.g., Student's t-test), then any additional assumptions made in using them should be verified

(e.g., testing for normality). These issues are discussed in detail in EPA QA/G-9 (1996).

Some alternative tests that may be considered in certain situations are discussed in Chapter 14.

For example, if the data are symmetric, the one-sample WSR test is generally more powerful

than the Sign test.

Table 4.5 Methods for Checking the Assumptions of Statistical Tests

Assumption Diagnostic

Spatial Independence Posting Plot

Symmetry Histogram, Quantile Plot, Skewness

Data Variance Sample Standard Deviation, Kurtosis

Power is Adequate Retrospective Power Chart

4.5 Draw Conclusions From the Data

Perform the calculations required for the statistical tests and document the inferences drawn as a

result of these calculations. The specific details for conducting the statistical tests are given in

Chapters 5, 6 and 7.

In each survey unit, there are two types of measurements made: (1)'direct measurements or

samples at discrete locations and (2) scans. The statistical tests are only applied to the

measurements made at discrete locations. When the data clearly show that a survey unit meets or

exceeds the release criterion, the result is often obvious without performing the formal statistical

analysis. Table 2.3 in Section 2.5 discussed those circumstances where a conclusion can be

drawn from a simple examination of the data.

Sections 2.5.6 and 2.5.7 discuss the elevated measurement comparison (EMC) and the

investigation levels that flag a locations for further study in order to determine whether the

survey unit meets or exceeds the release criterion.

NUREG-1505 4-16
ANALYSIS

This report has been fairly explicit about the steps that should be talken to show that a survey unit

meets release criteria. Less has been said about the procedures that should be used if at any point

the survey,,unit fails. This is primarily because there are many different ways that a survey unit

may fail the final status survey. The overall level of residual radioactivity may not pass the

nonparametric statistical tests. Further investigation following the elevated measurement

comparison may show that there is a large enough area with a concentration too high to meet the

dose criterion. Investigation levels may have been flagged during scanning that indicate

unexpected levels of residual radioactivity for the survey unit classification. It is impossible to

enumerate all of the possible reasons for failure, their causes, and their remedies.

When a survey unit fails the release criterion, the first step is to review and confirm the data that

led to the decision. Once this is done, the extent of the residual radioactivity is that caused the

failure should be determined. Once the cause of failure has been remediated, determine the

additional data, if any, needed to document that the survey unit meets the release criterion.

For example, a Class 2 survey unit passes the nonparametric statistical tests, but has several

measurements on the sampling grid that exceed the DCGLw. This is unexpected in a Class 2 area,

and according to Table 2.4, these measurements are flagged for further investigation. Additional

sampling confirms that there are several areas where the concentration exceeds the DCGL w This

indicates that the survey unit was mis-classified. However, the scanning technique that was used

was sufficient to detect'residual radioactivity at the DCGLEMC calculated for the sample grid. No

areas exceeding the DCGL'EMC where found. Thus, the only difference between the final status

survey actually done, and that which would be required for Class 1, is that the scanning may not

have covered 100% of the survey unit area. In this case, it would be reasonable to simply increase

the scan coverage to 100%. If no areas exceeding the DCGLEMc are found, the survey unit has, in

effect, met the release criteria as a Class 1 survey'unit.

A second example might be a Class 1 Survey unit which passes the nonparametric statistical

tests, but which contains some ar'eas that were flagged for investigation during scanning. Further

investigation, sampling and analysis indicates one area is truly elevated. This area has a

concentration that exceeds the DCGLw by a factor greater than the area factor calculated for its

actual size. This area is remediated, and remediation control sampling shows that the residual

radioactivity was removed, and no other areas were contaminated with removed material. It may

be reasonable in that case, to simply document the original survey and the additional remediation

data. It is not clear that further final status survey data would provide any useful information.

As a last example, consider a Class 1 area which fails the nonparametric statistical tests.

Confirmatory data indicates that the average concentration in the survey unit does exceed the

DCGLw substantially over a majority of its area. There would appear to be little alternative to

remediation of the entire survey unit, followed by another final status survey.

These examples are meant to illustrative of the actions that may be necessary to secure the

release of a survey unit that has initially failed to meet the release criterion. The DQO process

should be revisited so that a plan can be made for attaining the original objective: to safely

release the survey unit by showing that it meets the release criteria. Whatever data is necessary to

meet this objective will be in addition to the final status survey data already in hand. All of the

data, and only the data, necessary to meet the objective should be required.

4-17 NUREG-1505

5 SIGN TEST: CONTAMINANT NOT PRESENT IN BACKGROUND

The statistical test discussed in this section is used to compare each survey unit directly with the

applicable release criterion. With only the set of survey unit measurements being analyzed, the

Sign test used here is called a one-sample test. This section applies if

(1) radionuclide-specific measurements are made to determine the concentrations, and

(2) the background concentration of the radionuclide is negligible.

Otherwise, the methods of Chapter 6 and 7 should be used. Together the above conditions

eliminate the need for a reference. The residual radioactivity concentrations in the survey unit are

compared directly to the DCGLw value. The background concentration of the radionuclide need

not be zero, but this background amount will be included with the residual radioactivity when

analyzing the survey results. The amount that is considered negligible depends on the fraction of

the DCGLw that it represents, and how much residual radioactivity is actually in the survey unit.

The risk of the survey unit failing because the background concentration of the radionuclide is

included with the residual radioactivity total should be weighed against the savings obtained by

not having to make reference area measurements. Sites need not be contiguous areas, however

the statistical test are generally applied to individual survey units that cover contiguous areas.

The Sign test is designed to detect uniform failure of remedial action throughout the survey unit.

This test does not assume that the data follow any particular distribution, such as normal or

log-normal. In addition to the Sign Test, the DCGLEMC for the Elevated Measurement

Comparison (EMC)-described in Chapter 8-is compared to each measurement to ensure none

exceeds the DCGLEMC. If a measurement exceeds this DCGLEMC, then additional investigation is

recommended-at least locally-to determine the actual areal extent of the elevated

concentration.

5.1 Introduction

The use of the Sign test in Scenario A and Scenario B is described in the next two sections,

illustrated with example data. The same data will be used in both cases. We consider a survey

unit that has been remediated, but may have some residual radioactivity. The DCGLwfor the

radionuclide in question has been determined to be 15.9. (The particular radionuclide and units of

measurement are irrelevant to the example, and will be left arbitrary.) From data collected during

the remediation, it is estimated that the standard deviation of the measurements in the survey unit

made during the final status survey should be about 3.3.

During the DQO process it was agreed that the decision error rates should be both set equal to

0.05 initially, and determine if a reasonable survey design cduld meet these. It was estimated that

the costs of additional remediation would be moderate down to concentrations of about 11.5, but

would rise sharply below that. On the basis of these considerations, a chart of the desired

probability that the survey unit passes was developed and is shown in Figure 5.1.

5-1 NUREG-1505
SIGN TEST

The lower bound of the gray region is 11.5, and the DCGLw is 15.9, so A = 15.9-11.5 = 4.4.

Since a is estimated at 3.3, A/( = 4.4/3.3 = 1.3. From Table 3.2, for a = f3 = 0.05, 21 samples are

required.

U, 100

a'

In

' 806.

C

S60

= 40

20

1.0 I I I

8 10 12 14 16 18 20

True Concentration in Survey Unit

Figure 5.1 Desired Probability That.the Survey Unit Passes

The data were taken on a triangular grid. The posting plot is shown in Figure 5.2. It is clear from

this plot that there is residual radioactivity that is higher near the center of the survey unit and

that diminishes as the survey unit boundary is approached. It is near zero on the west side.

80 N

70 N

60 N

50 N

40 N

30 N

20 N

1ON

ON

-1.1

0.4

0.2

2.0

4.7

16.21

19.5 12.9

23.2

15.5

21.4

26.3

23.6

14.9

17.4

28.7

10.4

28.4

10.8

18.5

30 E 40 E

5.8

OE 1OE 20 E

Figure 5.2 Posting Plot of Survey Unit Data

NUREG-1505 5-2
SIGN TEST

The one reported negative value stands out. Negative values can occur whenever an instrument,

analysis or blank background is subtracted to obtain a net reading for a sample ', which is then

converted to a concentration.

Summary statistics for these data are shown in Table 5.1.

Table 5.1 Summary Statistics for Example Data of Figure 5.2

Mean 14.3

Standard Error 2.07

Median 15.5

Standard Deviation 9.5

Sample Variance 90.0

Kurtosis -1.09

Skewness -0.18

Range 29.8

Minimum -1.1

Maximum 28.7

Count 21

The mean is 14.3, which in this case is actually less than the median, 15.5. This is the influence

of the few low values on the west part of the survey unit, and is also the cause for the slight

negative skewness. The range, 29.8, is only about three times the standard deviation, 9.5. This

standard deviation, however, is three times larger than the value of 3.3 assumed in the survey

design. If the null hypothesis is accepted, the effect of this higher standard deviation on the

power of the test should be investigated. Recall that lower power in Scenario A means that more

survey units with concentrations that are actually lower than the DCGLw are apt to fail the test.

In Scenario B, lower power means that more survey units with concentrations greater than the

DCGLw are apt to pass the test. A retrospective power curve, calculated as described in Chapter

10, can be used to decide if the error rates achieved are acceptable. A retrospective power

analysis is not necessary when the null hypothesis is rejected, since the Type I error rate, ax, is

fixed at the design value when the critical value for the test is detehnined. However, taking

additional samples to increase the power will increase the Type I error rate, unless provision for a

two-stage test is made during the survey design. This is discussed further in Section 14.2.

A histogram of the data is shown in Figure 5.3. Except for the one negative value in the "zero"

bin, the distribution appears reasonably symmetric.

(I For samples with no residual radioactivity, negative values should occur about 50% of the time. It does

not imply the existence of negative concentrations. Due to random fluctuations, the measurement merely happened

to be lower during the sample analysis than it was during the background determination.

5-3 NUREG-1505
SIGN TEST

10

9

8

7

6

5

4

3

2

1

0

.4

l0ll I

0 10 20 30 40

Figure 5.3 Histogram of Survey Unit Data

Further insight into the distribution of the data can be obtained using a ranked data plot. Table

5.2 shows the survey unit data ranked in increasing order. From this table it is already apparent

that there is a large gap between the 6th and 7th ranked values. This is even clearer in the ranked

data plot of Figure 5.4. It seems that there is a mixture of two distributions' in the survey unit

data. One is a distribution of low values near the western edge of the survey unit, and one of

higher values in the rest of the survey unit. This partially explains the negative skewness and the

large standard deviation.

Table 5.2 Ranked Data for Example of Figure 5.2

Rank 1 2 3 4 5 6 7

Measurement -1.1 0.2 0.4 2 4.7 5.8 10.4

Rank 8 9 10 11 12 13 14

Measurement 10.8 12.9 14.9 15.5 16.2 17.4 18.5

Rank 15 16 17 18 19 20 21

Measurement 19.5 21.4 23.2 23.6 26.3 28.4 28.7

NUREG-1505 5-4
SIGN TEST

30

25

.o 20

C)

0

-5

0 5 10

Rank

15 20

Figure 5.4 Ranked Data Plot of Survey Unit Data

5.2 Applying the Sign Test: Scenario A

The hypothesis tested by the Sign test under Scenario A is:

Null Hypothesis:

H0: The median concentration of residual radioactivity in the survey unit is greater than the

DCGLw.

versus

Alternative Hypothesis:

H.: The median concentration of residual radioactivity in the survey unit is less than the LBGR.

In order to use the one-sample Sign test, background concentrations of the radionuclide of

concern are considered to be either zero or insignificant in comparison to the DCGLw Thus,

there is no reference to background in statement of the null and alternative hypothesis. The null

hypothesis is assumed to be true unless the statistical test indicates that it should be rejected in

favor of the alternative. The parameter of interest is the mean concentration. The median is equal

to the mean when the measurement distribution is symmetric, and is an approximation otherwise.

575 NUREG-1505
SIGN TEST

The null hypothesis states that the probability of a measurement less than the DCGLwis less than

one-half, i.e., the 50th percentile (or median) is greater than the DCGL, The median is the

concentration that would be exceeded by 50% of the measurements. Note that some individual

survey unit measurements may exceed the DCGLw even when the survey unit as a whole meets

the release criterion. In fact, a survey unit that averages close to the DCGLwmight have almost

half of its individual measurements greater than the DCGLw. Such a survey unit may still meet

the release criterion.

The hypothesis specifies a release criterion in terms of a DCGLw which is calculated as described

in Section 3.3. The test should have sufficient power (1- 3, as specified in the DQOs) to detect

residual radioactivity concentrations at the Lower Boundary of the Gray Region (LBGR). If a is

the standard deviation of the measurements in the survey unit, then A/a expresses the size of the

shift (i.e., A = DCGLw-LBGR) as the nurfiber of standard deviations that would be considered

large for the distribution of measurements in the survey unit. The procedure for determining A/a

was given in Section 3.8.1.

The Sign test is applied as follows in Scenario A:

(1) List the survey unit measurements, XA,,i = 1, 2, 3..., N. If a measurement is listed as "less

than" a given value, insert that value for the measurement.

(2) Subtract each measurement, X,, from the DCGLw to obtain the differences:

Di =DCGLw - Xi, i = 1, 2, 3..., N.

(3) If any difference is exactly zero, discard it from the analysis, and reduce the sample size, N,

by the number of such zero measurements.

(4) Count the number of positive differences. The result is the test statistic S+. N6te that a

positive difference corresponds to a measurement below the DCGLw and contributes

evidence that the survey unit meets the release criterion.

(5) Large values of S+ indicate that the null hypothesis is false. The value of S+ is compared to

the table of critical values in Section A.3. If S+ is greater than the critical value, k, in that

table, the null hypothesis is rejected.

For the example survey unit data, the calculations are shown in Table 5.3. Notice that when the

data are ranked, it is really only necessary to observe the rank of the smallest measurement below

the DCGLw in order to determine S+. However, the differences D, = DCGL, - Xi= 15.9 - Xi are

also shown. The number of positive differences, S+ = 11. The critical value of S+ for the Sign

Test with a = 0.05 and N= 21 is 14. Since S+ is less than 14, the null hypothesis cannot be

rejected. The survey unit has failed the test.

NUREG-1505 5-6
SIGN TEST

Table 5.3 Calculations for Sign Test in Scenario A

Rank 7IMeasurement: DCGLw - Measurement J Sign

1 -1.1 14.8 +

2 .0.2 15.7 +

3 0.4 15.5 +

4 2.0 13.9 +

5 . 4.7 .. 11.2 +

6 " 5.8 10.1 +

7 10.4 5.5 +

8 10.8 5.1 +

9 12.9 3.0 +

10 14.9 1.0 +

11 15.5 0.4 +

12 16.2 -0.3

13 17.4 -1.5

14 18.5 -2.6

15 19.5 -3.6

16 .21.4 -5.5 -

17 23.2 -7.3 -

18 23.6 -7.7

.19 26.3 -10.4

20 28.4 -12.5 -

21,.: 28.7 . -12.8 ._ - _

5.3 Applying the Sign Test: Scenario B -0

The hypothesis tested by the Sign test under Scenario B is:

Null Hypothesis:

H.: The median concentration of residual radioactivity in the'survey unit is less than the LBGR.

versus

Alternative Hypothesis:

Ha:The median concentration of residual radioactivity in the survey unit is greater than the

DCGLw.

In order to use the one-sample Sign test, background concentrations of the radionuclide of

concern are considered to be either zero or insignificant in comparison to the LBGR. Thus,

5-7 NUREG-1505
SIGN TEST

there is no reference to background in statement of the null and alternative hypothesis. The null

hypothesis is assumed to be true unless the statistical test indicates that it should be rejected in

favor of the alternative. The Type I error rate, a, is the probability that a survey unit with residual

radioactivity at the LBGR will fail to be released. The power, 1 - P3, is the probability that a survey

unit with residual radioactivity at the DCGLw will fail to be released. The parameter of interest is

the mean concentration. The median is equal to the mean when the measurement distribution is

symmetric, and is an approximation otherwise.

The Sign test is carried out for Scenario B in a manner very similar to that for Scenario A:

(1) List the survey unit measurements, X,, i = 1, 2, 3..., N. If a measurement is listed as "less

than" a given value, insert that value for the measurement.

(2) Subtract the LBGR from each measurement, X,, to obtain the differences:

Di = Xi - LBGR, i = 1, 2, 3....,N.

(3) If any difference is exactly zero, discard it from the analysis, and reduce the sample size, N,

by the number of such zero measurements.

(4) Count the number of positive differences. The result is the test statistic S+. A positive

difference corresponds to a measurement above the LBGR and is evidence that the median

concentration of residual radioactivity survey unit may exceed it.

(5) Large values of S+ indicate that the null hypothesis is false. The value of S+ is compared to

the table of critical values in Section A.3. If S+ is greater than the critical value, k, in that

table, the null hypothesis is rejected.

For the example survey unit data, the calculations are shown in Table 5.4. Notice that when the

data are ranked, it is really only necessary to observe the rank of the smallest measurement below

the LBGR in order to determine S- = N - S+. However, the differences D, = X1 - LBGR =

Xi - 11.5 are also shown. The number of positive differences, S+ = 13. The critical value of S+

for the Sign Test with a = 0.05 and N = 21 is 14. Since S+ is less than 14, the null hypothesis

cannot be rejected. The survey unit has passed the test. However, it remains to determine the

power of the test. Since the observed standard deviation is much greater than that estimated for

the test design, it is likely that this survey unit passed simply because there was insufficient

power to detect residual radioactivity at the DCGLw. The power calculation for this example is

given in Section 10.1.

5.4 Interpretation of Test Results

Once the results of the statistical tests are obtained, the specific steps required to achieve site

release will depend on the procedures described in the regulatory guide. The following are

suggested considerations for the interpretation of the test results with respect to the release limit

established for the site.

NUREG-1505 5-8
SIGN TEST

Table 5.4 Calculations for Sign Test in Scenario B

Rank T Measurement Measurement -LBGR Sign

1 -1.1 -12.6 -

2 0.2 -11.3 -

3 0.4 -11.1 -

4 2.0 -9.5 -

5 4.7 -6.8 -

6 5.8 -5.7 -

7 i0.4 -1.1

8 10.8 -0.7

9 12.9 1.4 +

10 14.9 3.4 +

11 15.5 4.0. +

12 16.2 4.7 +

13 17.4 5.9 +

14 18.5 7.0 +

15 19.5 8.0 +

16 21.4 9.9 +

17 23.2 11.7 +

18 23.6 12.1 +

19 26.3 14.8 +

20 28.4 16.9 +

21 28.7 17.2 +

5.4.1 If the Null Hypothesis Is Not Rejected

Whenever the null hypothesis is not rejected, it is important to complete the analysis by

performing a retrospective power analysis for the test. In Scenario A, this will ensure that further

remediation is not required simply because the final status survey was not sensitive enough to

detect residual radioactivity below the LBGR. In Scenario B, this will ensure that a survey unit is

not released simply because the final status survey was not sensitive enough to detect residual

radioactivity above the guideline level. The power analysis may be performed as indicated in

Chapter 10, using the actual values of the number of measurements, N, and their observed

measurement standard deviation s in place of a. In some cases, a site specific simulation of the

retrospective power may be warranted when sufficient power cannot be demonstrated by any of

the other suggested methods.

5-9 NUREG-1505
SIGN TEST

5.4.2 If the Null Hypothesis Is Rejected

If the null hypothesis for the Sign test is rejected in Scenario A, it indicates that the residual

radioactivity in the survey unit is less than the DCGLw. However, it may still be necessary to

document the concentration of residual radioactivity. It is generally best to use the average

measured concentration for this purpose.

If the null hypothesis is rejected in Scenario B it indicates that the residual radioactivity in the

survey unit exceeds the LBGR. In this case it is important to determine not only the average

concentration of residual radioactivity in the survey unit, 8, but also whether this amount exceeds

the release criteria. When the data are normally distributed, the average concentration is generally

the best estimator for 8. However, when the data are not normally distributed, other estimators

are often better for the same reasons that nonparametric tests are often better than the

corresponding parametric tests. These methods are discussed by Lehmann and D'Abrera (1975).

When the estimate for 6 is below DCGLw, the survey unit may be judged sufficiently

remediated, shbject to ALARA consideiations. Otherwise, further remediation will generally be

required.

The treatment of data that are below the limit of detection will be an important component of

these calculations. Whenever possible, the actual results of the measurement should be reported,

with an associated total uncertainty that includes both random and systematic errors. Replacing

values below the MDC with the MDC value will cause 8 to be overestimated.

NUREG-1505 5-10
6 WILCOXON RANK SUM TEST:

CONTAMINANT PRESENT IN BACKGROUND

The statistical tests discussed in this section will be used to compare each survey unit with an

appropriately chosen, site-specific reference area. Each reference area should be chosen on the

basis of its similarity to the survey unit, as discussed in Section 2.2.7.

In Scenario A, the comparison of measurements from the reference area and survey unit is made

using the Wilcoxon Rank Sum (WRS) test (also called the Mann-Whitney test).

Under Scenario B, the comparison of measurements in the reference area and survey unit is made

using two nonparametric statistical tests: the WRS test and the Quantile test. The WRS and

Quantile tests are both used because each test detects different residual contamination patterns in

the survey units. Because two tests are used, the Type I error rate, ca, specified during the DQO

process is halved for the individual tests. The Quantile test is discussed in Chapter 7.

In addition to the statistical tests, the EMC is performed against each measurement to assure that

it does not exceed a specified investigation level. If any measurement in the remediated survey

unit exceeds the specified investigation level, then additional investigation is recommended, at

least locally, regardless of the outcome of the WRS or Quantile test.

The WRS test is most effective when residual radioactivity is uniformly present throughout a

survey unit. The test is designed to detect whether or not this activity exceeds the DCGLw. The

advantage of the nonparametric WRS test is that it does not assume that the data are normally or

log-normally distributed. The WRS test also allows for less than detectable measurements in

either the reference area or the survey unit. As a general rule, the WRS test can be used with up

to 40% less than detectable measurements in the reference area and the survey unit combined.

However, the use of less than values in data reporting is not encouraged. Wherever possible, the

actual result of a measurement, together with its uncertainty, should be reported.

6.1 Introduction

The use of the WRS test in Scenario A and Scenario B are described in the next two sections,

illustrated with example data. We consider a Class 2 survey consisting of interior drywall

surfaces, that may have some residual radioactivity. The DCGLw for the radionuclide in question

has been determined to be 160. (The particular radionuclide and units of measurement are

irrelevant to the example, and will be left arbitrary.) The background level is about 40. It is

estimated that the standard deviation of the measurements in the survey unit and the reference

area is about 6.

Since the DCGLw is so much larger than a, large sample sizes will not be needed even if the

acceptable error rates are set to low values. In this circumstance, the rule of thumb that A/a

should lie between one and three can be used to set the Lower Bound of the Gray Region

(LBGR). If, for example, A/a = 3, then A = 18, since a is estimated at 6. The Lower Bound of the

Gray Region is then LBGR = DCGLw - A = 160 - 18 = 142. If the decision error rates are both

6-j NUREG-1505
WRS TEST

set equal to 0.05 initially, then from Table 3.3, for a = P = 0.05 with A/a = 3, ten samples each

are required in the reference area and the survey unit.

Under Scenario B, the Type I error rate is halved, so cw = a/2 = 0.05/2 = 0.025. Then, again

from Table 3.3, for a = 0.025, P3 = 0.05, and A/a = 3, twelve samples each are required in the

reference area and the survey unit. The corresponding chart of the desired probability that the

survey unit passes is shown in Figure 6.1. Note that, although the probability that a survey unit at

the LBGR passes the WRS test in Scenario B is 97.5%, the overall probability of passing both

the WRS and Quantile tests is approximately 95%.

Since both scenarios are illustrated using the same set of data, the larger sample size will be used

for both.

100.

06o

'E

S40-

c o

U,

100 120 140 160 180 200

True Concentration Above Background In Survey Unit

Figure 6.1 Desired Probability That the Survey Unit Passes

The data were taken on a triangular grid('), and the posting plot is shown in Figure 6.2. For this

example the concentration of the radionuclide of interest is given in arbitrary units. It is clear

from this plot that there is residual radioactivity above background in the survey unit.

Summary statistics for these data are shown in Table 6.1. The mean and median are fairly close

in both the reference area and the survey unit. The standard deviations of the data are slightly

larger than estimated during the survey design, but the ratio A/a remains above 2, so the impact

on the power of the tests should not be severe. The range of the data is between 3 and 4 standard

(I)A random start systematic grid is used in Class 2 and 3 survey units primarily to limit the size of any

potential elevated areas. Since areas of elevated activity are not an issue in the reference areas, the measurement

locations can be either random or on a random start systematic grid.

NUREG-1505 6-2
WRS TEST

deviations, which is about right for these sample sizes (see Figure 4.1).

Reference Area

70 N 51

60 N 35 39

50 N 32

40 N 43 34

30 N 45

20 N 37 39

1ON 36

0 N M7 28

50 N

40 N

30 N

20 N

1ON

ON

Su Unit -y 

187 177

193 188

188 180

193 187 - 188 - 191

2071 1197 1 1

OE 10E 20E 30E

I

OE I1E 20E

Figure 6.2 Posting Plot of Reference Area and Survey Unit Data

Table 6.1 Summary Statistics for Example Data of Figure 6.2

Reference Area JJ Survey Unit

Mean 38.8 Mean 189.8

Median 38 Median 188

Std Dev 6.6 Std Dev 8.1

Kurtosis -0.4 Kurtosis 2.2

Skewness 0.3 Skewness 0.9

Range 23 Range 32

Minimum 28 Minimum 177

Maximum 51 Maximum 209

Count 12 Count 12

A histogram of the data is shown in Figure 6.3. The data distributions are fairly symmetric. The

survey unit and reference area distributions are clearly separated by an amount much larger than

the width of either. The difference in the medians is 188 - 38 = 150, and the difference in the

means is 189.8 - 38.8 = 151. Both of these values are very close to the DCGLW of 160. It is in

just such cases that the statistical tests are most useful in determining the significance of these

values.

6-3 NUREG-1505
WRS TEST

25 35 45 55 165 175 185 195 205,

Figure 6.3 Histograms of Reference Area and Survey Unit Data

Since there are equal numbers of data in the reference area and in the survey unit, a QuantileQuantile plot is easily constructed. Table 6.2 shows the reference area and survey unit data each

separately ranked in increasing order. The pairs of data from the reference area and the survey

unit with the same rank are plotted in Figure 6.4. This is the Quantile-Quantile plot. The position

of the medians is indicated by the solid bar and the central 50% of the data is enclosed in the

dashed box. The plot is fairly straight, and the slope is not greatly different from one; indicating:

the the shapes of the reference area and survey unit distributions are similar. However, the survey

unit distribution is shifted to values about 150 larger. Again, the significance of this relative to

the DCGLw is precisely what the WRS test is designed to determine.

Table 6.2 Ranked Data for Example of Figure 6.2

Rank jReference Areaj Rank J Survey Unit

1 28 1 177

2 32 2 180

3 34 3 187

4 35 4 187

5 36 5 188

6 37 6 188

7 39 7 188

8 39 8 191

9 43 9 193

10 45 10 193

11 47 11 197

12 51 1- 212

NUREG-1505 6-4
WRS TEST

210

C 205o

E

~200O

g 195

SE 190 , :• ~~*L ........ ,

>% 185

180

C)

.175

25 30 35 40 45 50 55

Reference Area Measurement

Figure 6.4 Quantile-Quantile Plot of Example Data

6.2 Applying the WRS Test: Scenario A

The hypothesis tested by the WRS test under Scenario A is:

Null Hypothesis:

H0: The median concentration in the survey unit exceeds that in the reference area by more than

the DCGLw.

versus

Alternative Hypothesis:

H.: The median concentration in the survey unit exceeds that in the reference area by less than

the LBGR.

The null hypothesis is assumed to be true unless the statistical test indicates that it should be

rejected in favor of the alternative. One assumes that any difference between the reference area

and survey unit concentration distributions is due to a shift in the survey unit concentrations to

higher values-i.e., due to the presence of residual radioactivity in addition to background. The

size of this shift is the difference in the mean concentrations. The median is equal to the mean

when the measurement distributions are symmetric, and is an approximation otherwise.

Note that some or all of the survey unit measurements may be larger than some reference area

measurements, while still meeting the release criterion. Indeed, some survey unit measurements

6-5 NUREG-1505
WRS TEST

may exceed some reference area measurements by more than the DCGLW. The result of the

hypothesis test determines whether or not the survey unit as a whole is deemed to meet the

release criterion. The EMC is used to screen individual measurements.

Assumptions underlying this test are that (1) the samples from the reference area and the survey

unit are independent random samples, and (2) each measurement is independent of every other

measurement-regardless of the set of samples from which it came.

The hypothesis specifies a release criterion in terms of a DCGLw which is calculated as

described in Section 3.3. The test should have sufficient power (1-3, as specified in the DQOs)

to detect residual radioactivity concentrations at the Lower Boundary of the Gray Region

(LBGR). If o is the standard deviation of the measurements in the survey unit, then A/0

expresses the size of the shift (i.e. A = DCGLw - LBGR) as the number of standard deviations

that would be considered large for the distribution of measurements in the survey unit. The

procedure for determining Alo was given in Section 3.8.1.

The WRS test is applied as follows under Scenario A:

(1) Obtain the adjusted reference area measurements, ZI , by adding the DCGLw to each

reference area measurement, Xj. Zi = Xi + DCGLw

(2) The m adjusted reference area sample measurements, Z,, and the n survey unit sample

measurements, Y,, are pooled and ranked in order of increasing size from I to N, where N =

m+n.

(3) If several measurements are tied (have the same value), they are all assigned the average rank

of that group of tied measurements.

(4) If there are t less than detectable values, they are all assigned the rank (t + 1)/2. If there is

more than one detection limit, all observations below the largest detection limit should be

treated as less than detected. If more than 40% of the data from either the reference area or

survey unit are less than detectable, the WRS test cannot be used. As stated previously, the

use of less than values in data reporting is not eqcouraged. Wherever possible, the actual

result of a measurement, together with its uncertainty, should be reported.

(5) Sum the ranks of the adjusted measurements from the reference area, W,. Note that since the

sum of the first N integers is N(N+1)12, one can equivalently sum the ranks of the

measurements from the survey unit, Ws, and compute W, = N(N + 1)/2 - W,.

(6) Compare W, with the critical value given in Table A.4 for the appropriate values of n, m, and

a. If W, is greater than the tabulated value, reject the hypothesis that the survey unit exceeds

the release criterion.

The data for the example are shown in column A of Table 6.3 In column B, the codeR was

inserted to denote a reference area measurement, and S to denote a survey unit measurement. In

column A, the data are simply listed as they were obtained. Column C contains the adjusted

data. The adjusted data are obtained by adding the DCGLw to the reference area measurements.

NUREG-1505 6-6
WRS TEST

The ranks of the adjusted data appear in Column D. They range from 1 to 24, since there is a

total of 12 + 12 measurements. The sum of all the ranks is N(N + 1)/2 = (24)(25)/2 = 300.

Column E contains only the ranks belonging to the adjusted reference area measurements. The

sum of the ranks of the adjusted reference area data is 199. From Table A.4, for cc = P = 0.05 and

n = m = 12, we find that the critical value is 179. Thus, the sum of the reference area ranks, 199,

is greater than the critical value, 179, and the null hypothesis that the survey unit concentrations

exceed the DCGLw is rejected. In Scenario A, the survey unit passes.

The analysis for the WRS test is very well suited to the use of a computer spreadsheet. The

spreadsheet formulas in Microsoft ExcelTM (1993) used for the example above are given in Table

6.4.

Table 6.3 WRS Test for Class 2 Interior Drywall Survey Unit

(Measurements from the reference area and the survey unit are denoted by R and S, respectively)

[ A B C D E

1 Data Area Adjusted Data Ranks Reference Area Ranks

2 47 R 207 22 22

3 28 R 188 6.5 6.5

4 36 R 196 15 15

5 37 R 197 16.5 16.5

6 39 R 199 18.5 18.5

7 45 R 205 21 21

8 43 R 203 20 20

9 34 R 194 13 13

10 32 R 192 10 10

11 35 R 195 14 14

12 39 R 199 18.5 18.5

13 51 R 211 24 24

14 209 S 209 23 _

16 197 S 197 16.5

16 188 S 188 16.5

17 191 S 191 9

18 193 S 193 11.5 _

19 187 S 187 3.5 _

20 188 S 188 6.5 _

21 180 S 180 2 _

22 193 S 193 11.5 _

23 188 S 188 6.5

24 187 S 187 3.5

25 177 S 177 1

26 Sum= 300 199

6-7 NUREG-1505
WRS TEST

Table 6.4 Spreadsheet Formulas Used in Table 6.3

I A IBI C D E

1 Data Area Adjusted Data Ranks Reference Ranks

2 47 R =IF(B2="R",A2+160,A2) "RANK(C2,$C$2:$C$25,1) =IF(B2='R",D2,0)

_+(COUNTIF($C$2:$C$25,02) - 1) / 2

3 28 R =IF(B3="R",A3+160,A3) =RANK(C3,$C$2:$C$25,1) =IF(B3="R*,D3,0)

I_ +(COUNTIF($C$2:$C$25,C3) - 1) / 2

4 36 R =IF(B4="R*,A4+160,A4) =RANK(C4,$C$2:$C$25,1) =IF(B4="R",D4,0)

+(COUNTI F($C$2:$C$25,C4) - 1) / 2

5 37 R =IF(B5="R",A5+160,A5) =RANK(C5,$C$2:$C$25,1) =IF(B5="R",D5,0)

+(COUNTIF($C$2:$C$25,C5) - 1) / 2

6 39 R =IF(B6="R",A6+160,A6) =RANK(C6,$C$2:$C$25,1) =IF(B6="R*,D6,0) +(COUNTIF($C$2:$C$25,C6) - 1) / 2

7 45 R =IF(B7=WR',A7+160,A7) =RANK(C7,$C$2:$C$25,1) =IF(B7="R",D7,0)

+(COUNTIF($C$2:$C$25,C7) - 1) / 2

8 43 R =IF(B8="R",A8+160,A8) =RANK(C8,$C$2:$C$25,1) =IF(B8="R",D8,O)

+(COUNTIF($C$2:$C$25,C8) - 1) / 2

9 34 R =IF(B9="R",A9+160,A9) =RANK(Cg,$C$2:$C$25,1) =IF(B9="R",D9,0)

+(COUNTIF($C$2:$C$25,cg) - 1)/2

10 32 R =IF(B10="R*,A10+160,A10) =RANK(C10,$C$2:$C$25,1) =IF(B10="R*,D10,0)

+(COUNTIF($C$2:$C$25,C10) - 1)/2

11 35 R =IF(B11="R",A11+160,A11) =RANK(C 1,$C$2:$C$25,1) =IF(B11="R",D11,0)

+(COUNTIF($C$2:$C$25,C1 1) - 1) / 2

12 39 R =IF(B1 2="R",A12+160,A12) =RANK(C1 2,$C$2:$C$25,1) =IF(B12="R",D12,0) +(COUNTIF($C$2:$C$25,C12) - 1) / 2

13 '51 R =IF(BI3="R",A13+160,A1 3) =RANK(C1 3,$C$2:$C$25,1) =IF(B13="R",D13,0)

+(COUNTIF($C$2:$C$25,C13) - 1) 12

14 209 S =IF(B1 4="R,A1 4+160,A1 4) =RANK(C1 4,$C$2:$C$25,1) =IF(B14=*R",D14,0)

+(COUNTIF($C$2:$C$25,C14) - 1) 12

15 197 S =IF(B1 5="R,A1 5+160,A1 5) =RANK(C1 5,$C$2:$C$25,1) =IF(B15="R",D15,0)

+(COUNTIF($C$2:$C$25,C15) - 1) / 2

16 188 S =IF(B1 6="R,A1 6+160,A1 6) =RANK(C1 6,$C$2:$C$25,1) =IF(B16=*R",D16,0)

+(COUNTIF($C$2:$C$25,C1 6) - 1) / 2

17 191 S =IF(B1 7='R,A1 7+160,A1 7) =RANK(C1 7,$C$2:$C$25,1) =IF(B17="R",D17,0)

+(COUNTIF($C$2:$C$25,C17) - 1) / 2

18 193 S =IF(B1 8=R",A1 8+1 60,A1 8) =RANK(CIB,$C$2:$C$25,1) =IF(B18="R",D18,0) +(COUNTIF($C$2:$C$25,018) - 1) / 2

19 187 S =IF(B1 9="R",A1 9+160,A1 9) =RANK(C1 9,$C$2:$C$25,1) =IF(B19="R",D19,0)

+(COUNTIF($C$2:$C$25,C19) - 1) / 2

20 188 S =IF(B20="R",A20+160,A20) =RANK(C20,$C$2:$C$25,1) =IF(B20=-"R",D20,0) +(COUNTIF($C$2:$C$25,C20) - 1) / 2

21 180 S =IF(B21="R",A21+160,A21) =RANK(C21,$C$2:$C$25,1) =IF(B21="R",D21,O) +(COUNTIF($C$2:$C$25,C21) - 1) / 2

22 193 S =IF(B22="R",A22+160,A22) =RANK(C22,$C$2:$C$25,1) =IF(B22=*R",D22,0)

+(COUNTIF($C$2:$C$25,C22) - 1) / 2

23 188 S =IF(B23="R",A23+160,A23) =RANK(C23,$C$2:$C$25,1) =IF(B23="R",D23,0)

+(COUNTIF($C$2:$C$25,C23) - 1) 12

24 187 S =IF(B24="R",A24+160,A24) =RANK(C24,$C$2:$C$25,1) =IF(B24="R",D24,0)

+(COUNTIF($C$2:$C$25,C24) - 1) / 2

25 177 S r=IF(B25="R",A25+160,A25) =RANK(C25,$C$2:$C$25,1) =IF(B25="R",D25,0)

+(COUNTIF($C$2:$C$25,C25) - 1) /2

26 Sum= 299 =SUM(E2:E25)

NUREG-1505 6-8
WRS TEST

Note that some spreadsheet programs assign the lowest rank in a group of ties to every member

of the group, rather than the average rank for the group. This can be corrected by adding to each

rank the value (g -1)/2, where g is the number of data points in the group. This is the function of

the COUNTIF statement in column D of Table 6.4.

6.3 Applying the WRS Test: Scenario B

Two tests are used in Scenario B to ensure that there is adequate power to detect a survey unit

that does not meet the release criterion. The concept of the statistical power of a test was

discussed in Section 2.3.2. The WRS test has more power than the Quantile test to detect

uniform failure of remedial action throughout the survey unit. The Quantile test has more power

than the WRS test to detect failure of remedial action in only a few areas within the survey unit.

These nonparametric tests do not require that the data be normally or log-normally distributed.

Measurements reported as non-detects may also be used with these tests, although this practice is

discouragedt 2). As a general rule, the WRS test can be used with up to 40% less than detectable

measurements in either the reference area or the survey unit. The Quantile test can be used even

when more than 50% of the measurements are below the limit of detection.

In addition, an elevated measurement comparison is conducted. This consists of determining if

any measurements in the remediated survey unit exceed a specified investigation level. If so,

then additional investigation is required, at least locally, regardless of the outcome of the WRS

and Quantile tests.

The hypothesis tested by the WRS test under Scenario B is:

Null Hypothesis:

HO: The difference in the median concentration of radioactivity in the survey unit and in the

reference area is less than the LBGR.

versus

Alternative Hypothesis:

H.: The difference in the median concentration of radioactivity in the survey unit and in the

reference area is greater than the DCGLw.

The Type I error rate, aw = 02, is the probability that a survey unit with residual radioactivity

(above background) at the LBGR will fail this test. The power, 1-3, is the probability that a

survey unit with residual radioactivity at the DCGLw will fail this test.

The WRS test is applied as follows under Scenario B:

(1) Obtain the adjusted survey unit measurements, Z1 , by subtracting the LBGR from each

(2) All actual measurement results (with an associated uncertainty) should be reported, even if they are

negative, so that unbiased estimates of averages can be calculated.

6-9 NUREG-1505
WRS TEST "

survey unit measurement, Y,. Zi = Y1 - LBGR

(2) The n adjusted survey unit measurements, Z , and the m reference area measurements, Xi,

are pooled and ranked in order of increasing size from 1 to N, where N = m+n.

(3) If several measurements are tied (have the same value), they are all assigned the average rank

of that group of tied measurements.

(4) If there are t less than detectable values, they are all assigned the rank (t + 1)/2. If there is

more than one detection limit, all observations below the largest detection limit should be

treated as less than detectable. If more than 40% of the data from either the reference area or

survey unit are less than detectable, the WRS test cannot be used.

(5) Sum the ranks of the adjusted measurements from the survey unit, W,. Note that since the

sum of the first N integers is N(N + 1)/2, one can equivalently sum the ranks of the

measurements from the reference area, W, and compute W, = N(N + 1)/2 - W,.

(6) Compare W, with the critical value given in Table A.4 for the appropriate values of n, m, and

az. If W, is greater than the tabulated value, reject the hypothesis that the difference in the

median concentration between the survey unit and the reference area is less than the LBGR.

The data for the example are shown in column A of Table 6.5 In column B, the code R was

inserted to denote a reference area measurement, and S to denote a survey unit measurement. In

column A, the data are simply listed as they were obtained. Column C contains the adjusted

data. The adjusted data are obtained by subtracting the LBGR from the survey unit

measurements.

The ranks of the adjusted data appear in Column D. They range from 1 to 24, since there is a

total of 12 + 12 measurements. The sum of all the ranks is N(N + 1)/2 = (24)(25)/2 = 300.

Column E contains only the ranks belonging to the adjusted survey unit measurements. The sum

of the ranks of the adjusted survey unit data is 194.5. From Table A.4, for cew = W2 = 0.025,

P = 0.05, and n = m = 12, we find that the critical value is 184. Thus, the sum of the adjusted

survey unit ranks, 194.5, is greater than the critical value, 184, and the null hypothesis that the

survey unit concentrations do not exceed LBGR is rejected. In Scenario B, the true survey unit

residual radioactivity is judged to be in excess of 142 above background.

The analysis for the WRS test is very well suited to the use of a computer spreadsheet. The

spreadsheet formulas in Microsoft ExcelTM (1993) used for the example above are given in

Table 6.6. Note that some spreadsheet programs assign the lowest rank in a group of ties to every

member of the group, rather than the average rank for the group. This can be corrected by adding

to each rank the value (g - 1)/2, where g is the number of data points in the group. This is the

function of the COUNTIF statement in column D of Table 6.6.

NUREG-1505 6-10
WRS TEST

Table 6.5 WRS Test Under Scenario B for Class 2 Interior Drywall Survey Unit

(Measurements from the reference area and the survey unit are denoted by R and S, respectively)

[ A B C D E

1 Data Area Adjusted Data Ranks Survey Unit Ranks

2 47 R 47 18

3 28 R 28 1 _

4 36 R 36 6 _

5 37 R 37 7

6 39 R 39 9.5

7 45 R 45 13

8 43 R 43 11

9 34 R 34 3

10 32 R 32 2

11 35 R 35 4.5

12 39 R 39 9.5

13 51 R 51 21

14 209 S 67 24 24

15 197 S 55 23 23

16 188 S 46 16 16

17 191 S 49 19 19

18 193 S 51 21 21

19 187 S 45 13 13

20 188 S 46 16 16

21 180 S 38 8 8

22 193 S 51 21 21

23 188 S 46 16 16

24 187 S 45 13 13

25 177 S 35 4.5 4.5

26 Sum= 300 194.5

6-11 NUREG-1505
WRS TEST

Table 6.6 Spreadsheet Formulas Used in Table 6.5

LIA BI C D IE_ __ _I

1 Data Area Adjusted Data Ranks Survey Unit

Ranks

2 47 R =IF(B2='S",A2-142,A2) =RANK(02,$C$2:$C$25,1) =IF(B2="S",D2,0)

+(COUNTI F($C$2:$C$25,C2) - 1) / 2

3 28 R =IF(B3='SA3-142,A3) =RANK(C3,$C$2:$C$25,1) =IF(B3="S",D3,0)

_________________+(COUNTIF($C$2:$C$25,C3) - 1) / 2

4 36 R =IF(B4="S",A4-142,A4) =RANK(C4,$C$2:$C$25,1) =IF(B4="S",D4,O)

+(COUNTIF($C$2:$C$25,C4) - 1) / 2

5 37 R =IF(B5="S',A5-142,A5) =RANK(C5,$C$2:$C$25,1) =IF(B5="S",D5,0)

I +(COUNTIF($C$2:$C$25,C5) - 1) 2

6 39 R =IF(B6="S",A6-142,A6) =RANK(C6,$C$2:$C$25,1) =IF(B6="S",D6,0)

+(COUNTIF($C$2:$C$25,C6) - 1)/ 2

7 45 R =IF(B7="S",A7-142,A7) =RANK(C7,$C$2:$C$25,1) =IF(B7="S",D7,O)

I +(COUNTIF($C$2:$C$25,C7) - 1) 2

8 43 R =IF(B8="S",A8-142,A8) =RANK(C8,$C$2:$C$25,1) =IF(B8="S",DB,0) ÷(COUNTIF($C$2:$C$25,C8) - 1) 2

9 34 R =IF(B9="S",A9-142,A9) =RANK(C9,$C$2:$C$25,1) =IF(B9="S",D9,0)

.+(COUNTIF($C$2:$C$25,C9) - 1) 2

10 32 R =IF(B1 0="S",A10-142,A10) =RANK(CIO,$C$2:$C$25,1) =IF(B1O="S",D10,0)

I_ +(COUNTIF($C$2:$C$25,C10) - 1)/ 2

11 35 R =IF(B11="S",A11-142,A11) =RANK(C 1,$C$2:$C$25,1) =IF(B11 ="S",D11,0)

_(COUNTIF($C$2:$C$25,C11) - 1) 2

12 39 R =IF(B12="S",A12-142,A12) =RANK(C12,$C$2:$C$25,1) =IF(B12="S",D12,0) +(COUNTIF($C$2:$C$25,C12) - 1) /2

13 51 R =IF(B13="S",A13-142,A13) =RANK(C13,$C$2:$C$25,1) =IF(B13="S",D13,0)

I +(COUNTIF($C$2:$C$25,C13) - 1) / 2

14 209 S =IF(B14="S",A14-142,A14) =RANK(C1 4,$C$2:$C$25,1) =IF(B14="S",D14,0)

__________________+(COUNTIF($C$2:$C$25,C1 4) - 1) /2

15 197 S =IF(B1 5="S",A1 5-142,A1 5) =RANK(C1 5,$C$2:$C$25,1) =IF(B15="S",D15,0)

+(COUNTIF($C$2:$C$25,Cl 5) - 1) / 2

16 188 S =IF(B16="S",A16-142,A16) =RANK(C1 6,$C$2:$C$25,1) =IF(B1 6="S",D16,0) +(COUNTIF($C$2:$C$25,C16) - 1) / 2

17 191 S =IF(B17="S",A17-142,A17) =RANK(C17,$C$2:$C$25,1) =IF(B17="S",D17,0) +(COUNTIF($C$2:$C$25,C17) - 1) / 2

18 193 S =1F(B18="S",A18-142,A18) =RANK(C18,$C$2:$C$25,1) =IF(B18--"S",D18,0)

,-+(COUNTIF($C$2:$C$25,C18) - 1) /2

19 187 S =IF(B19="S",A19-142,A19) =RANK(C19,$C$2:$C$25,1) =IF(B19="S",D19,0) +(COUNTIF($C$2:$C$25,C1 9) - 1) / 2

20 188 S =IF(B20="S",A20-142,A20) =RANK(C20,$C$2:$C$25,1) =IF(B20="S",D20,0)

I_ +(COUNTIF($C$2:$C$25,C20) - 1) / 2

21 180 S =IF(B21 ="S",A21-142,A21) =RANK(C21,$C$2:$C$25,1) =IF(B21="S",D21,0)

I_ +(COUNTIF($C$2:$C$25,C21) - 1) / 2

22 193 S =IF(B22="S",A22-142,A22) =RANK(C22,$C$2:$C$25,1) . =IF(B22="S",D22,0) +(COUNTIF($C$2:$C$25,C22) - 1) / 2

23 188 S =IF(B23="S",A23-142,A23) =RANK(C23,$C$2:$C$25,1) =IF(B23="S",D23,0)

I +(COUNTIF($C$2:$C$25,C23) - 1) / 2 r

24 187 S =IF(B24="S",A24-142,A24) -RANK(C24,$C$2:$C$25,1) =IF(B24="S",D24,0)

,+(COUNTIF($C$2:$C$25,C24) - 1) / 2. ____________

25 177 S =IF(B25-"S",A25-142,A25) =RANK(C25,$C$2:$C$25,1) =IF(B25="S",D25,0)

_+(COUNTI F($C$2:$C$25,C25) - 1) / 2

26 Sum= 299 • SUM(E2:E25)

NUREG-1505 6-12
WRS TEST

6.4 Interpretation of Test Results

Once the results of the statistical tests are obtained, the specific steps required to achieve site

release will depend on the procedures described in the regulatory guide. The following are

suggested considerations for the interpretation of the test results with respect to the release limit

established for the site.

6.4.1 If the Null Hypothesis Is Not Rejected

Whenever the null hypothesis is not rejected, it is important to complete the analysis by

performing a retrospective power analysis for the test. In Scenario A, this will ensure that further

remediation is not required simply because the final status survey was not sensitive enough to

detect the difference in mean radioactivity concentration between the survey unit and the

reference area when that difference is below the LBGR. In Scenario B, this will ensure that a

survey unit is not released simply because the final status survey was not sensitive enough to

detect the difference in mean radioactivity concentration between the survey unit and the

reference area when that difference is above the guideline level. The power analysis may be

performed as indicated in Chapter 10, using the actual values of the number of measurements, N,

and their observed measurement standard deviation s in place of a. In some cases, a site-specific

simulation of the retrospective power may be warranted when sufficient power cannot be

demonstrated by any of the other suggested methods. .

If the null hypothesis for the WRS test is not rejected in Scenario B, the Quantile test described

in Chapter 7 must also be performed.

6.4.2 If the Null Hypothesis Is Rejected

If the null hypothesis for the Sign test is rejected in Scenario A, it indicates that the residual

radioactivity in the survey unit is less than the DCGLw. However, it may still be necessary to

document the concentration of residual radioactivity. It is generally best to use the difference in

mean radioactivity concentration between the survey unit and the reference area for this purpose.

If the null hypothesis is rejected in Scenario B, it indicates that the residual radioactivity in the

survey unit exceeds the LBGR. In this case it is important to determine not only the difference in

mean radioactivity concentration between the survey unit and the reference area, 8, but also

whether this difference exceeds the release criteria. When the data are normally distributed, the

average concentration is generally the best estimator for 8. However, when the data are not

normally distributed, other estimators are often better for the same reasons that nonparametric

tests are often better than the corresponding parametric tests. These methods are discussed by

Lehmann and D'Abrera (1975). When the estimate for 8 is below DCOLw , the survey unit may

be judged sufficiently remediated, subject to ALARA considerations. Otherwise, further

remediation will generally be required.

6-13 NUREG-1505

7 QUANTILE TEST

The Quantile test was specifically developed to detect differences between the survey unit and

the reference area that consist of a shift, 8', to higher values in only a fraction, E, 0 < E < 1, of the

survey unit. It should be noted that, in general, this shift, 8', is not necessarily the same as the

shift 8 used for the WRS test. The Quantile test is only used in Scenario B. The Quantile test is

performed after the WRS test, if the null hypothesis for that test has not been rejected. Using the

Quantile test in tandem with the WRS test in Scenario B results in higher power to detect survey

units that have not been adequately remediated than either test has by itself.

7.1 Introduction

The specific hypothesis tested by the Quantile test (see Johnson et al., 1987; EPA 230-R-94-004,

1994) is:

Null Hypothesis:

H0: c = 0 or8' 6 LBGR

versus

Alternative Hypothesis:

Ha: E > 0 and 8' > LBGR

Simply put, the null hypothesis is that there is no residual radioactivity above the LBGR in any

part of the survey unit. The Quantile test is better at detecting situations in which only a portion,

E, of the survey unit contains excess residual radioactivity. The WRS test is better at detecting

situations in which any excess residual radioactivity is uniform across the entire survey unit.

7.2 Applying the Quantile Test

For the Quantile test, the appropriate page in Table A.7 is selected, according to the value Of ca =

xc/2.(') Find the nearest values of n, the number of measurements from the survey unit, and m,

the number of measurements from the reference area, that are tabulated. There are three numbers

associated with each tabulated pair of n and m values, namely r, k, and aQ.

The m measurements from the reference area and the n adjusted measurements from the survey

unit are pooled and ranked in order of increasing size from 1 to N, where N = m + n. This is the

same as steps (1) - (3) in Section 6.3 for the WRS test, and the same rankings can be used. If k

or more of the r largest measurements in the combined ranked data set are from the survey unit,

the null hypothesis is rejected For a survey unit that has failed the WRS test, it is not usually

necessary to also perform the Quantile test. It is done here as an illustration of the method.

Table 7.1 reproduces the data in Table 6.5, but with two added columns showing the sorted ranks

of the adjusted data, and the locition associated with each rank, i.e., R for reference area and S

for survey unit.

(1) Recall that since the Quantile test is performed in tandem with the WRS test, a Q=aw Ca/ 2, so that the

that the size of the two tests in tandem is approximately a = ao+atw

7-1 NUREG-1505
QUANTILE TEST

Table 7.1 Quantile Test Under Scenario B for Class 2 Interior Drywall Survey Unit

(Measurements from the reference area and the survey unit are denoted by R and S, respectively)

A B C D E I F G

Data Area Adjusted Ranks Survey Sorted Location

1" Data Unit Ranks Associated

Ranks With Sorted

Rank

2 47 R 47 18 - 1 R

3 28 R 28 1 - 2 R

4 36 R 36 6 - 3 R

5 37 R 37 7 - 4.5 R

6" 39 R 39 9.5 - 4.5 S

7 45 R 45 13 - 6 R

8 43 R 43 11 - 7 R

9 34 R 34 3 - 8 S

10 32 R 32 2 - 9.5 R

11 35 R 35 4.5 - 9.5 R

12 39 R 39 9.5 - 11 R

13 51 R 51 21 - 13 R

14 209 S 67 24 24 13 S

15 197 S 55 23 23 13 S

16 188 S 46 16 16 16 S

17 191 S 49 19 19 16 S

18 193 S 51 21 21 16 S

19 187 S 45 13 13 18 R

20 188 S 46 16 16 19 S

21 180 S 38 8 8 21 R

22 193 S 51 21 21 21 S

23 188 S 46 16 16 21 S

24 187 S 45 13 13 23 S

25 177 S 35 4.5 4.5 24 S

26 Sum = 300 194.5 1 1

On the second page of Table A.7, the closest entry to n = m = 12"is for n = m = 10. The values

of r = 7, k = 6 and 0•Q = 0.029 are found. Thus, the null hypothesis is rejected if 6 of the 7 lafgest

adjusted measurements come from the survey unit. From Table 7.1, we find that only 5 of the 7

largest adjusted measurements come from the survey unit.

The Quantile test as applied above yields only an approximate result. The values of n and m that

NUREG-1505 7-2
QUANTILE TEST

were used are close to, but not equal to, the actual values. The value of acQ will generally be

different from that listed in the table. It is prudent to check a few other entries that are near the

actual sample size. For n = m = 15, the values of r = 5, k = 5, and aQ = 0.021 are found. Thus,

the null hypothesis is rejected if all of the 5 largest adjusted measurements come from the survey unit. From Table 7.1, we find that only 4 of the 5 largest adjusted measurements come from the:

survey unit. For n = 15, m = 10 the values of r = 6, k = 6, and aQ = 0.028 are found. From

Table 7.1, we find that only 5 of the 6 largest adjusted measurements come from the survey unit.

For n = 10, m = 15, we have r = 6, k = 5, and aQ = 0.023. Since 5 of the 6 largest adjusted

measurements come from the survey unit, the null hypothesis would be rejected in this case. If

the results are ambiguous, the methods of the next section can be used to fine tune the test for the

sample sizes actually used..

The power of the Quantile test is more difficult to evaluate than that for the WRS test, since it

depends on the two parameters 6' and E. Setting specific values of these parameters in order to

define a specific alternative for evaluating the power would often be speculative at best. It will be

assumed that the power of the quantile test will be adequate for a range of values of 6' and e

when the sample size has been determined to assure adequate power for the WRS test.

If there are specific values of 6' and c that are identified as being of concern during the historical

site assessment, or prior surveys, power estimates can be obtained from tables in EPA 230-R-94-

004 (1994), which list the power for various combinations of E, 6'7o, aQo, m, n, r, and k. In those

tables, only cases where n = m are given. Since the power generally increases with sample size,

upper and lower bounds for the power can be estimated when m • n by consulting the entries for

sample sizes both equal to the smaller or larger of these numbers.

7.3 Calculation of ccQ for the Quantile Test

The Quantile test, as applied in Section 7.2 gives only an approximate result, since the values

used for n and m to find r and k were only approximately equal to the sample sizes actually used.

Therefore, the actual value of ao will be different from that listed in the table. Fortunately, it is

relatively easy to calculate the exact value of aQ for specific values of n, m, k, and r. The

number, k, out of the r largest measurements has a hypergeometric probability distribution when

the null hypothesis is true. The probability that k or more of the r largest measurements are from

the survey unit, when there is actually no residual radioactivity in the survey unit is:

Q = E ( ri~k ( n+m) i rk (7-1)

The symbol (n! is called a binomial coefficient. The symbol n!, called n factorial, (d 7) t (n-i) in!

is the product of the first n integers, n! = n(n - 1)(n - 2) ... (3)(2)(1). 0! is defined as equal to 1.

7-3 NUREG-1505
QUANTILE TEST

These calculations can be performed easily with any spreadsheet program that has the

hypergeometric function built in. Otherwise, the following approximation, correct to almost three

decimal places (Ling and Pratt, 1984), may be used:

1-'Q =(D(Z1_.) (7-2)

where

= 2 Vk(m - r + -(n - k.+ 1)(r - k + 1)

aV(n +m - 1)

4)(z) is the cumulative distribution function of a standard normal random variable tabulated in

Table A. 1. For the actual values of n and m, aQ can be calculated for different combinations of r

and k until a value sufficiently near the DQO is found.

Table 7.2 shows the calculations for the example data used in the previous section. For each of

the possible values of r considered, i.e., 5, 6, and 7, the values of aQ are calculated for each

possible value of k, from 0 up to r. The value of aLQ closest to the desired value of a/2 = 0.025

occurs for r = 5 and k = 5, where aQ = 0.0186. Using these values, and observing that only 4 of

the 5 largest measurements are from the survey unit, the null hypothesis is not rejected. The

survey unit passes the Quantile test, but it fails the WRS test (see Section 6.3). This may seem

paradoxical, but is actually consistent with the patterns of residual radioactivity each test is

designed to detect. This is discussed further in the next section.

Table 7.2 Example Calculation of tXQ for the Quantile Test

I A I BZ I C IDI E LL FL I G 111i1iV Ju I I K

1 r=5 r=6 r=7

2 k= Prob a k= Prob a k= Prob a

3 0 0.0186 1.0000 0 0.0069 1.0000 0 0.0023 1.000(

4 1 0.1398 0.9814 1 0.0706 0.9931 1 0.0320 0.997-

5 2 0.3416 0.8416 2 0.2427 0.9225 2 0.1510 0.965

6 3 0.3416 0.5000 3 0.3596 0.6798 3 0.3146 0.814

7 4 0.1398 0.1584 4 0.2427 0.3202 4 0.3146 0.500

8 5 0.0186 0.0186 5 0.0706 0.0775 5 0.1510 0.1859

9 6 0.0069 0.0069 6 0.0320 0.034

10 7 0.0023 0.002

11 mean k = 2.50 meank= 3.00 mean k= 3.50

12 std dev = 1.02 std dev = 1.08 1 std dev= 1.14

The spreadsheet formulas used for the example in Table 7.2 are shown in Table 7.3. In rows

1-12, only the formulas for columns A, B, and C are given, showing the calculations for r = 5,

provided the values of n and m are defined in the spreadsheet. The formulas for the other

columns are similar. One can get a feeling for the likelihood of the observed value of k by

calculating the mean and standard deviation for k when the null hypothesis is true. For n = m =

12, and r = 5, the expected value of k is 2.5 ± 1.0. Thus, the observed value, k= 4, is 1.5 standard

NUREG-1505 7-4
QUANTILE TEST

deviations above the mean. This is on the high end, but not quite high enough to reject the null

hypothesis.

Table 7.3 Spreadsheet Formulas Used in Table 7.2

A B C

1 r= 5

2 k= Prob a

3 0 =HYPGEOMDIST(A3, n, r, n+m) =1

4 1 =HYPGEOMDIST(A4, n, r, n+m) =1-SUM(B$3:B3)

5 2 =HYPGEOMDIST(A5, n, r, n+m) =1-SUM(B$3:B4)

6 3 =HYPGEOMDIST(A6, n, r, n+m) =1-SUM(B$3:B5)

7 4 =HYPGEOMDIST(A7, n, r, n+m) =1-SUM(B$3:B6)

8 5 =HYPGEOMDIST(A8, n, r, n+m) =1-SUM(B$3:7)

9

10

11 mean k= = (n',) I (m+n)

12 std dev = = sqrt( m*n*r'(m+n-1) I (m+n)A2)

7.4 Modified Example for the Quantile Test

It was noted in the previous section that for the example data of Table 7.1 under Scenario B, the

survey unit would fail the WRS test, yet pass the Quantile test. This occurred because the

residual radioactivity is more or less uniformly distributed across the survey unit. If the example

data is modified slightly, the result is different. Table 7.4 shows what would happen if 15 units of

residual radioactivity are subtracted from the first six survey unit measurements and added to the

last six survey unit measurements. The total residual radioactivity measured is unchanged. The

analysis shows, however, that the sum of the adjusted survey unit measurement ranks is now 178.

This is below the critical value of 184 from Section 6.3, and so the survey unit would pass the

WRS test. Now, however, all of the highest ranked five measurements are from the survey unit,

and so the survey unit would fail the Quantile test. This is because the spatial distribution of

residual radioactivity is not uniform over the survey unit, but is concentrated at higher values in

half of the survey unit. The purpose of using the two tests in tandem under Scenario B is to

discover survey units with residual radioactivity in excess of the LBGR. There would be no value

in using two tests if they always gave the same result.

7-5 NUREG-1505
QUANTILE TEST

Table 7.4 Quantile Test Under Scenario B: Modified Example Survey Unit

(Measurements from the reference area and the survey unit are denoted by R and S, respectively)

A B C D E F G

Data Area Adjusted Ranks Survey Sorted Location

Data Unit Ranks Associated

Ranks With Sorted

Rank

2 47 R 47 16 - 1 R

3 28 R 28 1 - 2 S

4 36 R 36 8.5 - 3 S

5 37 R 37 10 -4 R

6 39 R 39 11.5 - 5.5 R

7 45 R 45 15 - 5.5 S.

8 43 R 43 14 - 7 R

9 34 R 34 5.5 - 8.5 R

10 32 R 32 4 - 8.5 S

11 35 R 35 7 - 10 R

12 39 R 39 11.5 - 11.5 R

13 51 R 51 18 - 11.5 R

14 194 S 52 19 19 13 S

15 182 S 40 13 13 14 R

16 173 S 31 3 3 15 R

17 176 S 34 5.5 5.5 16 R

18 178 S 36 8.5 8.5 17 S

19 172 S 30 2 2 18. R

20 203 S 61 22.5 22.5 19 S

21 195 S 53 20 20 20 S

22 208 S 66 24 24 21 S

23 203 S 61 22.5 22.5 22.5 S

24 202 S 60 21 21 22.5 S

25 192 S 50 17 17 24 S

26 Sum= 300 178 1

NUREG-1505 7-6
8 ELEVATED MEASUREMENT COMPARISON

As discussed in Section 2.6, an Elevated Measurement Comparison is performed by comparing

each measurement from the survey unit to the DCGLEmO If the survey unit is being compared to

a reference area, the net survey unit measurement is first obtained by subtracting the mean of the

reference area measurements. A net survey unit measurement that equals or exceeds the

DCGLEMc is an indication that a survey unit may contain residual radioactivity in excess of the

release criterion.

In addition to direct measurements or samples at discrete locations, parts of each survey unit will

also be scanned. For the quantitative measurements obtained at discrete locations, performing the

EMC is a straightforward comparison of two numerical values. Some sophisticated scanning

instrumentation is also capable providing quantitative results with a quality approaching those

from direct measurements or samples. Other scanning measurements, however, may be more

qualitatiye. In that case, action levels should be established for the scanning procedure so that

areas with concentrations that may exceed the DCGLEMc are marked for a quantitative

measurement..

8.1 Introduction

The Elevated Measurement Comparison (EMC) against measurements taken on a systematic grid

are discussed in Section 8.1. The use of the EMC during scans is discussed in Section 8.2. Area

factors are discussed in Section 8.3, and an example is given in Section 8.4.

The statistical tests may not fail a survey unit when there are only a very few high measurements.

The EMC isused so that unusually large measurements will receive proper attention regardless

of the outcome of those tests-and any area that may have the potential for significant dose

contributions will be identified. The EMC is intended to flag potential failures in the remediation

process, and cannot be used to determine whether or not a site meets the release criterion until

further investigation is done.

The derived concentration guideline level for the EMC is: DCGLEMc = (FgS(DCGLw), where

F.d is the area factor for the area of the systematic grid area used (see Section 3.5.4). Note that

DCGLEMc is an a priori limit, established both by the DCGLw and by the survey design (i.e., grid

spacing and scanning MDC). The true extent of an area of elevated activity can only be

determined after performing the survey and then taking additional measurements if an elevated

measurement is found. Upon the completion of further investigation, the a posteriori limit,

DCGLjy, = (Factuai)(DCGLiy), can be established using the value of the area factor, F, appropriate

for the actual measured area of elevated concentration. The area that is considered elevated is

that bounded by concentration measurements at or below the DCGLw.

If residual radioactivity is found in an isolated area of elevated activity-in addition to residual

radioactivity distributed relatively uniformly across the survey unit-the unity rule can be used to

ensure that the total dose is within the release criterion:

8-1 NUREG-1505
EMC

8 + (average concentration in elevated area - 8) < 1

DCGLW (area factor for elevated area)(DCGLw)

If there is more than one elevated area, a separate term should be included for each. As an

alternative to the unity rule, the dose or risk due to the actual residual radioactivity distribution

can be calculated if there is an appropriate exposure pathway model available.

The preceding discussion primarily concerns Class 1 survey units. Measurements exceeding

DCGLw in Class 2 or Class 3 areas may indicate survey unit mis-classification. Scanning

coverage requirements for Class 2 and Class 3 survey units are less stringent than for Class 1

survey units.

If the investigation levels of Section 2.6 are exceeded, an investigation should (1) assure that the

area of elevated activity discovered meets the release criterion and (2) provide reasonable

assurance that other undiscovered areas of elevated activity do not exist. If further investigation

determines that the survey unit was misclassified with regard to contamination potential, a

resurvey using the method appropriate for the new survey unit classification may be appropriate.

8.2 Comparison Against Individual Measurements

The DCGLmc is calculated on the basis of the grid area, since this is about the same as the

largest circular area that has some chance of being missed when sampling on the grid. Figure 8.1

shows both a square and a triangular sampling grid. On the square grid, with grid area L2, the

small circular area with diameter L has an area of tn(L2)2 = 0.785L2. A circle with area L2 would

have a radius of 0.564L. From Figure 3.8, a circle of that radius has only about a 10% chance of

being missed. The triangular grid is a little more efficient. The grid area is the rhombus formed

from two of the triangles, and has area of 0.866L2-. A circle with that area has a radius of

0.525L, and, from Figure 3.8, has less than a 5% chance of being missed. The significance of this

is that when no measurement exceeds the DCGLEMo it is unlikely that there are areas remaining

that could cause the release criterion to be exceeded. The survey is planned in anticipation of a

negative result, and provides a quantitative measure of risk when no elevated measurements are

found, as well as an objective, dose-based definition of what is considered elevated.

When a measurement is found to exceed the DCGLEMO there is more work to be done before the

question of compliance can be answered. An individual elevated measurement on a systematic

grid could conceivably represent an area three to four times as large as the systematic grid area

used to define the DCGLEMC. This is the area bounded by the nearest neighbors of the elevated

measurement location, as shown by the large circles in Figure 8.1. However, the elevated area

may also be smaller than the grid area. Since the allowable concentration generally increases as

the elevated area decreases in size, further investigation is necessary to determine both the actual

area and average concentration. The boundary of the elevated area is defined by concentration

measurements at or below the DCGLW. Once the actual elevated area is found, the corresponding

area factor, Factuai is calculated in order to determine the release criterion for the elevated area:

DCGLA = (F,,ctu•)(DCGLw).

NUREG-1505 8-2
EMC

Figure 8.1 Square (top) and Triangular (bottom) Sampling Grids and Grid Areas

(Circular elevated areas of radius LJ2 and L are shown for comparison)

The problem remaining is to determine whether or not the average concentration in the elevated

area meets the DCGLJ.. This is essentially the same problem as the original one of determining

whether or not the average concentration in the survey unit meets the DCGL, This is not to

suggest that it is necessary to define the elevated area as a separate survey unit and conduct a new

survey to determine its compliance with the DCGLE. For cases in which the decision is too close

to call, it may be useful to keep this analogy in mind when planning a resolution to the problem.

It will also be useful in planning the investigation of the elevated area.

There will be many types and sizes of elevated areas. In many cases, it may be obvious whether

or not the elevated area exceeds the release criterion based on the measurements taken during the

investigation, without performing an additional survey or performing additional statistical tests.

Obviously, -if the elevated area mean exceeds the DCGL A, the survey unit fails. If the elevated

area mean is less than the DCGLEby more than the standard error of the mean, ALARA

8-3 8-3 NUREG-1505
EMC

considerations will usually determine whether or not further remediation is necessary. As with

any measurements, the DQO process should be used in planning the investigation of the elevated

area, and what decisions will be made based on the results.

Some other considerations that may arise are:

(1) The variability of concentrations in the elevated area is likely to exceed that of any

background variations, so additional reference area measurements will not usually be

needed. If the survey unit is being compared to a reference area, the boundary of the

elevated area should be determined by measurements at or below the DCGL wadded to

the mean reference area measurement. The elevated area mean minus the mean reference

area concentration should not exceed the DCGLEA.

(2) There may be elevated areas within the elevated area: There may exist a smaller area

within the elevated area that has concentrations high enough to exceed the release

criterion when considered separately, even though the average concentration over the

entire elevated area is within the DCGLEA.

8.3 Comparison Against Scanning Measurements

The measurement results obtained during scanning are inherently more qualitative in nature than

those obtiined on the systematic grid. In Class 1 survey units, much of the survey design depends

on the ability to detect areas exceeding the DCGLESIc during scanning. This is the essence of the

requirement that the scanning MDC be below the DCGLEMc. In practice, this means that an

operating procedure for flagging suspect locations during scanning be devised to ensure that

potential elevated areas be investigated. Then, as is the case with measurements on the systematic

grid that exceed the DCGLEMCo the suspect area must be investigated to determine the area and

average concentration of the elevated area. In many cases, it would be prudent to set the criteria

for flagging elevated areas conservatively. If this is done, and subsequent quantitative

measurements indicate that the DCGLEMC is not actually being exceeded, nothing further would

generally be necessary unless for ALARA considerations. If measurements above the stated

scanning MDC are found by sampling or by direct measurement at locations that were not

flagged by the scanning survey, this may indicate that the scanning method did not meet the

DQOs.

Scanning requirements for Class 2 and Class 3 survey units are less stringent both with regard to

the coverage and sensitivity. This is possible because of the screening process necessary to show

that these areas are not highly contaminated (Sections 2.2.2 and 2.2.3). For this reason, the

investigation levels are lower than in Class 1 areas (Section 2.5.7).

8.4 Area Factors

Area factors have been discussed in Sections 2.2.1 and 3.8.2. These area factors should be

calculated using dose pathway models and assumptions that are consistent with those used to

calculate the DCGLw. In this section, examples of area factors for both indoor and outdoor

survey units are given.

NUREG-1505 8-4
EMC

The example outdoor area factors listed in Table 8.1 were calculated using RESRAD for

Windows 5.70 (ANL/EAD/LD-2). For each radionuclide, all dose pathways were calculated

assuming an initial concentration of 1 pCi/g. The default area of contamination in RESRAD 5.7

is 10000 m2, so for this size area, the area factor for all radionuclides is equal to one. Area

factors for other size areas were computed by taking the ratio of the dose per unit concentration

calculated by RESRAD for the default 10000 m2 to that calculated for 1, 3, 10, 30, 100, 300,

1000, and 3000 m2. The other RESRAD default values were not changed except to adjust the

length parallel to aquifer to be consistent with area of contamination..

The area factors for selected radionuclides are plotted in Figure 8.2. There it can be seen that

radionuclides generally fall into three groups. Those that deliver dose primarily through internal

pathways, those that deliver dose primarily through the external pathway, and a few for which

both are important. Generally, the radionuclides that deliver dose via internal pathways (e.g.,

1

4C, 9°Sr) have the highest area factors. These area factors scale with the area in a manner

suggesting that it is the total inyentory of the radionuclides that is most important. The area

factors for radionuclides that deliver dose primarily through external gamma have lower area

factors, reflecting the fact that these radionuclides can deliver dose at a distance. In a mixture, it

will generally be these radionuclides that will have the limiting area factors. Fortunately, these

are also the radionuclides most easily detected using scanning techniques.

Notice that Figure 8.2 is plotted on a log-log scale. Linear interpolation on this figure

corresponds to logarithmic interpolation in Table 8.1 for areas between those listed. For example,

if the area factor for 24SAm is needed for 25 m2, the table lists 96.3 for 10 m2 and 44.2 for 30 m2.

To interpolate, take the base 10 logarithms of these numbers:

loglo (10) = 1

loglo (30) = 1.477

loglo (25) = 1.398

loglo (13.4) =1.127

loglo (4.99) = 0.698.

The interpolation is done using these values:

loglo (A25) ' logl0 (13.4)

+ [ logl0 (25) - log10 (10)] ([loglo (4.99) - (log1 o (13.4)]i [loglo (30) - loglo (10)])

= 1.127 + [1.398 - 1)] [ 0.698 - 1.127] / [1.477 - 1]

1.127 + [0.398] { [ - 0.429] / [0.477]1

=0.769

Therefore, A2s = 10(0.769) = 5.88.

Example indoor area factors listed in Table 8.2 were calculated using RESRADBUILD for

Windows 2.11 (ANL/EAD/LD-3, 1994). For each radionuclide, all dose pathways were

calculated assuming an initial concentration of 1 pCi/m2. The default area of contamination in

8-5 NUREG-1505
EMC

RESRAD BUILD is 36 M2. The other areas compared to this value were 1, 4, 9, 16, or 25 n2. No

other changes to the RESRAD BUILD default values were made. Dose was computed for one

receptor, who spent 100% of time in the contaminated room. The area factors were then

computed by taking the ratio of the dose per unit concentration calculated by RESRAD BUILD

for the default 36 M2 to that calculated for the other areas listed. Thus, if the guideline limit

concentration for residual radioactivity distributed over 36 m2 is multiplied by this value, the

resulting concentration distributed over the specified smaller area delivers the same average

dose. There are obviously many other exposure scenarios which may result in different area

factors.

Table 8.1 Example Outdoor Area Factors

Nuclide 1000Dm2 3000 m2 1000 m2 300 m2 100 m' 30 m2 10 m2 3 m2 1 m2

Am-241 1.00 1.01 1.01 1.20 1.86 4.99 13.4 40.2 109

C-14 1.00 2.09 3.06 4.84 8.40 23.6 65.7 207 609

Cd-109 1.00 1.03 1.04 1.11 1.42 3.05 7.61 22.1 63.0

Ce-144 1.00 1.03 1.04 1.11 1.21 1.49 2.05 4.24 9.30

Co-57 1.00 1.03 1.05 1.11 1.19 1.46 1.99 4.06 8.69

Co-60 1.00 1.04 1.06 1.13 1.23 1.52 2.12 4.39 9.81

Cs-134 1.00 1.07 1.10 1.19 1.30 1.61 2.22 4.57 10.1

Cs-137 1.00 1.10 1.14 1.28 1.41 1.75 2.41 4.98 11.0

Eu-152 1.00 1.03 1.05 -1.10 1.19 1.47 2.03 4.20 9.28

Fe-55 1.00 2.12 3.12 9.97 27.1 71.6 149 284 484

H-3 1.00 1.06 1.08 1.38 2.18 5.97 16.4 51.3 150

1-129 1.00 1.19 1.34 1.90 3.14 8.92 25.0 79.1 233

Mn-54 1.00 1.03 1.05 1.12 1.22 1.50 2.08 4.30 9.52

Na-22 1.00 1.05 1.08 1.13 1.22 1.51 2.07 4.28 9.44

Nb-94 1.00 1.01 1.05 1.18 1.27 1.56 2.15 4.42 9.77

Ni-63 1.00 1.46 1.68 5.59 16.6 54.2 155 464 1180

Pu-238 1.00 1.02 1.04 1.82 2.50 3.26 4.24 6.01 8.88

Pu-239 1.00 1.02 1.03 1.83 2.51 3.28 4.26 6.07 8.94

Ru-106 1.00 1.01 1.02 1.07 1.36 2.65 7.23 14.9 32.7

Sb-125 1.00 1.03 1.05 1.10 1.18 1.45 1.99 4.10 8.94

Sr-90 1.00 1.17 1.23 4.04 11.9 37.1 98.7 285 729

Tc-99 1.00 1.02 1.07 1.54 2.55 7.16 20.0 62.8 185

Th-232 1.00 1.03 1.05 1.47 1.75 2.24 3.12 6.08 12.3

U-235 1.00 1.01 1.19 2.18 3.84 10.3 15.9 30.2 58.8

U-238 1.00 1.01 1.04 1.43 2.27 5.73 11.1 18.3 30.5

Zn-65 1.00 1.31 1.45 1.81 2.07 2.62 3.64 7.62 17.0

NUREG-1505 8-6
EMC

Figure 8.2 Example Outdoor Area Factors

The indoor area factors for selected radionuclides are plotted in Figure 8.3. There is not as much

variation between radionuclides as there is with the outdoor area factors. All of the area factors

scale nearly with the size of the contaminated area. As with the outdoor area factors, the

radionuclides that deliver dose primarily through internal pathways have higher area factors than

those that deliver dose primarily through the external pathway. The area factors for radionuclides

that deliver dose primarily through internal pathways scale with the area in a manner suggesting

8-7 NUREG-1505
EMC

that it is the total inventory of the removable fraction of these radionuclides that is most

important.

Table 8.2 Example Indoor Area Factors

Nuclide 36 m2 25 m2 16 m' 9 m2 4 m2 I m,

Co-60 1.0 1.2 1.6 2.5 5.5 22.7

Cs-137 1.0 1.2 1.7 2.6 5.7 23.5

H-3 1.0 1.4 2.3 4.0 9.0 36.0

Ni-63 1.0 1.4 2.3 4.0 9.0 36.0

Pu-239 1.0 1.4 2.3 4.0 9.0 36.0

Ra-226 1.0 1.4 2.1 3.6 8.0 32.2

Sr-90 1.0 1.4 2.2 4.0 8.9 35.7

Th-232 1.0 1.4 2.3 4.0 9.0 36.0

U-235 1.0 1.4 2.2 4.0 9.0 35.8

Zn-65 1.0 1.2 1.6 2.5 5.4 22.3

I-.

LL. 10.0

U

0m

--- CO-6O

--- "CS-131

:-S-,.'.. H 4 '

•NI-63

• -K-PU-2-31

--- RA.221

-S-- 8R-90

- TH-239

- U-235

- ZN-65

10 100

Area (square meters)

I

Figure 8.3 Example Indoor Area Factors

NUREG-1505 8-8
EMC

The area factors for radionuclides that deliver dose primarily through external gamma have lower

area factors, reflecting the fact that this dose can be delivered at a distance. Thus, in a mixture, it

will generally be these radionuclides that will usually have the limiting area factors. However,

the effect is not as large indoors as outdoors.

8.5 Example

A concrete room, 5 meters by 6 meters, had been contaminated with '

37

Cs and 6Co, and

subsequently remediated. The floor and the bottom 2 meters of the walls were to be surveyed as a

Class 1 survey unit. Measurements were to be made for 100s at each grid point with a 16 cm2

GM counter with a 10.1% efficiency. The DCGLw for both'nuclides were within about 10%, so

the lower was taken to conservatively apply to both. This DCGLw, about 1100 dpm per 100 cm2,

translated into 30 counts per 100s with this detector. During the DQO process it was determined

that Scenario A would be used with a = 0.05, f3 = 0.025, and A = 10 counts. The average

background readings for this type of building on site had been about 60 ± 10 counts, so the

estimated a = 10. From Table 3.3, it was found that 39 measurements each were required in the

reference area and the survey unit. This was rounded up to 40. The survey unit area is 61 m', so

the spacing, L, on a triangular grid is L = [61/ (0.866N)]' = [61/ (34.6)]" = 1.3 m, using

N= 40. The grid area is 0.866 L2 = 0.866 (1.3)2 = 1.5 m2 . Interpolating into Table 8.2 gives an

area factor for 1.5 m2 of 15. This results in a DCGLEMC = 15 (DCGLw) = 16500 dpm per 100 cm2,.

or 450 GM counts per 100s. This level is easily seen while scanning, so no additional grid

measurements will be needed in order to find elevated areas.

When the random start triangular grid was laid out in the survey unit, 50 measurement locations

were identified. When there are more locations identified than are required, they are all sampled

and reported. In the reference area, the grid lay out was terminated when 40 locations were

found. The data are shown in Table 8.3.

The mean and standard deviation of the reference area measurements was 58 ± 10. For the survey

unit, these were 88 ± 92. The difference of the means, 88 - 58, is just at the DCGLw of 30.

However the median in the reference area is 59 while that in the survey unit is only 58. This is an

indication that the survey unit data is fairly symmetric, but that the survey unit mean is being

driven up by a few very high measurements. This can be seen even more clearly in the combined

ranked data plot of Figure 8.4. For this plot the reference area measurements, adjusted by adding

the DCGLw to each, are combined with the survey unit measurements. The measurements are

then plotted against their rank in the combined data set, using different symbols for the reference

area points and the survey unit points. This is an easier diagnostic plot to.use than the QuantileQuantile plot when the reference area and survey unit have different numbers of data points.

From this plot it can be seen that the majority of the survey unit measurements fall below the

adjusted reference area measurements, but that there are eight survey unit measurements that are

much higher. One of those measurements exceeds the DCGLEMo which is equivalent to 450 GM

counts per 100s. Thus, further investigation of this survey unit will be required before it could be

released, regardless of the outcome of the WRS test.

The sum of the adjusted reference area ranks, shown in Table 8.3, is 2432. This is greater than

the critical value of 2023 given by the equation following Table A.3 for n = 50, m = 40, and

a = 0.05. Thus, the hypothesis that the survey unit as a whole uniformly exceeds the DCGLw is

8-9 NUREG-1505
EMC

rejected. Whether or not the survey unit may be released is now dependent on the results of the

investigation of the elevated measurements that were found.

A posting plot of the survey unit data is shown in Figure 8.5. In terms of GM counts, the elevated

area is defined by the average reference area measurement plus the DCGL w which is

58 + 30 = 88. The shaded area in Figure 8.5 encloses the measurements exceeding 88 GM counts

per 100s. This area, which also encloses all of the measurements exceeding the DCGL Emo covers

almost 16 m2 . From Table 8.2, the area factor for `aTCs is 1.7 and that for 60Co is 1.6. For a

mixture of the two radionuclides, the smaller value is used. Thus, in this case the DCGLE = 1.6

(DCGLw) = 1760 dpm per 100 cm2, or 48 GM counts per 100s. The average of the ten

measurements in the shaded area is 216.8 GM counts per 100s, which is 216.8 - 58 = 158.8 GM

counts per 100s above the reference area average. Thus, the survey unit does not meet the release

criterion, and may not be released without further remediation.

This example illustrates how the nonparametric statistical tests used in combination with the

elevated measurement comparison work to assure that the release criterion is met.

500"

450 * Reference Area + DCGL

400 t3Survey Unit

350- t

300

250 0

200

150 0

100" C 0 • 0a r nnt•:]

50 r' 0 0CP 13 0 0 0 13 3r3 0 0 u

0 - 1 I

0 10 20 30 40 50 60 70 80 90

Figure 8.4 Combined Ranked Data Plot of Reference Area and Survey Unit Measurements

NUREG-1505 8-10
EMC

Table 8.3 -Data for Indoor Survey Unit and Reference Area

(GM counts per 100s)

Ref Survey Adjusted Ref Survey

Data Data Ref Data Ranks Ranks

55 56 88 55 23.5

50 46 .83 48. 7

63 55 96 70.5 22

51 58 84 52 27

43 68 76 -39.5 35.5

50 51 83 48 .16

50 83 83 48 48

50 89 83 48 56

50 47 83 48 9

53 134 86 54 86

35 410 68 35.5 89

50 78 83 48 42.5

43 52 76 39.5 18.5

73 40 106 79 2.5

63 111 96 70.5 83

75 340 108 80 88

60 50 o93 65 13.5

58 43 91 59 6

70 62 103 77.5 30.5

61 495 94 67.5 90

68 63 101 76 32

57 47 . 90 57 9

63 52 96 70.5 18.5

60 254 93 65 87

58 78 91 59 42.5

59 67 92 62 33

47 97 80 44 73.5

58 59 91 59 29

76 49 109 81.5 11.5

61 118 94 67.5 84

70 120 103 77.5 85

67 77 100 75 41

60 42 93 65 4.5

76 53 109 81.5 20.5

52 68 85 53 35.5

59 68 92 62 35.5

41 36 74 38 1.

63 62 96 70.5 30.5

64 56 97 73.5 23.5

59 58 92 62 27

______ 47 _ ____ 9

40 ______ 2.5

50 _____ 13.5

57 ______ 25

_____ 58 ___________ 27-

49 ___________ 11.5

______ 53 ______ _____ 20.5

_____ .51 ______ .16

42 *4.5

51 16

_. _ 243?2 16631

8-11 8-11 NUREG-1505
EMC

Figure 8.5 Posting Plot of Indoor Concrete Survey Unit

NUREG-1505 8-12
9 SAMPLE SIZE

9.1 Sample Size and Decision Errors

To determine the number of samples to collect, acceptable values of the Type I error rate (a) and

Type H error rate (P3) must be specified as part of a statistical test. The process for doing this was

discussed in Section 3.7. If there are many survey units and each unit requires a separate

decision, even if H0 is true approximately 100a% of the times the test is conducted, the null

hypothesis will be incorrectly rejected. If a smaller value of a is used, the number of times this

can be expected to happen decreases proportionately. On the other hand, larger values of a will

reduce the number of samples initially required from each survey unit.

The power (1- 3) is the ability of a statistical test to detect when the null hypothesis is indeed

false and should be rejected. A test should have high power, i.e., small 13, but smaller specified

values of P3 require a larger number of measurements.

The number of samples depends not only on a and f3, but also on the width of the gray region

relative to the measurement variability, A/a. This parameter essentially describes the resolution

of the decision problem. When the resolution is high, only a few measurements are needed.

When the resolution is low, many more measurements may be required, even though the

specified a and P3 are unchanged.

The DQO steps desciibed in Sections 3.7 and 3.8 are used to balance the cost of sampling against

the risks involved with any potential decision errors. It is important to realize, however, that the

value of a is fixed at the desired value when the critical value for the test statistic is determined

from the provided tables and used in the test. If the sample size is larger (or smaller) than

planned, the effect will be an increase (or decrease) in the power, 1- j3. Similarly, if the

measurement standard deviation is larger (or smaller) than anticipated, the effect will be a

decrease (or increase) in the power.

The consequences of increasing or decreasing the power depends on whether Scenario A or

Scenario B is being used. In Scenario A, where the null hypothesis is that the survey unit does

not meet the release criterion, high power means that a survey unit that meets the release criterion

has a high probability of passing the test. In Scenario B, where the null hypothesis is that the

survey unit meets the release criterion, high power means that a survey unit that does not meet

the release criterion has a high probability of failing the test.

In most cases, the sample sizes required can be determined using Tables 3.2 and 3.3. In the

following sections, the assumptions made and the calculations performed in creating these tables

are described. Methods for modifying the calculations under alternative assumptions are also

given. It must be emphasized that relatively little effort is required to perform the suggested

sample size determinations compared to the time and expense involved in collecting and

analyzing samples. This is a key advantage to using the DQO process to determine sample sizes.

9-1 NUREG-1505
SAMPLE SIZE

9.2 Sample Size Calculation for the Sign Test Under Scenario A

For the Sign test, the number of samples, N, required from the survey unit can be approximated

from a formula given by Noether (1987):

N - (Z1 -+Z 1 -) 2 (9-1)

4 (p -0.5)2

where:

a = specified Type I error rate

13 = specified Type II error rate

Z =. 100(1- a) percentile of the normal distribution

ZI.p = 100(1- 13) percentile of the normal distribution

p = estimated probability that a random measurement from the survey unit will be less than the

DCGLw when the survey unit median is actually at the LBGR. pt 0.5

Commonly used values for a and 13, and the corresponding values of ZI., (or Z,.,) may be found

from Table 9.1. Other values can be obtained using any table of the cumulative standard normal

distribution function, such as that in Appendix A.

Table 9.1 Some Values of Z,.. and Z,.p Used To Calculate the Sample Sizes

S(or 13) IZ 1.,, (or Z,-p)

0.01 2.3268

0.025 1.9604

0.05 1.6452

0.10 1.2817

0.20 0.8415

The numerator of Equation 9.1, (Z,-,, + Z1 _p )', depends on a and 13, but not on A or a. In

addition, it only depends on the pair of values used for a and 13, and is the same if these values

are reversed. This can be seen in the symmetry of Table 9.2, where the value of (Z,-. + Z1 _p )2 is

given for each pair of the values of a and 13 listed in Table 9.1. As will be seen, these are also the

minimum samples sizes for each pair of a and 13 values.

The denominator of Equation 9.1 depends on the parameterp, but not on a or 13. The definition.

of the parameter p states that it is the estimated probability that a random measurement from the

survey unit will be less than the DCGLw when the survey unit median is actually at the LBGR.

This is illustrated in Figure 9.1. The value of I -p expresses the likelihood that measurements

exceeding the DCGLw will be observed, even if half of the concentration distribution is below

the LBGR. This likelihood is higher when the measurement standard deviation is large compared

NUREG-1505 9-2
SAMPLE SIZE

to the width of the gray region. Some assumptions about the data distribution have to be made in

estimating p. If it were possible to specify p exactly, there would be no need to do the survey.

Table 9.2 Some Values of (Z-._ + Zjp )' Used To Calculate Sample Sizes

I 0.01 1 0.025 1 0.05 1 0.1 0.2

0.01 22 19 16 14 11

0.025 19 16 13 11 8

0.05 16 13 11 9 7

0.1 14 11 9 7 5

0.2 11 8 7 5 3

The relative width of the gray region, A/a, is especially useful for estimating the parameter p. If

the data are even approximately normally distributed, then p can be estimated from

DCGL.

P I f e -(x-LBGR) 2/2o 2dx

- f e (xLBGRe)2O2 dx

A

f efe-x2/ dx

(9-2)

Values of p as a function of A/c, computed from Equation 9-2, can be found in Table 9.3, or in

the table of the cumulative normal distribution (Appendix A, Table A. 1).

The factor 1/[4(p - 0.5)2 ] in Equation 9-1 can be viewed as a multiplier applied to the sample

sizes given in Table 9.2. If p = 1, this factor is one. Figure 9.2 shows the dependence of both p

and the sample size multiplier on the resolution of the decision problem, A/a. Increasing A/a

beyond about three has little effect on reducing the sample size multiplier. Decreasing A/a below

about one causes the sample size multiplier to rise dramatically. The range from the smallest (3)

to largest (22) sample size in Table 9.2 is about a factor of seven. When A/a is less than about

0.5, the sample size multiplier exceeds seven. Thus, small values of A/a can have a bigger impact

on increasing the sample size than the choice of acceptable decision error rates.

9-3 NUREG-1505
SAMPLE SIZE

Dis

of

Wis

is

Scenario A

-tMibution

measurements

'en true median

it the LBGR

p is the area under

the unshaded part of

the distribution.

1-p

LBGR DCGL

Figure 9.1 The Parameterp for the Sign Test Under Scenario A

The assumption of normality is not critical in the above calculations, since it is only being used

to estimate an efficient sample size. However, if a different distribution is considered more

appropriate, it can be used. Values of p for other probability distributions with density function

ftx), mean equal to the LBGR, and standard deviation o, can be computed from

DCGLW

p= f f(x)dx

-00

(9-3)

NUREG-1505 9-4
SAMPLE SIZE

Table 9.3 Values ofp for Use in Computing Sample Size for the Sign Test

A/ p I /oy p WaF p A/o p

0.1 0.53983 1.1 0.86433 2.1 0.98214 3.1 0.99903

0.2 0.57926 1.2 0.88493 2.2 0.9861 3.2 0.999331

0.3 0.61791 1.3 0.9032 2.3 0.98928 3.3 0.99952

0.4 0.65542 1.4 0.91924 2.4 0.9918 3.4 0.99966

0.5 0.69146 1.5 0.93319 2.5 0.99379 3.5 0.99977

0.6 0.72575 1.6 0.9452 2.6 0.99534 4.0 0.99997

0.7 0.75804 1.7 0.95544 2.7 0.99653 5.0 1.00000

0.8 0.78815 1.8 0.96407 2.8 0.99745

0.9 0.81594 1.9 0.97128 2.9 0.99813

1.0 0.84135 2.0 0.97725 3.0 0.99865

In some situations, it may be possible to estimate p directly from remediation control survey

data, since it is simply an estimate of the proportion of the final status survey measurements that

are expected to fall below the DCGLw.

Figure 9.2 Sample Size Multiplier and the Parameterp Versus A/a

9-5 NUREG-1505
SAMPLE SIZE

One can also calculate p from the estimated odds that a random measurement is less than the

DCGLw versus that it is above the DCGLw .If these odds are r,:r2, then p = r, /(r1 + r2). For

example, if the odds that a random measurement is less than the DCGLw are 3:2, then

p = 3/(3+2) =3/5 = 0.6.

Whatever method is used to estimate p, it is important not to overestimate it, since that will result

in a sample size inadequate to achieve the desired power of the test. The dependence of the

sample size multiplier, l/[4(p - 0.5)2], onp is shown in Figure 9.3.

100

Co,

a10-

E

Co

0.5 0.6 0.7 0.8 0.9

Probability, p

Figure 9.3 Dependence of Sample Size onp

As an illustration, consider the example given in Section 5.1. For that example, the

DCGLw = 15.9, the LBGR = 11.5, ot = P = 0.05, and Y = 3.3. From Table 9.2,(Z_,,. + Z,. )2 = 11

when a = = 0.05. This is the minimum sample size required for those values of the acceptable

error rates. The width of the gray region, A = DCGLw - LBGR = 15.9 - 11.5 = 4.4, so

A/a = 4.4/3.3 = 1.3.

From Table 9.3, the value of p using the normal approximation is 0.903199. Thus the factor

1/[4(p, - 0.5)2] 1/[4(0.903199 - 0.5)2]

= 1/[4(0.403199)2]

= 1/[4(0.162569)]

= 1/0.650278

1.54

So, the minimum sample size of 11 is increased by a factor of 1.54 to 16.9. This would normally

be rounded up to 17. However, because Equation 9-1 is an approximation, it is prudent to

increase this number moderately. An increase of 20% is recommended. This increases the

NUREG-1505 9-6
SAMPLE SIZE

number of samples to 1.2(16.9) = 20.28, which is rounded up to 21. This is the number that

appears in Table 3.2.

The effect of increased variability in the measurement data will be an increase in the required

sample sizes. As A/a becomes smaller, p also becomes smaller. This decreases the denominator

of Equation 9-1, increasing the sample size N accordingly.

9.3 Sample Size Calculation for the Sign Test Under Scenario B

Under Scenario B, Equation 9-1 is also used to estimate the required sample size. The roles of a

and [3 are reversed, but this has-no effect on the numerator of Equation 9-1, so Table 9.2 may still

be used. The form of the denominator also remains the same, and Figure 9.3 still represents the

dependence of the sample size multiplier on p. However, the definition of the parameterp is

different. The definition of the parameter p under Scenario B is the estimated probability that a

random measurement from the survey unit will be greater than the LBGR when the survey unit

median is actually at the DCGLw. This is illustrated in Figure 9.4. The value of 1- p expresses

the likelihood that measurements less than the LBGR will be observed, even if half of the

concentration distribution is above the DCGLw. This likelihood is higher when the measurement

standard deviation is large compared to the width of the gray region.

If, as in Scenario A, we assume that the data are approximately normally distributed, the width of

the gray region, A/a = (DCGLw - LBGR)/o, can be used to estimate the parameter p:

p 1 e -(X -DCGLw) 2/2cr2dx

Sj e -(x-DCUt'12a2 dx

F2 (.BGR DCGLw) +DCGLW

001 _x• 212dx

LBGR-DCGLw

DCGLw -LBGR

e x - 12dx

(9-4)

This is the same as Equation 9-2. Even though the definition of p has changed, its value as a

function of A/a has not changed.

9-7 NUREG-1505
SAMPLE SIZE

Scenario B

Distribution

of measurements

when true median

is at the DCGL

1-P

p is the area under

the unshaded part of

the distribution.

LBGR DCGL

Figure 9.4 The Parameterp for the Sign Test Under Scenario B

The above calculation shows that the value of p computed from Equation 9-2, as found in Table

9.3, or in the table of the cumulative normal distribution (Appendix A) can be used in Scenario B

as well as Scenario A. Figure 9.2, expressing the dependence of sample size on A/0 is

unchanged, and that is why only one version of Table 3.2 is needed for both scenarios:

If a distribution other than normal is considered more appropriate for detennifiing p, the

following equation can be used. For a probability distribution with density function ftx), mean at

the DCGLw , and standard deviation a,

0o

p =f f(x)dx (9-5)

LBGR

NUREG-1505 9-8
SAMPLE SIZE

As in Scenario A, it may be possible to estimate p directly from remediation control survey data.

To do this, the data should first be shifted so that the median is at the DCGLw before the

proportion that falls above the LBGR is calculated.

p may also be estimated from the odds that a random measurement will be greater than the

LBGR versus that it is below the LBGR when the median is near the DCGL , If these odds are

r,:r2, then p = rl/(r,+r2). For example, if the odds that a random measurement is greater than the

LBGR when the median is near the DCGLW are 3:1, then p = 3/(1+3) =3/4 = 0.75. Notice that

since we are assuming that the true median is near the DCGL1 v the odds must be greater than 1:1.

Once a survey unit has been remediated, it may be somewhat unnatural to try to estimate the odds

this way.

9.4 Sample Size Calculation for the WRS Test Under Scenario A

For the WRS test, the total number of required samples from the reference area and survey unit

combined is estimated from (Noether, 1987):

N (Z- 1 .+ Z1 -))2 12 c (I - c) (P,. - 0.5)(96 (9-6)

where:

= specified Type I error rate

8 = specified Type II error rate

Zi.a= 100(l -a) percentile of the standard normal distribution function

Z1,6= 100(1 -,g) percentile of the standard normal distribution function

c proportion of measurements taken in the survey unit.

P,= estimated probability that a random measurement from the survey unit exceeds a random

measurement from the reference area by less than the DCGLw when the survey unit median

is at the LBGR above background. Pr t 0.5

The numerator of equation 9-6 is the same as that in equation 9-1. Therefore, the same methods

are used to calculate it as were discussed in Section 9.2. Table 9.2 gives commonly used values

of a and P3, together with the corresponding values of (Z,- + Z,•_ ). For planning purposes, c is

set equal to 0.5, so that Equation 9-6 becomes

,N =(ZI-,+ Zl-P)2 (9-6')

3 (Pr - 0.5)2

The denominator of Equation 9-6 differs from that in Equation 9-1 in three important ways. First,

the parameter Pr replaces the parameter p. It is a different probability, but because it is still a

probability, 0 < P•r 1. The factor (Pr - 0.5)2 cannot be larger than 0.25. Second, the constant

9-9 NUREG-1505
. SAMPLE SIZE

factor in the denominator is 3 rather than 4. This means that the sample size multiplier

11[3(Pr - 0.5)2] cannot be smaller than 4/3. Third, Equation 9.6 yields the total number of

samples required in both the survey unit and a reference area. N12 samples will be taken in each.

The definition of the parameter P, states that it is the estimated probability that a random

measurement from the survey unit exceeds a random measurement from the reference area by

less than the DCGLw when the survey unit median is above the reference area median by an

amount equal to the concentration value at the LBGR. This is illustrated in Figure 9.5.

Distribution

of reference area

measurements

with true median

atBKGD

Distribution

of survey unit

measurements

when true median

Is at the

LBGR + Background

;GL

BKGD BKGD + LBGR

Distribution of the difference

between a random survey

unit measurement and a

random reference area

measurement

P. is the area under the

unshaded

part of the distribution

1 - P, is the area

under the shaded part

of the distribution

V

LBGR DCGL

Figure 9.5 The Parameter P, for the WRS Test Under Scenario A

9-10
SAMPLE SIZE

As was done for the Sign test, the normal distribution may be used to facilitate the conversion of

the values of A/c to values of Pr in order to calculate the required sample sizes. The normal

distribution is not used to actually conduct the test. Values of P, are computed for a normal

distribution from the following equation:

Pr -Probability(U=X - Y < DCGL)

Sfcj ffx( + y)f$(y)dydu

=Djffe -(ueY-LBGR)2D2 ed

12r f e -(u-LlGR) 2

1402 du

DCGLw-LBGR

1__. f e-X12dX f

D( (9-7)

Values of Pr as a function of A/c are listed in Table 9.4. The probability P, and the sample size

multiplier 1/[3(Pr - 0.5)2] are shown as a function of A/a in Figure 9.6.

Table 9.4 Values of P, for Use in Computing Sample Size for the WRS Test

A/o P, A/o P, A/o P,7 A/a P,

0.1 0.528186 1.1 0.781662 2.1 0.931218 3.1 0.985811

0.2 0.556231 1.2 0.801928 2.2 0.940103 3.2 -0.988174

0.3 0.583998 1.3 0.821015 2.3 0.948062 3.3 0.990188

0.4 0.611351 1.4 0.838901 2.4 0.955157 3.4 0.991895

0.5 0.638163 1.5 0.855578 2.5 0.961450 3.5 0.993336

0.6 0.664313 1.6 0.871050 2.6 0.967004 4.0 0.997661

0.7 0.689691 1.7 0.885334 2.7 0.971881 5.0 0.999796

0.8 0.714196 1.8 0.898454 2.8 0.976143 6.0 0.999989

0.9 0.737741 1.9 0.910445 2.9 0.979848

1.0 0.760250 2.0 0.921350 3.0 0.983053

9-11 NUREG-1505
SAMPLE SIZE

10 .- ~1

/ - 0.95

/ / , 0.9

-/ / o0.85

0.80

N W

-10 5 0.75 Cr

.2 1~0.7

/ 0.6

/ 0.55

I I J 0.5

0 1 2 3 4

- ~A/oFigure 9.6 Dependence of Sample Size on A/a for WRS Test

(Values of p used for the Sign Test are shown in gray for comparison)

Notice that the only difference between Equation 9-7, and equation 9-4 for computing p for the

Sign test is that the standard deviation Y is replaced by VF2 times a. This is because the variance

of the difference of two independent measurements is the sum of the variances of the individual

measurements. If the variances of the individual measurements are about the same, i.e., 02, the

variance of their difference is 2 o2 . Thus, for a given value of A/a, Pr will always be less than p.

This also causes the total sample size required for the WRS test to be greater than that for the

Sign test. Values of p are shown for comparison to P, by the gray dashed line in Figure 9.6.

The combined effect of all the differences between Equation 9-7 and 9-4 is summarized in

Figure 9.7.

Values of Pr for distributions other than normal can be calculated from the following equation:

DCG4.o

Pr Probability(U=X - Y < DCGL) f.ffx(U + y)fr(yYU (9-8)

where Y is a random measurement from the reference area with density fy and X is a random

measurement from the survey unit with densityfx. However, in PNL-8989 (1993), Hardin and

Gilbert have found that using the values of P, from Equation 9-6 yielded good results when the

9-12
SAMPLE SIZE

distributions being tested were positively skewed, such as the log-normal.

3

2.5 ;2

0

1a .5 -

=) 1 -'---.

N - m••m

a$.5 Survey Unit Only

E

M 0 1 2 3 4

Cl)

A/a

Figure 9.7 Comparison of Sample Sizes Required for the WRS Test and the Sign Test

In some situations, it may be possible to estimate P,, directly from remediation control survey

data. It is an estimate of the proportion of time that a random survey measurement will exceed a

random reference area measurement by less than the DCGLw.

P, may also be estimated by the odds that a random survey measurement will exceed a random

reference area measurement by less than the DCGLwversus that a random survey measurement

will exceed a random reference area measurement by more than ihe DCGL w'. If these odds are

r,:r2 , then Pr = r,1/(r, + r2).

Whatever method is used to estimate P., it is important not to overestimate it, since that will

result in a sample size inadequate to achieve the desired power of the test. The dependence of the

sample size multiplier, 1/[3(Pr - 0.5)2], on Pr is shown in Figure 9.8.

As an illustration, consider the example given in Chapter 6.1. For that example, the

DCGLW = 160, the LBGR = 142, a = P3 =0.05, and a = 6. From Table 9.2,(Z1_- + Z,.p )2 = 11

when a = = 0.05. The width of the gray region, A = DCGLW - LBGR = 160 - 142 = 18, so

A/o = 18/6 = 3.

From Table 9.4, thevalue of Pr using the normal approximation is 0.983053. Thus the factor

1/[3(p - 0.5)2 ] = 1/[3(0.983053 - 0.5)2 ]

9-13 NUREG-1505
SAMPLE SIZE

= 1/[3(0.483053)2]

= 1/[3(0.233340)]

= 1/0.700021

= 1.43.

So, the minimum sample size of 11 is increased by a factor of 1.43 to 15.7. This would normally

be rounded up to 16, or 8 samples each in the reference area and the survey unit. However,

because Equation 9-6 is an approximation, it is prudent to increase this number moderately. An

increase of 20% is recommended. This increases the number of samples to 1.2(15.7) = 18.9,

which is rounded up to the next even integer, 20. Thus, 10 samples each in the reference area

and the survey unit are required. This is the number that appears in Table 3.3.

1000

100

10

Co

a..

E

0.

0.5 0.6 0.7 0.8 0.9 1

pr

Figure 9.8 Dependence of Sample Size on P,

9.5 Sample Size Calculation for the WRS Test Under Scenario B

Under Scenario B, Equation 9-6 is also used to estimate the required sample size. The roles of a

and P3 are reversed, but this has no effect on the numerator of Equation 9-6, so Table 9.2 may still

be used. However, since under Scenario B, both the WRS test and the Quantile test are used in

tandem, the value of cc decided on during the DQO process is halved for each test. Thus, the

Table 9.2 value for aw = a/2 and 03 is used.

The form of the denominator also remains the same, and Figure 9.8 still represents the

dependence of the sample size multiplier on P,. However, the definition of the parameter P, is

different. The definition of the parameter Pr under Scenario B is the estimated probability that

the difference between a random measurement from the survey unit and a random measurement

from the reference area will be greater than the LBGR when the survey unit median is actually at

the DCGLw above the background median. This is illustrated in Figure 9.9.

9-14
SAMPLE SIZE

Distribution

of reference area

measurements

with true median

at BKGD

Distribution

of survey unit

measurements

when true median

is at the

DCGL + BKGD

BKOD +LBOR BKGD BKGD+DCGL

Distribution of the difference

between a random survey unit

measurement and a random

reference area measurement

P, is the area under the unshaded

part of the distribution

1 - P, is the area under

the shaded part of the

distribution IJ -q ur

LBGR

V

DCGL

Figure 9.9 The Parameter P, for the WRS Test Under Scenario B

The.value of 1 -P, expresses the likelihood that differences less than the LBGR will be observed,

even if-half of the survey unit concentration distribution is above the background median by

more than the DCGLw. This likelihood is higher when the measurement standard deviation is

I

9-15 NUREG-1505
SAMPLE SIZE

large compared to the width of the gray region.

If, as in Scenario A, we assume that the data are approximately normally distributed, the width of

the gray region, A/a = (DCGLw - LBGR)l/, can be used to estimate the parameter P,:

Pr = Probability(U =X - Y > LBGR)

LBGR[.O +Yf(dYU

1. -(u y-DCGLw-BKGDI2 1 2 2 -(y-BKGDo) 2 I2dy2 u

= GR[f 2 7ja e 

1 e -(u-DCGLW)14o2 du

DCGLw-LBGR

f-1 e -x2

12dx

=~z~ V2 4 (9-9)

This is the same as Equation 9-7. Even though the definition of Pr has changed, its value as a

function of A/u has not changed. Therefore, the value of Pr computed from Equation 9-7, as

found in Table 9.4, can be used in Scenario B as well as in Scenario A. Figure 9.6, expressing

the dependence of sample size on A/a is unchanged, and that is why only one version of Table

3.3 is needed for both scenarios.

Values of P, for distributions other than normal can be calculated from the following equation:

r=Probability (U =X - Y > LBGR) = f [ffx(u + y)fy(.y)dy dA (9-10)

where Y is a random measurement from the referenece area with density fy and X is a random

measurement from the survey unit with densityfx. ""

9-16
SAMPLE SIZE

As in Scenario A, it may be possible to estimate P, directly from remediation control survey data.

To do this, the data should first be shifted so that the median is at the DCGLw before the

proportion that fall above the LBGR is calculated.

Pr may also be estimated from the odds that a random measurement will be greater than the

LBGR versus that it is below the LBGR when the median is near the DCGLw If these odds are

rl:r 2, then Pr = r1(rl+r2). For example, if the odds that a random measurement is greater than the

LBGR when the median is near the DCGLw, are 3: 1, then Pr = 3/(1+3) =3/4 = 0.75. Notice that

since we are assuming that the true median is near the DCGLw, the odds must be greater than

1:1. However, once a survey unit has been remediated, it may be somewhat unnatural to try to

estimate the odds this way.

9-17. NUREG-1505

10 POWER CALCULATIONS FOR THE STATISTICAL TESTS

10.1 Statistical Power and the Probability of Survey Unit Release

The concept of the statistical power of a test was introduced in Section 2.3.2. The use of this

concept in optimizing the design of final status surveys was discussed in Section 3.8.1. The

power of a statistical test is defined as the probability that the null hypothesis is rejected when it

is false. It is 1- 3, where P3 is the Type II error of the test.

The statistical power will have different implications for survey unit release, depending on

whether Scenario A or B is used. The same information can be expressed slightly differently. In

this report, it is expressed as the probability that the survey unit passes the statistical test, i.e., the

result of the test is the decision that the survey unit may be released.

The relationship between this probability and the Type I and Type II errors was given in

Table 3.1. Figures 3.9 through 3.12 show this probability as a function of the true residual

radioactivity concentration for selected values of a and P3 over a range of sample sizes. In many

cases, it will be sufficient to check the curve in these figures that corresponds most closely to the

situation at hand. In the following sections, the assumptions made and the calculations performed

in creating these figures are described.

10.2 Power of the Sign Test Under Scenario A

Recall that for the Sign test in Scenario A, the test statistic, S+, was equal to the number of

survey unit measurements below the DCGLw. If S+ exceeds the critical value k, then the null

hypothesis that the median concentration in the survey unit exceeds the DCGLw is rejected, i.e.,

the survey unit passes this test. The probability that any single survey unit measurement falls

below the DCGLw is found from Equation 9-2 or 9-3. The probability that more than k of the N

survey unit measurements fall below the DCGLw 'is simply the following binomial probability:

N k IN [p]'[l.p]N= 1 _ ( N [p],[l _p]N-i 1 _ k-Np (D i =k+l N__p F0 "I (_. _p), (01

The indicated approximation is generally used when both Np and N(1 -p) are five or greater.

(Z(z) is the cumulative standard normal distribution function given in Table A. 1.

Withp calculated as in Section 9.2, Equation 10-1 yields the probability that the null hypothesis

is rejected when the true median of the residual radioactivity concentration in the survey unit is at

the LBGR. This is the power of the test at the.LBGR.

The probability, p(C), that any single survey unit measurement falls below the DCGLw when the

survey unit median concentration is at any other value, C, can be determined by simply replacing

the value of the LBGR in Equation 9-2 with the value of C:

10-1 NUREG-1505
POWER

DCGLW

p(C) f e_(x_)n/2a2 dx

C + (DCGLw - C)

1 f e -(x- C)/2c2dx

(DCGLW- C)

a

f e -X2

/2 d

( (DCGLW Q (10-2)

Note that if C = DCGLw, p(C) = 0.5. The assumption of normality is not critical in the above

calculations, since it is only being used to estimate the power. However, if a different distribution

is considered more appropriate, Equation 9-3 can be used to calculate p(C).

When the value of p(C) from Equation 10-2 is inserted in Equation 10-1, we obtain the

probability that the null hypothesis is rejected at the concentration C. When C = DCGLw, this

probability is the probability of a Type I error, a"). This calculation can even be performed for

values of C greater than the DCGLw. The probability obtained is still the probability that the

null hypothesis is rejected, i.e., that the survey unit passes the test.

If the probability that the null hypothesis is rejected (calculated from Equation 10-1) is plotted

against the concentration, C, the result is called a power curve. When the power calculation is

performed at the design stage, using an estimated value of o, it is called a prospective power

curve. When the calculation is performed after the survey, using the standard deviation of the

survey unit measurements as an estimate of a, it is called a retrospective power curve.

To illustrate the construction of a power curve, consider the example of Chapter 5. The DCGLw

for this example was 15.9 and the LBGR was 11.5. The DQOs for a = P = 0.05 resulted in a

sample size of N = 21, using the estimate that a = 3.3. From Table A.3, the critical value for the

Sign test with N=21 and a = 0.05 is k = 14. This is all of the information necessary to construct

the prospective power curve. To construct the retrospective power curve, we use the standard

deviation of the measurement data, 9.5, as the estimate of a.

The results of these calculations are shown in Table 10.1 and Figure 10.1.

0 )The value of a actually obtained from Equation 10-1 should be close to that specified in the DQOs. It

may not exactly equal that value when the sample sizes are small, since the critical value, k, can only take integer

values.

NUREG-1505 10-2
POWER

Table 10.1 Example Power Calculations: Sign Test Scenario A

Prospective RetrosDective

C (DCGLw - C)/a p(C)) power (DCGLw - C)Io p(C) power

(Eq. 10-2) (Eq. 10-1) (EA. 10-2) (Eq. 10-1)

0 4.82 1.0000 1.000 1.67 0.9525 1.000

5 3.30 0.9995 1.000 1.15 0.8749 0.989

6 3.00 0.9987 1.000 1.04 0.8508 0.972

7 2.70 0.9965 1.000 0.94 0.8264 0.942

8 2.39 0.9916 1.000 0.83 0.7967 0.884

9 2.09 0.9817 1.000 0.73 0.7673 0.802

10 1.79 0.9633 1.000 0.62 0.7324 0.679

11 1.48 0.9306 1.000 0.52 0.6985 0.544

11.5 1.33 0.9082 '0.998 0.46 0.6772 0.459

12 1.18 0.8810 0.991 0.41' 0.6591 0.390

13 0.88 0.8106 0.914 0.31 0.6217 0.262

14 0.58 0.7190 0.627 0.20 0.5793 0.151

15 0.27. 0.6064 0.217 0.09 0.5359 0.076

15.9 0.00 " 0.5000 0.039 0.00 0.5000 0.039

16 -0.03 0.4880 0.031 -0.01 0.4960 0.036

17 -0.33 0.3707 0.001 -0.12 0.4522 0.014

18 -0.64 0.2611 0.000 -0.22 0.4129 0.005

19 -0.94 0.1736 0.000 -0.33 0.3707 0.001

20 -1.24 0.1075 0.000 -0.43 0.3336 0.000

0.8 --

-0.6 -

0 4

0.4 --

0.2 Prospective 44

02 .... Retrospective 44

0

0 5 10 15 20

Concentration

Figure 10.1 Example Power Curves: Sign Test Scenario A

10-3 NUREG-1505
POWER

Notice that the increase of a due to a higher than anticipated measurement standard deviation

causes the retrospective power curve to differ considerably from the prospective power curve. In

Table 3.3, we see that A/c = (15.9 - 11.5)/9.5 = 0.46 results in a much larger required sample

size (over 100) to achieve the desired power. Recall that in this example, S+ = 11, which is

smaller than the critical value k = 14. Thus the null hypothesis was not rejected. The survey unit

did not pass. We now see that this might have been a consequence of having insufficient power

rather than the survey unit actually exceeding the release criterion. The lack of power was due to

underestimating the measurement variability.

10.3 Power of the Sign Test Under Scenario B

Recall that for the Sign test in Scenario B, the test statistic, S+, was equal to the number of

survey unit measurements above the LBGR. If S+ exceeds the critical value k, then the null

hypothesis that the median concentration in the survey unit is less than the LBGR is rejected, i.e.

the survey unit does not pass. The probability that any single survey unit measurement falls

below the DCGLw, is found from Equation 9-4 or 9-5. The probability that more than k of the N

survey unit measurements fall above the LBGR is simply the following binomial probability:

N (N) [p]Y[l p]_ = 1- k (N) [p]'[a p]N_ - _-Np(0- k 

i=k+l i=0 VNp (I -p)

The indicated approximation is generally used when both Np and N(1 -p) are five or greater.

1'(z) is the cumulative standard normal distribution function given in Table A. 1.

With p calculated as in Section 9.3, this is the probability that the null hypothesis is rejected

when the true median of the residual radioactivity concentration in the survey unit is at the

DCGLw. This is the power of the test at the DCGLw.

The probability, p(C), that any single survey unit measurement falls above the LBGR when the

survey unit median concentration is at any other value, C, can be determined by simply replacing

the value of the DCGLw in Equation 9-4 with the value of C:

A1 f e_-(x_22d

0

LBGR

1 f e -(x - C)92o2

°dx

%• 0 (LBGR-C)+C

1 e _x2

12dx

Vfi2" LBGR -C

a

NUREG-1505 10-4
POWER

C -LBGR

-f ex 2

1dx

=((C~-LBGR) (10-4)

Note that if C = LBGR, p(C) 0 )(0) ; 0.5. The assumption of normality is not critical in the

preceding calculations, since it is only being used to estimate the power. However, if a different

distribution is considered more appropriate, Equation 9-5 can be used to calculate p(C).

When the value of p(C) from Equation 10-4 is inserted into Equation 10-2, we obtain the

probability that the null hypothesis is rejected at the concentration, C. When C = LBGR, this

probability is the probability of a Type I error, a(2). This calculation can even be performed for

values of C less than the LBGR. The probability obtained is still the probability that the null

hypothesis is rejected, i.e., that the survey unit passes the test, but it is not normally referred to as

the power.

If the probability that the null hypothesis is rejected (calculated from Equation 10-3) is plotted

against the concentration, C, the result is called a power curve. When the power calculation is

performed at the design stage, using an estimated value of a, it is called a prospective power

curve. When the calculation is performed after the survey, using the standard deviation of the

survey unit measurements as an estimate of a, it is called a retrospective power curve.

To illustrate the construction of a power curve, consider the example of Chapter 5. The DCGLw

for this example was 15.9 and the LBGR was 11.5. The DQOs for a = P = 0.05 resulted in a

sample size of N = 21, using the estimate that d = 3.3. From Table A.3, the critical value for the

Sign test with N =21 and x = 0.05 is k = 14. This is all of the information necessary to construct

the prospective power curve. To construct the retrospective power curve, we use the standard

deviation of the measurement data, 9.5, as the estimate of a. The results of these calculations are

shown in Table 10.2 and Figure 10.2.

Notice that the increase of a due to a higher than anticipated measurement standard deviation

causes the retrospective power curve to differ considerably from the prospective power curve.

A/a = (15.9 - 11.5)/9.5 = 0.46 results in a much larger required sample size to achieve the

desired power. Recall that in this example, S+ = 13, which is smaller than the critical value,

k = 14. Thus the null hypothesis was not rejected. The survey unit passes. We now see that this

might have been a consequence of having insufficient power rather than the survey unit actually

meeting the release criterion. The lack of power was due to underestimating the measurement

variability.

(2) The value of a actually obtained from Equation 10-2 should be close to that specified in the DQOs. It

may not exactly equal that value when the sample sizes are small, since the critical value, k, can only take integer

values.

10-5 NUREG-1505
POWER

Table 10.2 Example Power Calculations: Sign Test Scenario B

Prospective Retrospective

C (C - LBGR)lu p(C) power (C - LBGR)lo p(C) power

(Eq. 10-3) (Eq. 10-4) (Eq. 10-2) (Eq. 10-1)

0 -3.48 0.0003 0.000 -1.21 0.1131 0.000

5 -1.97 0.0244 0.000 -0.68 0.2483 0.000

6 -1.67 0.0475 0.000 -0.58 0.2810 0.000

7 -1.36 0.0869 0.000 -0.47 0.3192 0.000

8 -1.06 0.1446 0.000 -0.37 0.3557 0.001

9 -0.76 0.2236 0.000 -0.26 0.3974 0.003

10 -0.45 0.3264 0.000 -0.16 0.4364 0.009

11 -0.15 0.4404 0.010 -0.05 0.4801 0.026

11.5 0.00 0.5000 0.039 0.00 0.5000 0.039

12 0.15 0.5596 0.112 0.05 0.5199 0.057

13 0.45 0.6736 0.445 0.16 0.5636 0.119

14 0.76 0.7764 0.830 0.26 0.6026 0.207

15 1.06 0.8554 0.976 0.37 0.6443 0.336

15.9 1.33 0.9082 0.998 0.46 0.6772 0.459

16 1.36 0.9131 0.999 0.47 0.6808 0.474

17 1.67 0.9525 1.000 0.58 0.7190 0.627

18 1.97 0.9756 1.000 0.68 0.7517 0.750

19 2.27 0.9884 1.000 0.79 0.7852 0.854

20 2.58. 0.9951 1.000 0.89 0.8133 0.919

21 2.88 0.9980 1.000 1.00 0.8413 0.962

22 3.18 0.9993 1.000 1.11 0.8665 0.984

23 3.48 0.9997 1.000 1.21 0.8869 0.993

24 3.79 0.9999 1.000 1.32 0.9066 0.998

25 4.09 1.0000 1 .000 1.42 0.9222 0.999

In Scenario A, the power and the probability that the survey unit passes the test age equivalent. In

Scenario B, the power is equivalent to the probability that the survey unit does not pass. To plot

the probability that the survey unit passes, the power is subtracted from 1. The result is shown in

Figure 10.3.

10.4 Power of the Wilcoxon Rank Sum Test Under Scenario A

Recall that for the Wilcoxon Rank Sum (WRS) test in Scenario A, the test statistic, W, was

equal to the sum of the ranks of the reference area measurements adjusted for the DCGLw. If Wr

exceeds the critical value W,, then the null hypothesis that the median concentration in the

survey unit exceeds that in the reference area by more than the DCGLw is rejected, i.e., the

survey unit passes this test.

The power of the WRS test is very difficult to calculate exactly. However, a good approximation

is available (Lehmann and D'Abrera, 1975,p(C) Chapter 2, Section 3, pp. 69-75).

NUREG- 1505 10-6
POWER

1.0

0.8

0.6

I. 0.4 /

/

0.2 Prospective

- - Retrospective /

0

0 5 10 15

Concentration

70

/

/

/

I

/

I

20 25

Figure 10.2 Example Power Curves: Sign Test Scenario B

1.0 00

0

0

(A

M

I.

0

.0

S

Ia.

0.4+

0.2

Prospective

- - Retrospective

0 -- I I . . . . , I . 1 , 0 1 , I

0 5 10 15 20 25'

Concentration

Figure 10.3 Probability Example Survey Unit Passes: Sign Test Scenario B

10-7 NUREG-1505
POWER

If the distribution of the Mann-Whitney form of the WRS test statistic is approximated by a

normal distribution, the probability that the null hypothesis will be rejected when the alternative

is true can be calculated from:

Power = -IP[W,-0.5 -O.5m(m+l)-E(10-5)

.lVarCWmw) (10-5)

where W, is the critical value found in Table A.4 for the appropriate values of the Type I error, a,

the number of survey unit measurements, n, and the number of reference area measurements, m.

E(WMw) and Var(WMw) are the mean and variance of the Mann-Whitney form of the WRS test

statistic. Values of (D(z), the standard normal cumulative distribution function, are given in

Table A.1.

The Mann-Whitney form of the WRS test statistic is WMw = W, - 0.5m(m+ 1). It is obtained by

subtracting from Wr its minimum value, 0.5m(m+ 1). The mean of WMw is

E(WMw) = mnp, (10-6)

where p, is the probability that any single measurement from the survey unit exceeds a single

measurement from the reference area by less than the DCGLw. This probability depends on the

difference in median concentration between the survey unit and the reference area. When this

difference is equal to the LBGR, then p, is equal to P, as calculated from Equation 9-7. For other

values of the difference median concentration between the survey unit and the reference area, C,

we simply replace the LBGR in Equation 9-7 with C:

P,(C) = Probability(U =X - Y < DCGL)

DC1ffr( D + yy)dy}du

fC 1 -(u.y-c-CBKGD) 2 2o2 I -(y-BKGDI 2 dyru

DCGL

1 f e _(uC)214 2du

VeV o

NUREG-1505 10-8
POWER

DCGLw- C

= 1 _•2 /2dx

= V~(10-7) a 

Note that if C = DCGLw, then p(C) = 0.5. The assumption of normality is not critical in the

preceding calculations, since it is only being used to estimate the power. However, if a different

distribution is considered more appropriate, Equation 9-8 can be used to calculate p I(C).

The variance of WMw is:

Var(WMw) = mnp1 (1 -p1 ) +mn(n-')(p2 -pr) +mn(m-1)(p3 -p2) (10-8)

P2 is the'probability that two random measurements from the survey unit will each exceed a

single random measurement from the reference area by less than the DCGLw; and P3iS the

probability that a single random measurement from the survey unit will exceed each of two

random measurements from the reference area unit by less than the DCGLw. When the

difference in the concentration distributions of the survey unit and the reference area

measurements consists of a shift in the median, and the measurement distributions are

symmetric, then P2 = P3. Then the variance of Wmw simplifies to

Var(WMw) = mnpl(1-p,) +mn(n+m-2)(P2-Pr) (10-9)

If the measurement distributions are normal, then P2 is equal to the probability that two correlated

standard normal random variables (i.e., with mean = 0 and variance = 1), with correlation

coefficient 0.5, are both less than (DCGLw - C)I(aV2-). This probability also depends on the

difference in median concentration, C, between the survey unit and the reference area. Even with

the simplifications employed, the values of p2'are not easy to calculate. Table 10.3 provides

values of p, and P2 as a function of (DCGLw - C)/o that can be used in calculating the mean and

variance of Wmw. Nomographs of bivariate normal probabilities that can also be used for this

purpose are given in Abramowitz and Stegun (1972).

The power calculated using Equations 10-5 through 10-8 is an approximation. This

approximation was compared against the power simulations for the WRS test reported by Gilbert

and Simpson (PNL-7409, 1992). It was found that the approximation is sufficiently accurate to

determine if the sample design achieves the DQOs.

10-9 NUREG-1505
POWER

Table 10.3 Values ofp, and P2 for Computing the Mean and Variance of WMW (3)

(DCGLw - C)Io Pi P2 (DCGLw - C)Io Pi P2

-6.0 0.000010 0.000000 0.7 0.689691 0.544073

-5.0 0.000204 0.000010 0.8 0.714196 0.574469

-4.0 0.002339 0.000174 0.9 0.737741 0.604402

-3.5 0.006664 0.000738 1.0 0.760250 0.633702

-3.0 0.016947 0.002690 1.1 0.781662 0.662216

-2.5 0.038550 0.008465 1.2 0.801928 0.689800

-2.0 0.078650 0.023066 1.3 .0.821015 0.716331

-1.9 0.089555 0.027714 1.4 0.838901 0.741698

-1.8 0.101546 0.033114 1.5 0.855578 0.765812

-1.7 0.114666 0.039348 1.6 0.871050 0.788602

-1.6 0.128950 0.046501 1.7 0.885334 0.810016

-1.5 0.144422 0.054656 1.8 0.898454 0.830022

-1.4 0.161099 0.063897 1.9 0.910445 0.848605

-1.3 0.178985 0.074301 2.0 0.921350 0.865767

-1.2 0.198072 0.085944 2.1 0.931218 0.881527

-1.1 0.218338 0.098892, 2.2 0.940103 0.895917

-1.0 0.239750 0.113202 2.3 0.948062 0.908982

-0.9 0.262259 0.128920 2.4 0.955157 0.920777

-0.8 0.285804 0.146077 2.5 0.961450 0.931365

-0.7 0.310309 0.164691 2.6 0.967004 0.940817

-0.6 0.335687 0.184760 2.7 0.971881 0.949208

-0.5 0.361837 0.206266 2.8 0.976143 0.956616

-0.4 0.388649 0.229172 2.9 0.979848 0.963118

-0.3 0.416002 0.253419 3.0 0.983053, 0.968795

-0.2 0.443769 0.278930 3.1 0.985811 0.973725

-0.1 0.471814 .0.305606 3.2 0.988174 0.977981

0.0 0.500000 0.333333 3.3 0.990188 0.981636

0.1 0.528186 0.361978 3.4 0.991895 0.984758

0.2 0.556231 0.391392 3.5 0.993336 0.987410

0.3 0.583998 0.421415 4.0 0.997661 0.995497

0.4 0.611351 0.451875 5.0 0.999796 0.999599

0.5 0.638163 0.482593 6.0 0.999989 0.999978

0.6 0.664313 0.513387

When the values ofpI(C) and p2(C) and the resulting values of E(WMwv) and Var(WMw) are

inserted in Equation 10-5, we obtain the probability that the null hypothesis is rejected at

concentration C. When C = DCGLw, this probability is the probability of a Type I error, a.

(

3

OThis table may also be used for Scenario B when (DCGLw - C)Io is replaced by (C - LBGR)Io.

(4) The value of a actually obtained from Equation 10-5 should be close to that specified in the DQOs. It

may not exactly equal that value when the sample sizes are small, since the critical value, k, can only take integer

values.

NUREG-1505 10-10
POWER

The preceding calculations can even be performed for values of C greater than the DCGLW. The

probability obtained is still the probability that the null hypothesis is rejected, i.e., that the survey

unit passes the test.

If the probability that the null hypothesis is rejected (calculated from Equation 10-5) is plotted

against the concentration, C, the result is called a power curve. When the power calculation is

performed at the design stage, using an estimated value of a, it is called a prospective power•

curve. When the calculation is performed after the survey, using the standard deviation of the

survey unit measurements as an estimate of a, it is called a retrospective power curve.

To illustrate the construction of a power curve, consider the example of Chapter 6. The DCGLw

for this example was 160 and the LBGR was 142. The DQOs for a = = 0.05 resulted in a

sample size of n = m = 10, using the estimate that a = 6. Twelve samples each were actually

taken from the survey unit and the reference area. From Table A.4, the critical value for the WRS

test with n = m = 12 and a = 0.05 is W, = 179. This is all of the information necessary to

construct the prospective power curve. To construct the retrospective power curve, we use the

larger of the standard deviations of the measurement data from the survey unit and the reference

area, 8. 1 as the estimate of a.

The results of these calculations are shown in Table 10.4 and Figure 10.4. In the figure it can be

seen that the retrospective power is slightly less than that specified in the DQOs. However, in

this example, the null hypothesis was rejected, so the question of the power is moot. The

retrospective power calculation is really only necessary when the null hypothesis is not rejected.

In that case, it is important to know that it was not rejected simply because there was insufficient

power. When the null hypothesis is rejected in spite of insufficient power, the survey designer

can consider himself lucky, but the conclusion is still statistically valid.

Table 10.4 Example Prospective Power Calculation: WRS Test Scenario A

C (DCGLw - C)/o .P1 P2 E(WMw) Var(WMw) SD(Wuw) z Power

136 4.00 0.997661 0.995497 143.7 0.9 - 0.9 -46.21 1.00

139 3.50 0.993336 0.987410 143.0 3.2 1.8 -23.96 1.00

142 3.00 0.983053 0.968795 141.6 10.0 3.2 -12.98 1.00

145 2.50 0.961450 0.931365 138.4 27.4 5.2 -7.24 1.00

148 2.00 0.921350 0.865767 132.7 63.9 8.0 -4.02 1.00

151 1.50 0.855578 0.765812 123.2 124.9 11.2 -2.03 0.98

154 1.00 0.760250 0.633702 109.5 202.8 14.2 -0.63 0.74

157 0.50 0.638163 0.482593 91.9 271.9 16.5 0.52 0.30

160 0.00 0.500000 0.333333 72.0 300.0 17.3 1.65 0.05

163 -0.50 0.361837 0.206266 52.1 271.9 16.5 2.93 0.00

166 -1.00 0.239750 0.113202 34.5 202.8 14.2 4.63 0.00

169 -1.50 0.144422 0.054656 20.8 124.9 11.2 7.13 0.00

172 -2.00 0.078650 0.023066 11.3 63.9 8.0 11.15 0.00

10-11 NUREG-1505
POWER

1.0 S

N

0.9--

0.8--

0.7

s.0.6-

00.50"

0.4--

0.3-- Prospective

0.2 --- - Retrospective 0.1 •l•o':-• "

136 139 142 145 148 151 154 157 160 163 166 169 172

Concentration

Figure 10.4 Example Power Curves: WRS Test Scenario A

10.5 Power of the Wilcoxon Rank Sum Test Under Scenario B

Recall that for the WRS test in Scenario B, the test statistic, W5 , was equal to the sum of the

ranks of the survey unit measurements adjusted for the LBGR. If W, exceeds the critical value

We, then the null hypothesis that the median concentration in the survey unit exceeds that in the

reference area by less than the LBGR is rejected, i.e., the survey unit does not pass this test.

The power of the WRS test in Scenario B can be approximated in a manner similar to that used

in Scenario A, using Equations 10-5, 10-6 and 10-9:

Power, - 0.W-O5 -O'5m(m+I1)-- E(Wmw).

Power = 1 - Va

PiV&(mw)

E(WMw) = mnp,

Var(WMw) = mnp1(l -p 1) +mn(n+m-2)(p2 -p;)

NUREG-1505 10-12
POWER

W, is the critical value found in Table A.4 for the appropriate number of survey unit

measurements, n, and number of reference area measurements, m. Since under Scenario B, both

the WRS test and the Quantile test are used in tandem, the value of the Type I error,' a, decided

on during the DQO process, is halved for each test. Thus, the Table A.4 value for value of W, for

aw = a/2 is used. E(WMw) and Var(Wuw) are the mean and variance of the Mann-Whitney form

of the WRS test statistic for Scenario B, namely WMW = W, - n(n+l)/2. Values of 4(z), the

standard normal cumulative distribution function, are given in Table A. 1.

In Scenario B, P, is the probability that any single measurement from the survey unit exceeds a

single measurement from the reference area by more than the LBGR .This probabilitydepends

on the difference in median concentration between the survey unit and the reference area. When

this difference is equal to the DCGLw, then p, is equal to P, as calculated from Equation 9-9. For

other values of the difference median concentration between the survey unit and the reference

area, C, we simply replace the DCGLw in Equation 9-9 with C:

Pr(C) Probability(U =X - Y > LBGR)

= + y)jfr(y)dy du

C -.fiBGR

_ 4 f o _x 2 a2dx

= •( C-LBGR) (10-10)

This is the same as Equation 10-7, with (DCGLw - C) replaced by (C - LBGR). Although the

definition of p, has changed, its value may still be found from Table 10.3 when (C - LBGR)Io

is substituted for (DCGLw - C)Io. Note that if C = LBGR then p, (C) = 0.5. The assumption of

normality is not critical in the above calculations, since it is only being used to estimate the

power. However, if a different distribution is considered more appropriate, Equation 9-10 can be

used to calculate p,(C).

10-13 NUREG-1505
POWER

In Scenario B, P2 is the probability that two random measurements from the survey unit will

each exceed a single random measurement from the reference area by more than the LBGR; and

p3 is the probability that a single random measurement from the survey unit will exceed each of

two random measurements from the reference area unit by more than the LBGR. When the

difference in the concentration distributions of the survey unit and the reference area

measurements consists of a shift in the median, and the measurement distributions are

symmetric, then P2 = P3. If the measurement distributions are normal, then P2 is equal to the

probability that two correlated standard normal random variables (i.e., with mean = 0 and

variance = 1), with correlation coefficient 0.5, are both less than (C - LBGR)I(ovW ). This

probability also depends on the difference in median concentration, C, between the survey unit

and the reference area. Again, values of P2 may be obtained from Table 10.3 when (C-LBGR)/o

is substituted for (DCGLw - C)lo.

Although the power calculated as above is an approximation, this approximation has been

compared against the power simulations for the WRS test reported by Gilbert and Simpson

(PNL-7409, 1992). It was found that the approximation is sufficiently accurate to determine if the

sample design achieves the DQOsWhen the values of p,(C) and P2(C) from Table 10.3, and the resulting E(WMw) and Var(WMw) are

inserted in Equation 10-5, we obtain the probability that the null hypothesis is rejected at the

concentration C. When C = DCGLw, this probability is the probability of a Type I error,

= -w a/2.(5). This calculation can even be performed for values of C less than the LBGR . The

probability obtained is still the probability thai the null hypothesis is rejected, i.e., that the survey

unit passes the test, but it is not usually referred to as the power.

If the probability that the null hypothesis is rejected (calculated from Equation 10-5) is plotted

against the concentration, C, the result is called a power curve. When the power calculation is

performed at the design stage, using an estimated value of a, it is called a prospective power

curve. When the calculation is performed after the survey, using the standard deviation of the

survey unit measurements as an estimate of a, it is called a retrospective power curve.

To illustrate the construction of a power curve, consider the example of Chapter 6. The DCGLw

for this example was 160 and the LBGR was 142. The DQOs for cw = ad2 = 0.025, and P = 0.05,

result in a sample size of n = m = 12, using the estimate that a = 6. From Table A.4, the critical

value for the WRS test with n = m = 12 and a = 0.025 is W, = 184. This is all of the information

necessary to construct the prospective power curve. To construct the retrospective power curve,

we use the larger of the standard deviations of the measurement data from the survey unit and the

reference area, 8.1, as the estimate of a.

The results of these calculations are shown in Table 10.5 and Figure 10.5.

(5) Thd value of a actually obtained from Equation 10-5 should be close to that specified in the DQOs. It

may not exactly equal that value when the sample sizes are small, since the critical value, k, can only take integer

values.

NUREG-1505 10-14
POWER

Table 10.5 Example Prospective Power Calculation: WRS Test Scenario B

C (C-LBGR)/o V p- E(W.uw) Var(W.;) SD(W.,W) z Power

136 -1.0 0.239750 0.113202 34.5 202.8 14.2 4.98 0.00

139 -0.5 0.361837 0.206266 52.1 271.9 16.5 3.24 0.00

142 0.0 0.500000 0.333333 72.0 300.0 17.3 1.93 0.03

145 0.5 0.638163 0.482593 91.9 271.9 16.5 0.82 0.20

148 1.0 0.760250 0.633702 109.5 202.8 14.2 -0.28 0.61

151 1.5 0.855578 0.765812 123.2 124.9 11.2 -1.58 0.94

154 2.0 0.921350 0.865767 132.7 63.9 8.0 -3.40 1.00

157 2.5 0.961450 0.931365 138.4 27.4 5.2 -6.29 1.00

160 3.0 0.983053 0.968795 141.6 10.0 3.2 -11.40 1.00

163 3.5 0.993336 0.987410 143.0 3.2 1.8 -21.15 1.00

166 4.0 0.997661 0.995497 143.7 0.9 0.9 -40.85 1.00

172 5.0 0.999796 0.999599 144.0 0.0 0.2 -175.07 1.00

In the figure it can be seen that the retrospective power is slightly less than that specified in the

DQOs. However, in this example the null hypothesis was rejected, so the question of the power

is moot. The retrospective power calculation is really only necessary when the null hypothesis is

accepted. In that case it is important to know that it was not accepted simply because there was

insufficient power. When the null hypothesis is rejected in spite of insufficient power, the survey

designer can consider himself lucky, but the conclusion is still statistically valid.

1.0

0.9.

0.8

0.7-

a0.6

00.5-

0.4.

0.3

0.2

0.1-

0,: II : i : II: iii III Ii: :

136 139 142 145 148 151 154 157 160 163 166 169 172

Concentration

Figure 10.5 Example Power Curves: WRS Test Scenario B

In Scenario A, the power and the probability that the survey unit passes the test are equivalent. In

10-15 NUREG-1505
POWER

Scenario B, the power is equivalent to the probability that the survey unit does not pass. Thus,

the probability that the survey unit passes is one minus the power. The result is plotted in Figure

10.6.

1.0-

0.9

Co0 0.8 " I

0.7

0.6-

0.5

0.4"°

- Prospective 0.3 ,,,, " " Retrospectlv

O0.2--

0.1

0

136 139 142 145 148 151 154 157 160 163 166 169 172

Concentration.

- Figure 10.6 Probability Example Survey Unit Passes: WRS Test Scenario B

.NUREG-1505 10-16
11 MULTIPLE RADIONUCLIDES

There are two cases to be considered when dealing with multiple radionuclides, namely (1) when

the radionuclide concentrations have a fairly constant ratio throughout the survey unit, or (2)

when the concentrations of the different radionuclides appear to be unrelated in the survey unit.

In statistical terms, we are concerned about whether the concentrations of the different

radionuclides are correlated or not. A simple way to judge this would be to make scatter plots of

the concentrations against each other, and see if the points appear to have an underlying linear

pattern. The correlation coefficient can also be computed to see if it lies nearer to zero than to

one. One could also perform a curve fit and test the significance of the result. Ultimately,

however, sound judgment must be used in interpreting the results of such calculations. If there is

no physical reason for the concentrations to be related, they probably are not. Conversely, if there

is sound evidence that the radionuclide concentrations should be related because of how they

were treated, processed or released, this information should be used.

11.1 Using the Unity Rule

In either of the two above cases, the unity rule described in Section 3.3 is applied. The difference

is in how it is applied. Suppose there are n radionuclides. If the concentration of radionuclide i is

denoted by Ci, and its DCGLw is denoted by Di, then the unity rule for the n radionuclides states

that

C11D, + C21D2 + C31D3 + ... + C.ID,, • 1 (11-1)

This will ensure that the total dose due to the sum of all the radionuclides does not exceed the

release criterion. Note that if Dmin is the smallest of the DCGLs, then

(C1 + C2 + C3 + C,. )/Dmi. < C1ID, + C21D2 + C31D3 + ... + C.ID. (11-2)

So that the smallest DCGL may be applied to the total activity concentration, rather than using

the unity rule. While it is an option to consider, in many cases this approach will be too

conservative to be useful.

11.2 Radionuclide Concentrations With Fixed Ratios

If there is an established ratio among the concentrations of the n radionuclides in a survey unit,

then the concentration of every radionuclide can be expressed in terms of any one of them, e.g.,

radionuclide #1. The measured radionuclide is often called a surrogate radionuclide for the

others.

If C2 = R2C1, C3 = R3C1, ..., Ci = RiC1 , ..., C, = RC,,C

then

C11D, + C21D2 + C31D3 + - + C,.ID,, = C, ID, + R2C1 ID2 + R3C1 ID3 + ... + RC, ID,.

= C1 (lID 1 + R21D2 + R31D3 + ... + RID,.

= CI/Dto, (11-3)

11-1 NUREG-1505
MULTIPLE RADIONUCLIDES

where

Dttal = 1/ [liD1 + R21D2 + R31D3 + + R,,ID,,]. (11-4)

Thus, DtotaI is the DCGLw for the surrogate radionuclide when the concentration of that

radionuclide represents all radionuclides that are present in the survey unit. Clearly, this scheme

is applicable only when radionuclide-specific measurements of the surrogate radionuclide are

made. It is unlikely to apply in situations where the surrogate radionuclide appears in

background, since background variations would tend to obscure the relationships between it and

the other radionuclides.

Thus, in the case in which there are constant ratios among radionuclide concentrations, the

statistical tests are applied as if only the surrogate radionuclide were contributing to the residual

radioactivity, with the DCGLw for that radionuclide replaced by D1t

0 . For example, in planning

the final status survey, only the expected standard deviation of the concentration measurements

for the surrogate radionuclide is needed to calculate the sample size.

For the elevated measurement comparison, the DCGLEMC for the surrogate radionuclide is

replaced by

Etotd I=/ [lE, + R2 /E2 +R 3 IE3 + + Rn1En], (11-5)

where E, is the DCGLEMc for radionuclide i.

11.3 Unrelated Radionuclide Concentrations

If the concentrations of the different radionuclides appear to be unrelated in the survey unit, then

there is little alternative to measuring the concentration of each radionuclide and using the unity

rule. The exception would be in applying the most restrictive DCGLw to all of the radionuclides,

as mentioned in Section 11.1.

Since the release criterion is

C1D 1 + C2 ID2 + C3 ID3 + ... + C/ ID, < 1.0 (11-6)

the quantity to be measured is the weighted sum, T = C1 ID, + C2 ID 2 + C3 ID3 + + Cn ID,.

The DCGLw for T is 1.0. In planning the final status survey, the measurement standard deviation

of the weighted sum, T, is estimated by

d(T) = [ C)ID]2 1(+ [o(C2)/D2 2 + [0(C 3)ID3 ] 2 + ... + [o([C.)ID ]2, (1 1-7)

since the measured concentrations of the various radionuclides are assumed to be uncorrelated.

For the elevated measurement comparison, the inequality .

C1/E1 + C21E2 + C3/E3 + ... + Cn/En < 1.0 (11-8)

NUREG-1505 11-2
MULTIPLE RADIONUCLIDES

is used, where E1 is the DCGLImc for radionuclide i. For scanning,' most restrictive DCGL.Mc

should generally be used.

When some of the radionuclides also appear in background, the quantity T = C, IDI + C21ID2 + C31ID3+ -. + C,, ID, "

mustalso be measured in an appropriate reference area. If radionuclide i does not appear in

background, set C1 = 0 in the calculation of T for the reference area.

Note that if there is a fixed ratio between the concentrations of some radionuclides, but not

others, a combination of the method of this section with that of the previous section may be used,

using the appropriate value of Dtota, with the concentration of the measured surrogate radionuclide

to replace the corresponding terms in Equation 11-7.

11.4 Example Application of WRS Test to Multiple Radionuclides

This section contains an example application of the nonparametric statistical methods in this

report to sites that have residual radioactivity from more than one radionuclide. Consider a site

with both WCo and 1"Cs contamination. `"7Cs appears in background from global atmospheric

weapons tests at a typical concentration of about 1 pCi/g. Assume that the DCGLwfor To is 2

pCilg and that for 131Cs is 1.4 pCi/g. In disturbed areas, the background concentration of '3aCs

can vary considerably. An estimated spatial standard deviation of 0.5 pCi/g for `3Cs will be

assumed. During remediation it was found that the concentrations of the two radionuclides were

not well correlated in the survey unit. 0Co concentrations were more variable than the '37cs

concentrations, and 0.7 pCi/g is assumed for its standard deviation. Measurement errors for both

6°Co and '37Cs using gamma spectrometry will be small compared to this. For the comparison to

the release criteria, the weighted sum of the concentrations of these radionuclides is computed

from the following:

Weighted sum = (Co concentration)/(6WCo DCGLw) + (13'Cs concentration)/(' 37Cs DCGLW)

= (6°Co concentration)/(2) + ('37Cs concentration)/(1.4)

The variance of the weighted sum, assuming that the 6°Co and '37Cs concentrations are spatially

unrelated is

a2 = [(60

Co standard deviation)/(6°Co DCGLW)] 2 + [(137Cs standard deviation)/(' 37Cs DCGLw)]2

= [(0.7)/(2)]2 + [(0.5)/(1.4)]2 = 0.25.

Thus a = 0.5. The DCGLw for the weighted sum is one. Scenario A will be used, i.e., the null

hypothesis is that the survey unit exceeds the release criterion. During the DQO process, the

LBGR was set at 0.5 for the weighted sum, so that A = DCGLw - LBGR =1.0 - 0.5 = 0.5, and

A/a = 0.5/0.5 = 1.0. The acceptable error rates chosen were a = f3 = 0.05. To achieve this, 32

samples each are required in the survey unit and the reference area.

The weighted sums are computed for each measurement location in both the reference area and

the survey unit. The WRS test is then performed on the weighted sum. The calculations for this

example are shown in Table 11.1.

11-3 NUREG-1505
MULTIPLE RADIONUCLIDES

Table 11.1 Example WRS Test for Two Radionuclides

_ Reference Area Surve Unit Weighted Sum Ranks

_ 

37Cs 60Co 1

37Cs "Co Ref Survey Adj Ref Survey Adj Ref

1 2 0 1.12 0.06 1.43 0.83 2.43 1 56

2 1.23 0 1.66 1.99 0.88 2.18 1.88 43 21

3 0.99 0 3.02 0.56 0.71 2.44 1.71 57 14

4 1.98 0 2.47 0.26 1.41 1.89 2.41 23 55

5 1.78 0 2.08 0.21 1.27 1.59 2.27 9 50

6 1.93 0 2.96 0.00 1.38 2.11 2.38 37 54

7 1.73 0 2.05 0.20 1.23 1.56 2.23 7 46

8 1.83 0 2.41 0.00 1.30 1.72 2.30 16 52

9 1.27 0 1.74 0.00 0.91 1.24 1.91 2 24

10 0.74 0 2.65 0.16 0.53 1.97 1.53 27 6

11 1.17 0 1.92 0.63 0.83 1.68 1.83 13 18

12 1.51 0 1.91 0.69 1.08 1.71 2.08 15 32

13 2.25 0 3.06 0.13 1.61 2.25 2.61 47 63

14 1.36 0 2.18 0.98 0.97 2.05 1.97 30 28

15 2.05 0 2.08 1.26 1.46 2.12 2.46 39 58

16 1.61 0 2.30 1.16 1.15 2.22 2.15 45 41

17 1.29 0 2.20 0.00 0.92 1.57 1.92 8 25

18 1.55 0 3.11 0.50 1.11 2.47 2.11 59, 35

19 1.82 0 2.31 0.00 1.30 1.65 2.30 11 51

20 1.17 0 2.82 0.41 0.84 2.22 1.84 44 19

21 1.76 0 1.81 1.18 1.26 1.88 2.26 22 48

22 2.21 0 2.71 0.17 1.58 2.02 2.58 29 62

23 2.35 0 1.89 0.00 1.68 1.35 2.68 3 64

24 1.51 0 2.12 0.34 1.08 1.68 2.08 12 33

25 0.66 0 2.59 0.14 0.47 1.92 1.47 26 5

26 1.56 0 1.75 0.71 1.12 1.60 2.112 10 38

27 1.93 0 2.35 0.85 1.38 2.10 2.38 34 53

28 2.15 0 2.28 0.87 1.54 2.06 2.54 31 61

29 2.07 0 2.56 0.56 1.48 2.11 2.48 36 60

30 1.77 0 2.50 0.00 1.27 1.78 2.27 17 49

31 1.19 0 1.79 0.30 0.85 1.43 1.85 4 20

32 1.57 0 2.55 0.70 1.12 2.17 2.12 42 40

Avg 1.62 0 2.28 0.47 1.16 1.86 2.16 Sum= Sum=

Std Dev 0.43 0 0.46 0.48 0.31 0.36 0.31 799 1281

In Scenario A, the DCGLw (i.e., 1.0) is added to the weighted sum for each location in the

reference area. The ranks of the combined survey unit and adjusted reference area weighted sums

are then computed. The sum of the ranks of the adjusted reference area weighted sums is then

compared to the critical value for n = m = 32, a = 0.05, which is 1162 (see formula following

NUREG-1505 11-4.
MuLTIPLE RADiONUCLIDES

Table AA4). In Table 11.1, the sum of the ranks of the adjusted reference area weighted sums is

1281. This exceeds the critical value, so the null hypothesis is rejected. The survey unit meets the

release criterion. The difference between the mean of the weighted sums in the survey unit and

the reference area is 1.86 - 1.16 = 0.7. Thus, the estimated dose due to residual radioactivity in

the survey unit is 70% of the release criterion.

11-5 NUREG-1505
t.
12 MULTIPLE SURFACES

12.1 Choosing Survey Units

Three criteria mentioned in Section 2.2 constrained the choice of survey units:

(1) Classification by Contamination Potential: Survey units are composed of areas with similar

usage, contamination, and remediation histories that determine the requirements of the final

status survey.

(2) Congruity With the Dose Model Used: When the release criteria are dose-based, the survey

unit configuration should be consistent with those assumed in the dose model used.

(3) Data Variability: The measurement data variability, T, within survey units should be

minimized so that acceptable decision error rates can be obtained with efficient sample sizes.

These criteria should guide the selection of survey units during the DQO process. However, there

are situations in which it will necessary to balance the requirements of one criterion against the

requirements of another. In those circumstances, one should be guided by the ultimate objective

of the final status survey, namely to make the correct decision on whether the survey unit meets

the release criterion.

As an example, consider a room with a concrete floor, one wall with tile, another with wallboard,

a third with glass doors and windows, and a fourth with a large blackboard. There are at least five

different surfaces, with potentially five different levels of residual radioactivity. Using only

criteria (1) and (3), it might seem important to treat these as five distinct survey units. However,

this is not only very inefficient, it may not even be the best solution. It is unlikely that any dose

model treats a contaminated blackboard by itself. Modeling the five areas separately and

combining the results may not be as faithful a representation as treating all the surfaces together

as one contaminated room.

In this chapter, we discuss some of the factors to be considered during the DQO process to

optimize the choice of survey units. We also introduce the concept of performing a Sign test on

paired measurements when there are many diverse background materials present in a survey unit.

12.2 Combining Dissimilar Areas Into One Survey Unit

The primary disadvantages in separating very small areas of dissimilar contamination potential

into distinct survey units are that such small survey units will not generally conform well to dose

models, and that the resulting sample densities may be unreasonably high. The disadvantage of

combining such areas is that the resulting survey unit will have a larger spatial concentration

variability, requiring a larger sample size than if it were more uniform. However, the total sample

size may not be as large as would be needed for separate survey units. This possibility can be

investigated and resolved during the DQO process.

12-1 12-I NUREG-1505
MULTIPLE SURFACES

Consider an area such as shown in Figure 12.1: a 25 m by 100 m gravel parking lot, with a paved

walkway across a 25 m by 100 m lawn to a building. The walkway covers about 400 m2 of the

lawn area. There are many ways that this area could be divided into survey units, depending on

the level of expected 'contamination.

Figure 12.1 Example Survey Units: Case #1

Example Case #1: 6Co was handled in hoods that were vented from the roof of the building.

There is the possibility of a small amount of residual radioactivity, but at levels anticipated to be

far below the DCGL. There is virtually no possibility of isolated elevated areas. All of the areas

would be considered Class 2. Surface samples will be analyzed for 'Co by Ge gamma

spectrometry. The total area (5000 mi) is within the parameters used by the dose model.

Although it is possible that average residual radioactivity levels between the walk, the gravel, and

the lawn are different, there is probably no reason to divide these areas into separate survey units.

The walk may have almost no contamination because of runoff. Judgmental scans should

probably be made along the edges of the walk and along the side of the building.

Example Case #2: The same as example 1, except that there had been a spill in the gravel parking

lot. The area of the spill is shown as the dark area in Figure 12.2. During remediation of the spill,

the surface material of the parking lot was disturbed by earthmoving equipment in the

crosshatched area, approximately 1700 n2. In this case, the disturbed area around the spill

should be considered a Class 1 area. The surrounding area is still Class 2. There is the possibility

of designating the entire parking lot as a Class 1 survey unit, but it might actually be more

reasonable to include the uncontaminated part of the parking lot along with the other Class 2

areas. This will yield a higher sampling density in the actual contaminated area, even though it

increases the variability in the Class 2 survey unit.

NUREG-1505 12-2
MULTIPLE SURFACES

Figure 12.2 Example Survey Units: Case #2

Example Case #3: The same as example 2, except that the spill was on the walkway, as shown in

Figure 12.3. The entire paved walkway was removed during remediation, and a substantial

portion of the lawn and parking lot were disturbed in the process. The entire area (5000 m2)

should probably be designated Class 1, but is too large to contain only one survey unit. In this

case, it may be reasonable to divide the area into two survey units- the former lawn area and the

f former parking lot. It is probably not practical, or prudent, to try to separate the small undisturbed

parts of the lawn and the parking lot into a third survey unit.

------------ ------------

C~~~ (. -C C -C _' 

Figure 12.3 Example Survey Units: Case #3

12-3 NUREG-1505
MULTIPLE SURFACES

These example cases are only meant to illustrate the considerations that may dictate how survey

units may be designated, and what tradeoffs may be involved. What is actually done in any

specific situation will depend on site-specific information from historical site assessments, and

prior scoping, characterization, and remediation control survey results.

12.3 Using Paired Observations for Survey Units with Many Different Backgrounds

In this report, we have discussed using the Sign test for residual radioactivity that does not appear

in background when radionuclide specific measurements are made, and otherwise using the WRS

test. However, there are cases when one. may wish to use the Sign test even when the

radionuclide appears in background and/or radionuclide specific measurements are not made. An

obvious instance would be when background is such a small fraction of the DCGL that including

it is unlikely to affect the decision errors. An example would be '

37

Cs residual radioactivity in an

area where the concentrations from global fallout are small. It may be more cost-effective to

simply compare the total 137Cs concentration in the survey unit to the DCGL using the Sign test

rather than to attempt to find a matching reference area.

Another case in which the Sign test may be more appropriate is when there are many different

materials within what would otherwise logically be a single survey unit. As indicated at the

beginning of this chapter, to divide such a survey unit into separ.4te parts, each requiring its own

reference area is not only impractical, but may be inconsistent with the dose models used to

determine the DCGLs.

Consider once again, the example case #1 shown in Figure 12.1. Suppose the residual

radioactivity of concern is 22.Ra rather than 'Co. We will call this example case #4. When 6Co

was the concern, only the variability in a material's potential for retaining or accumulating this

radionuclide was important. If 22.Ra is the concern, then the variability in the background

concentration of 226

Ra in the materials is an additional, perhaps more important concern in

forming survey units. In Figure 12.4, the same area as for case #1 is again shown, but with

consideration of this additional factor.

With an eye towards potential differences in background 226

Ra, it becomes important that:

" The walkway was paved with different concrete at two different times.

* The parking lot was expanded using a different type of gravel.

• Part of the lawn was graded with fill from another location.

* Soil and mulch were used for the plant beds next to the building.

What were previously three potentially different survey units are now possibly as many as nine

different survey units. The contamination potential is still Class 2, as in case #1, and on that basis

alone this area might be designated as a single survey unit. This disparity of effort between case

#1, using the Sign test for 'Co, and case #4 using the WRS test for 226

Ra is tremendous.

Fortunately, there is a third option- to use the Sign test with paired observations. Each

measurement in the survey unit is paired with an observation on a suitable reference material..

The Sign test is then performed on the difference. The tradeoff is the higher variability of the

differences compared to a single measurement.

NUREG-1505 12-4
MULTIPLE SURFACES

W ho ........

-------------------------- - -X. . - . .............. ............ -'-Z , Xi

"t"L"t, Z.,Z XFigure 12.4 Example Survey Units: Case #4

In case #1, an estimate of the variability of the measurements from the survey unit, a, is needed

to determine the relative width of the gray area, A/a, .which then in turn is used to determine the

required sample size, N. Suppose the survey unit measurements are designated by Y,, for

i = I to N. From each of these Y1, a paired measurement, X,, on an appropriate reference material

is subtracted. The Sign test is performed on the difference Y1 - X,. Thus it is the variability of

these differences, i -Xi), that is required in order to determine the required sample size. This

variability has three components:

G2 2 +2 2 2 + 2

Y'-xI °r Yr X= (R + jB XO• (12-1)

where

oa, is the standard deviation of the measurements on the reference material,

oy is the standard deviation of the measurements in the survey unit,

aBi is the standard deviation of the background in the survey unit material, and

a. is the standard deviation of the residual radioactivity in the survey unit.

If the reference material is truly representative, then oxi = oBi, so that

,2 , 2

a =iXi a + 2 GBJ (12-2)

Better precision may be possible if the average of mjmeasurements made on thejth reference

material is subtracted from each measurement from the survey unit made on that material.

The Sign test performed the differences:

12-5 NUREG-1505
MULTIPLE SURFACES

MY-j - in X (12-3) mi q=1

fork= 1 toni.

The variability of these differences is

O~'6 = +R B + ,Inj= OR +( nj) Osj (12-)

Note that when '°Co was the contaminant, the only component of variability was a = OR.

To estimate the sample size needed for case #1, suppose the DCGL for 'Co is 2 and that

OR = 0.7. If the LBGR is set at 1, then A/a = (2-1)/0.7 = 1.4. For acceptable error rates of

a = P = 0.05, the sample size found in Table 3.2 is 20. Suppose, for the sake of illustration, that

for case #4, the DCGL for 226

Ra is also 2, and oR = 0.7. If the 226

Ra background standard

deviation is about 0.5, then standard deviation of the difference of matched pairs of

measurements,

°r-xj = aoR + 2o 7 = (0.7)2+ 2(0.5)2 = /0.49+ 0.5 = .99" 1 (12-5)

Thus, A/o = 1, and for the same LBGR of 1 and acceptable error rates of a = = 0.05, the

sample size found in Table 3.2 is 29 measurements in the survey unit. An additional matching 29

measurements on reference materials are also needed, for a total of 58 measurements. To

simplify the above calculations, a single estimate of the standard deviation of background

measurements was used for all materials. It would be prudent to use the largest anticipated

standard deviation. Note that no assumption about the average of the background concentrations

in the different materials was made. This may vary considerably from one material to another.

Indeed, if the average concentration does not vary significantly, it would be better to perform the

WRS test using a. reference area with a composition that is reasonably well-matched to the survey

unit. Suppose in case #4, that the average concentrations of 226

Ra did not vary much. Then the

standard deviation of background in the reference area is still about 0.5, and the variability in the

survey unit is simply

= FR +2 = (0.7)2+ (0.5)2 = /0.49+ 0.25 -- .i 0.86 (12-6)

Thus, A/a = 1.155, and for the same LBGR of 1 and acceptable error rates of a = = 0.05, the

sample size found by interpolating in Table 3.3 is about 26 measurements each required in the

NUREG-1505 12-6-
MULTIPLE SURFACES

survey unit and reference area for a total of 52 measurements. The WRS test requires fewer

measurements. An equivalent observation is that for the same number of measurements, the

WRS test has greater power. The essential difference is whether the reference measurements can

be considered independent of the survey unit measurements, or whether they must be matched

together according to material type.

Notice that Equations 12-2, 12-4, and 12-6 differ primarily in the factor multiplying 0B . Using

the Sign test with a single matched reference measurement, this factor is 2. Using the WRS test,

this factor is 1. If the mean of njmeasurements on the jth material is used this factor is (n, + 1)n/ j.

This equals 2 when nj=l, and approaches 1 as njbecomes large.

The question remains as to how the measurements should be taken in the survey unit and from

the reference materials. The measurements in the survey unit should be taken according to the

regular procedure recommended for that class of survey unit, i.e., on a random start systematic

grid for Class 1 and Class 2, and randomly for Class 3. This is essentially the same as sampling

according to the proportional area of each material in the survey unit. Matching reference area

samples should be taken randomly on the chosen reference material.

12-7 NUREG-1505
'a
13 DEMONSTRATING INDISTINGUISHABILITY FROM BACKGROUND

Thus far in this report the emphasis has been on conducting final status surveys that demonstrate

that any residual radioactivity in a survey unit is within the release criterion. In these cases,

Scenario A is generally preferred for the survey design. In some cases, however, it may be more

appropriate to demonstrate indistinguishability from background. Demonstrating I

indistinguishability from background using Scenario B will be a useful option when the residual

radioactivity consists of radionuclides that appear in background, and the variability of the

background is relatively high. Background variability may be considered high when differences

in estimated mean concentration measured in potential reference areas are comparable to

screening level DCGLs.

13.1 Determining Significant Background Variability

In Section 2.2.7, the concept of a reference area was introduced. Any difference in the

concentrations between the reference area and the survey unit is assumed to be due to residual

radioactivity. It is not possible to determine whether or not an observed difference is actually due

to variations in the mean background concentrations between these areas.

When the variations in mean background among different potential reference areas are small

compared to the width of the gray region, they can often be neglected. In such cases, the choice

of reference area will not materially affect the decision on whether or not to release a survey unit

to which it is compared.

As the variations in mean background among different potential reference areas become

comparable in magnitude to the width of the gray region, they can no longer be ignored. When

the reference area has a higher mean background than the survey unit, the survey unit will be

more likely to pass, and when the reference area has a lower mean bLackground than the survey

unit, the survey unit will be more likely to fail. Since any difference in background activity

between the survey unit and the reference area is attributed to residual radioactivity, the choice of

reference area may materially affect the decision on whether or not to release a survey unit to

which it is compared.

As an example, consider Figure 13.1, which illustrates a DQO specification for a survey design.

The gray region and acceptable rates for decision errors are shown by the solid curve. Suppose

the reference area happens to have a lower mean concentration than the actual background

concentrations in the survey unit. This difference is depicted by the double-headed arrow. The

values of residual radioactivity concentrations in this survey unit will appear larger than they

actually are by the amount of that difference. The result is that the actual probability that the

survey unit passes is represented by the dashed curve, which is shifted downward from the solid

curve by the difference in mean background between the survey unit and the reference area. This

means, for example, that when the true residual radioactivity concentration is at the LBGR there

is only about a 65% probability of passing this survey unit, rather the 95% probability specified

in the DQO.

Exactly how much, and in which direction the probabilities shift will depend on the particular

13-1 NUREG-1505
DEMONSTRATING INDISTINGUISHABILITY

reference area. Under such circumstances, whether a survey unit passes or fails may depend

more on the particular reference area chosen than on the amount of residual radioactivity that it

contains. This leads to a quantitative definition of what it means for a survey unit to be

indistinguishable from background. It is expressed in terms of the potential for variations in

reference area mean concentrations to impact decision error rates. First, it is necessary to

establish that there is significant variability among potential reference areas. A procedure for

doing this is discussed in the next section. If it shown that significant variability exists, this

information is used to define a level of residual radioactivity concentration that is

indistinguishable from background variations. Section 13.4 discusses how this can be used to

plan a final status survey using Scenario B.

100 0

60 Probability of

0passing Is P lower If

= 40 -- reference -"

area mean Is

.lower than

" 20 survey unit

(U background

a. 0 0

0 LBGR DCGL 1.5 DCGL

True Concentration of Residual Radioactivity in Survey Unit

Figure 13.1 Impact of Background Variability on Decision Errors

13.2 Determining if Reference Areas Have Significantly Different Background Levels

In this section, we focus our attention on potential reference areas. Consider all the reference

areas to which a particular survey unit may be compared according to the criteria set out in

Section 2.2.7. To determine whether there are significant differences among the reference area

background means, the reference area measurements are expressed as:

xi = P'+ Zi+ (13-1)

where

xUj= the jth measurement in the ith reference area, for j = 1 to n,, the number of measurements in

the ith reference area, and for i = 1 to k, the number of reference areas

" = the mean concentration over all reference areas

pi = the difference between the overall mean and the mean in the ith reference area

zU = the contribution of random spatial and measurement variability to the jth measurement in

the ith reference area

13-2 NUREG-1505
DEMONSTRATING INDISTINGUISHABILITY

is an unknown constant. The pi are distributed across reference areas with mean zero and

standard deviation c(, but within reference area i, pi has a fixed value. The z0 have mean zero and

standard deviation u. The measurements within reference area i have mean f+ pi, and variance

o2. The reference area means are distributed around the overall mean f with variance deviation

W2. Thus, if there is no variability in the reference area means, then w2 0. The measurement

variability within each reference area is a2, and it is the same whether or not there is a significant

difference among the reference area means.

If the p, and the z are assumed to be normally distributed, the above corresponds to a random

effects one-way analysis of variance model, sometimes called Model II. o2 and a2 are called the

components of variance. The null hypothesis, H0 : (.o2 = 0, versus the alternative, H.: Wo2 > 0, is

tested parametrically with an F-test. The non-parametric equivalent is the Kruskal-Wallis test.

Before collecting data for the Kruskal-Wallis test, the acceptable Type I error rate, a ,, must be

specified. This is the acceptable probability of concluding that the reference areas have different

average concentrations, when in fact they are the same. In setting a.n, it is important to consider

that the risk involved in a Type I error may be much smaller than the risk of a Type II error.

To perform the Kruskal-Wallis test, all of the measurements from the reference areas are pooled

and ranked. For every measurement, x0, there is a corresponding rank, rj. There will be

N = nj + n2 + ... + nk measurements in all. The sum of all of the ranks is N(N+I)12. Therefore

the average rank is N(N+I)l(2N) =(N+l)12. If the distribution of measured concentrations in each

reference area is the same, then the average rank for each reference area, should also be about the

same, i.e., (N+1)12.

Let

n.

-i- E•_r R=Rni (13-2)

nl, j=1

be the average rank in reference area i. The quantity R - (N+1)12 is a measure of how

different this reference area is from the others.

The Kruskal-Wallis statistic is a weighted sum of the squares of these differences over all of the

reference areas:

N(N+I) n -(N+l)2 N(N 12 ( R52 1 ) - 3(N+l) (13-3) E •J = (j N N(N+I) i=I. 3(+1

The weights in Equation 13-3 have been chosen so that the probability that K exceeds a given

value K,, may be approximated by a chi-squared distribution with k-I degrees of freedom:

Prob(K > K) = 1 xk.m(K#) k- (13-4)

13-3 NUREG-1505
DEMONSTRATING INDISTINGUISHABILITY

2 2 The critical value K, is determined from setting Xk-.(KC) = aCW. Values of XkI(K- ) for

typical values of ccw are tabulated in Table 13.1. If the value of K computed from Equation 13-3

exceeds Kc, then the null hypothesis is rejected.

Table 13.1 Critical Values, K, for the Kruskal-Wallis Test

___aKW

k-1 0.01 0.025 0.05 0.1' 0.2

1 6.6 5.0 3.8 2.7 1.6

2 9.2 7.4 6.0 4.6 3.2

3 11.3 9.3 7.8 6.3 4.6

4 13.3 11.1 9.5 7.8 6.0

5 15.1 12.8 11.1 9.2 7.3

6 16.8 14.4 12.6 10.6 8.6

7 18.5 16.0 14.1 12.0 9.8

8 20.1 17.5 15.5 13.4 11.0

9 21.7 19.0 16.9 14.7 12.2

10 23.2 20.5 18.3 16.0 13.4

For example, suppose there are four reference areas under consideration, and ten measurements

are made in each. The data and the ranks are shown in Table 13.2. The same type of spreadsheet

functions that were used for the WRS test can also be used to calculate the ranks for the KruskalWallis test.

Using Equation 13-2,

K- N(N+I) 12 (_, Ri21n) - 3(N+l) 1

_ 12 (1552/10+1212/10+ 2 552/10+ 2 8 92/10 _ 3(41)

40(41)

= 12 12 - (2402.5+1464.1+6502.5+8352.1) - 123 = (18721.2) - 123.0 = 14.0

1640 1640

13-4 NUREG-1505
DEMONSTRATING INDISTINGUISHABILITY

With k-1 = 3, this value of K, is greater than the highest value of K, in Table 13.1, viz., 11.3 for

axw = 0.01 . The null hypothesis is rejected. It is concluded that these reference areas do have

significantly different concentration distributions.

Table 13.2 Example Data for the Kruskal-Wallis Test

Measurements Ranks

Areal Area2 Area3 Area4 Areal Area2 Area3 Area4

1 0.27 1.04 2.45 3.77 6 13 27 39

2 1.87 0.39 0.34 2.63 1 20 9 8 31

3 0.97 2.07 3.06 4.05 10 23 37 40

4 1.01 -0.57 2.83 1.72 11 2 35 19

5 2.08 1.97 1.09 1.50 24 21 14 17

6 1.62 -0.22 0.26 2.47 18 3 5 29

7 0.30 1.39 2.80 1.42 7 15 34 16

8 1.98 0.05 2.77 2.47 22 4 33 28

9 2.18 -0.75 2.42 2.76 25 1 26 32

10 1.02 2.50 2.86 3.35 12 30 36 38

Mean 1.33 0.79 2.09 2.61 Sum 155 121 255 289

StdDev 0.71 1.17 1.09 0.91 Total 820

13.3 Establishing the Concentration Level That Is Indistinguishable

Once it is decided that there are significant differences among the potential reference areas for a

survey unit, a measure of the variability among these reference areas is needed.

The sample mean of the measurements in area i is

ni

X, - (13-5)

ni j=1

which provides an estimate of the mean concentration in reference area i, namely 'f+ Pii.

The overall sample mean

k ni

i=1 j=I

EC

13-6)

k

ni

13-5 NUREG-1505
DEMONSTRATING INDISTINGUISHABILITY

is an estimate of the overall mean background concentration, E.

The sample variance of the measurements in area i is

ni ni

si • (x - 1(n -1)= E (zi - 1(n, -1) j=1 j=1 (13-7)

which is an estimate of 02.

Since 02, is assumed to be the same in each reference area, these estimates can be pooled into the

following estimate:

SW k jn 1 1

(n1

i=1 j=I i=1

k

(=1

(13-8)

In the analysis of variance, s" is called the mean square within reference areas

The mean square between reference areas is

k k Eni(pi -5ýC)2 nii2 ki- 1l i = l - k-i

± Xi) 2/ Etn, ki=lj=l i=1

k-1

The righthand sides of Equations 13-8 and 13-9 may appear imposing, but essentially only

involve:

(1) the sum of the squares of all the measurements,

k ni

Ei(13-10) i=1 j=I

(1) The sum of the squared differences from the overall mean is composed of two parts:

k k n, k ni k n, k

E (x ~ x.-X) + E E. (5ý )2 - L xjF + iX )

i= j=I i=lj=l i=l j=l i=l j=l i=1

The first term on the right hand side is the sum of squares within reference areas and the second term on the right

hand side is the sum of squares between reference areas. The mean square is obtained by dividing the sum of

squares by the degrees of freedom. For the means square within reference areas this is the total number of data

points minus the number of reference areas. For the means square between reference areas this is the number of

reference areas minus one.

13-6 NUREG-1505
DEMONSTRATING INDISTINGUISHABILITY

(2) the square of the sum of all the measurements,

k nl \

and

(3) the sum of the squares of the reference area averages weighted by the number of

measurements,

k

i=l

The component of variance, c 2 , is estimated by

(13-11)

(13-12)

k

N - ni2 IN

where no = (k-i) 0 (k-1)

(Z2 2 2 (b ( )no (13-13)

no is usually slightly less than the average number of samples taken in each reference area,'

k

F = kI

k i=

(13-14)

If the number of measurements in each reference area is the same, n I = n2 = ... = nk = n, then,

k

= (k-1)

_ N-kn2 IN

(k-1)

kn - kn 2 Ikn

(k-1)

kn -n

(k-1)

_ n(k- 1)

(k-1)

(13-15)

The calculation of (Z2 for the example data of Table 13.2 proceeds using the sums and squares

shown in Table 13.3.

The sums (1), (2), and (3) together with Sb2 and sw 2 are calculated from Table 13.3 as follows:

From Equation 13-10,

(1) = sum of squares = 22.28 + 18.50 + 54.30 + 75.80 = 170.88.

From Equation 13-11,

(2) = square of the sum = (13.30 + 7.87 + 20.88 + 26.14)2= (68.19)2 =4649.88.

13-7 NUREG61505
DEMONSTRATING INDISTINGUISHABILITY

From Equation 13-12,

(3) = weighted sum of the squares of the averages 10(1.332+ 0.792+ 2.092+ 2.612)

= 10(1.77 + 0.62 + 4.36 + 6.83) = 10(13.58) = 135.8.

From Equation 13-8,

s.2 = (170.88 - 135.8)I(N-k) = (170.88 - 135.8)/36 = 0.97.

From Equation 13-9,

Sb2 = (135.8 - (4649.88/N))/(k- 1) = (135.8 - (4649.88/40))/(3) = 6.52.

Finally, we have

(s2 - Sw)Ino = (6.52 - 0.97)/10 = 0.55

Table 13.3 Calculation of 6' for the Example Data

(13-16)

Measurements Measurements Squared

Area 1 Area 2 Area 3 Area 4 Area 1 Area 2 Area 3 Area 4

1 0.27 -1.04 2.45 3.77 0.07 1.08 6.00 14.21

2 1.87 0.39 0.34 2.63 3.50 0.15 0.12 6.92

3 0.97 2.07 3.06 4.05 0.94 4.28 9.36 16.40

4 1.01 -0.57 2.83 1.72 1.02 0.32 8.01 2.96

5 2.08 1.97 1.09 1.50 4.33 3.88 1.19 2.25

6 1.62 -0.22 0.26 2.47 2.62 0.05 0.07 6.10

7 0.30 1.39 2.80 1.42 0.09 1.93 7.84 2.02

8 1.98 0.05 2.77 2.47 3.92 0.00 7.67 6.10

9 2.18 -0.75 2.42 2.76 4.75 0.56 5.86 7.62

10 1.02 2.50 2.86 3.35 1.04 6.25 8.18 11.22

sum 13.30 7.87 20.88 26.14 22.28 18.50 54.30 75.80

average 1.33 0.79 2.09 2.61

avg sqd 1.77 0.62 4.36 6.83

Although the analysis of variance using the F-test requires the assumption that the data are

normally distributed, the calculation of G.2 does not. Therefore, the values of the mean squares S2

and S 2 that are generated by most statistical computer programs for ANOVA can be used for

these calculations. Table 13.4 shows an ANOVA for the example data generated by a spreadsheet

program. The entry for the mean square within groups, 0.97, is the same as was found in Table

13.3 for sW2. Similarly, the entry for the mean square between groups, 6.52, is the same as was

found in Table 13.3 for Sb2. The F-statistic, which is simply the ratio Sb2 /Sw2 , is also shown. Of

course, if the data from each reference area are consistent with the assumption of normality, the

F-test may simply be used instead of the Kruskal-Wallis test.

13-8 NUREG-1505
DEMONSTRATING INDISTINGUISHABILITY

Table 13.4 Analysis of Variance for Example Data

Source of Sum of Degrees of Mean Square F Statistic

Variation Squares Freedom

Between

Groups 19.56 3 6.52 6.69

Within

Groups 35.08 36 0.97

Total 54.65 39

13.4 Using the Concentration Level That Is Indistinguishable in the WRS Test

Recall from Section 3.6, that in Scenario B, the hypotheses being tested are

Null Hypothesis:

HO: The mean concentration of residual radioactivity in the survey unit is indistinguishable from

background up to a level specified by the LBGR.

versus

Alternative Hypothesis:

Ha: The mean concentration of residual radioactivity in the survey unit distinguishable from

background is in excess of the DCGLw.

In this scenario, a Type I decision error, with associated probability a, is made when a survey

unit fails when it should pass. A Type II decision error, with associated probability [3, is made

when a survey unit passes when it should fail. To set these decision errors, an appropriate gray

region is needed. The lower bound on this gray region is the concentration level above

background that may be considered distinguishable from background.

If the null hypothesis of Kruskal-Wallis test has been rejected, the mean background levels

among reference areas varies about the overall mean g with a standard deviation estimated by C,.

-The difference in concentration that is distinguishable above background variability may be

expressed in terms of an appropriate multiple of 6). For example, if the reference area means are

normally distributed, the probability that the survey unit mean is more than two standard

deviations away from the overall mean is about 5%. Regardless of how the data are distributed,

Chebyshev's Inequality states that the probability that the true mean background in the survey

unit differs from the overall mean background by more than t standard deviations is less than lit2.

Therefore, the probability that the survey unit mean is more than t = 2 standard deviations away

from the overall mean is less than lt 2= 1/4 = 25%. The probability that the survey unit mean is

more than t = 3 standard deviations away from the overall mean is less than l1t2= 1/9 = 11%. In

most cases, it is reasonable to assume that the true probabilities will fall somewhere between the

value calculated for the normal distribution and that established by Chebyshev's inequality. The

multiple that of Ca that is used in a specific application should be decided during the DQO

process, but a factor of three is a reasonable default.

13-9 NUREG-1505
DEMONSTRATING INDISTINGUISHABIITY

For the example data W2 = 0.55 so C = 0.74. Thus, differences smaller than 36 = 2.22 would not

be considered distinguishable from background variations. Notice that in Table 13.2, the

difference in the means between reference areas #4 and #2 is 2.61 - 0.79 = 1.82.

The WRS test is applied as described in Section 6.3, using the decided upon multiple of 6) as the

LBGR, and thewidth of the gray region equal to the DCGLw. The hypotheses tested by the

WRS under Scenario B are restated as

Null Hypothesis:

H0: The difference in the median concentration of radioactivity in the survey unit and in the

reference area is less than the LBGR.

versus

Alternative Hypothesis:

Ha: The difference in the median concentration of radioactivity in the survey unit and in the

reference area is greater than the DCGL,

The Type I error rate caw = ac/2(2), is the probability that a survey unit with a difference from the

reference area equal to the LBGR will fail the test. The power, 1 - P3, is the probability that a

survey unit with a difference at the DCGLw above the LBGR will fail the test. For example, the

desired probability for the survey unit passing might look similar to Figure 13.2.

100

80

60

40

C.

CL

o 0. 20

36) 36) + DCGLw

True Concentration of Residual Radioactivity in Survey Unit

Figure 13.2 Example DQOs for the Probability That the Survey Unit

Passes Versus the Concentration Difference Between the

Survey.Unit and the Reference Area

(2) Recall that since the Quantile test is performed in tandem with the WRS test, a w= a Q= a/2, so that the

that the size of the two tests in tandem is approximately a = aQ+aw

13-10 NUREG-1505
DEMONSTRATING INDISTINGUISHABILITY

The Quantile test is also performed as described in Section 7.2, with ccQ= ad2, and with the

LBGR equal to the decided upon multiple of 6.

All of the reference area measurements taken for the Kruskal-Wallis test should also be used in

the WRS and Quantile tests. In most cases, no additional reference area samples will be needed.

If additional samples are required, they should be obtained randomly, with all of the reference

areas being equally likely to be sampled.

13.5 Determining the Number of Reference Areas and the Number of Samples

In applying the methods of this chapter, it is natural to ask is how many reference areas should be

studied, and how many samples should be taken in each. As was seen in Section 3.8.1, the

number of samples needed depends on both the probability of a Type I error (a) and the

probability of a Type II error (P3) that are deemed acceptable for the test. Unfortunately, the power

(1-P) of the Kruskal-Wallis test involves functions that are "too complicated to be useful"

(Lehmann and D'Abrerra, 1975). However, it has been shown that the efficiency of the KruskalWallis test relative to the F-test is the same as the efficiency of the WRS test to the t-test

(Andrews, 1954). This means that one can get an approximate idea of the power of the KruskalWallis test by calculating the power of the F-test.

The power of the F-test is (Brownlee, 1960, p.268):

1 -P = Probability[ F(fjf2) > -a-V114 F1 2)A (13-17)

where

O2

= 1 + n- ,and F1 o2 -,(fJf 2) is the 1- aw percentile of the Fdistribution withf, = k - 1

andf2 = kn -k degrees of freedom. Through ý, the power depends on the ratio of the variance

components. Under the null hypothesis, C02 = 0, so (p = 1.

As an example, consider the data from Table 13.2. If one wished to detect a situation in which

W,2 = o22 then 4) =1+ 10(1)= 11. If aw= 0.1,then Fl_(flf 2) = F0.

9(3,36) =2.243. The

power, 1- P3, is then the probability that the F-statistic with 3 and 36 degrees of freedom exceeds

2.243/4 = 2.243/11 = 0.2039. P3 is the probability that the F-statistic with 3 and 36 degrees of

freedom is less than 0.2039. This probability is about 13 = 0.1, so the power is about 0.9.

The results of this calculation for other numbers of samples and reference areas are shown in

Table 13.5 for accw = 0.05, 0.10 and 0.20. Although this is only an approximation, and the

actual power of the Kruskal-Wallis test would be slightly lower, this table indicates that with four

reference areas each with between 10 and 20 samples in each should generally be adequate. Since

the risk of not detecting background variations that are actually present (a Type II error) could

involve the impossible task of remediating background, choosing a higher value for atv than for

P would often be justified. From Table 13.5, when k = 4, this implies that aKW = 0.1 is a

reasonable default, and in some circumstances even larger values could be considered.

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DEMONSTRATING INDISTINGUISHABILITY

Table 13.5 Power of the F-test When cw = a'

Number of Number Total 4) Power Power Power

reference of samples Number when when when

areas in each of a Kw = 0.05 a Kw = 0.1 a Kw = 0.2

(k) (n) Samples

(kn) I

2 10 20 11 53.4% 60.7% 69.3%

2 15 30 16 61.3% 67.4% 74.5%

2 20 40 21 66.1% 71.5% 77.7%

2 30 60 31 72.1% 76.5% 81.7%

3 10 30 11 74.0% 79.7% 85.7%

3 15 45 16 81.8% 85.9% 90.1%

3 20 60 21 86.1% 89.2% 92.4%

3 30 90 31 90.5% 92.7% 94.9%

4 10 40 11 85.3% 89.3% 93.0%

4 15 60 16 91.4% 93.8% 96.0%

4 20 80 21 94.2% 95.8% 97.3%

4 30 120 31 96.7% 97.6% 98.5%

5 10 50 11 91.8% 94.3% 96.5%

5 15 75 16 95.9% 97.2% 98.3%

5 20 100 21 97.6% 98.4% 99.0%

5 30 150 31 98.9% 99.2% 99.5%

6 10 60 11 95.4% 97.0% 98.3%

6 15 90 16. 98.1% 98.8% 99.3%

6 20 120 21 99.0% 99.4% 99.6%

6 30 180 31 99.6% 99.8% 99.9%

13.6 Determining When Demonstrating Indistinguishability Is Appropriate

The methods of this chapter were developed specifically to address potential difficulties with

demonstrating compliance with dose-based release criteria at sites with spatially variable

background concentrations of natural radionuclides. Generally, the use of Scenario A is preferred

since it involves fewer assumptions in its application and requires only one statistical test. The

null hypothesis for Scenario A is such that when the residual radioactivity in the survey unit is

very close to the DCGLw, the default decision is to not release the survey unit without further

investigation. This provides additional assurance that the release criteria will not be exceeded.

However, when the variability in background is high, this assurance comes at too high a price,

namely the possibility of requiring remediation of survey units containing only background

concentrations of radionuclides. Since the default decision in Scenario B is to release the survey

unit, a second statistical test, the Quantile test, is used to detect non-uniform concentrations of

13-12 NUREG-1505
DEMONSTRATING INDISTINGUISHABILITY

residual radioactivity that may be excess of the release criterion, but that might be missed by the

WRS test..

It is not possible to anticipate every circumstance in which these methods might be considered

applicable. The suitability of these methods to specific situations should be determined during

the DQO process. Two factors that should be considered in making this determination are:

(1) Have reasonable efforts been made to reduce measurement uncertainty. e.g., by use of

radionuclide-specific methods?

(2) Have reasonable efforts been made to reduce spatial variability by choosing homogeneous

survey units with well-matched reference areas?

Once it is determined that the methods of this chapter are appropriate, the error rates for the

Kruskal-Wallis test should be set. The Kruskal-Wallis test is used to determine whether the

spatial variability of average reference area background concentrations is significant. The

significance level is . If adequate consideration has been given to the decision to demonstrate

indistinguishability, a-, need not be set to too low a value. Indeed, if it is felt that background

variability should always have the benefit of the doubt, the Kruskal-Wallis test need not be

conducted. Not conducting the test is essentially the same as setting a, = 1.0. Table 13.5 could

still be used as a guide in determining the number of reference areas, and the number of

measurements in each, that will be used to estimate 6 according to the procedures of

Section 13.3.

Finally, the appropriate multiple of Z to be used as the LBGR should be determined. The

discussion of this issue in Section 13.4 can be used as a guide.

13-13 NUREG-1505

14 ALTERNATIVES AND MODIFICATIONS

14.1 Alternative Statistical Tests

The nonparametric statistical tests described in this report are expected to perform well in a wide

variety of circumstances. However, in some situations alternative methods can be considered. As

mentioned in Section 2.4.1, there are many statistical tests that can be used for determining

whether or not a survey unit meets the release criteria. Any one test may perform better or worse

than others, depending on the hypotheses to be tested, i.e., the decision that is to be made and the

alternative, and how well the assumptions of the test fit the situation. Some possible alternatives

are discussed below.

In evaluating statistical tests, generally one chooses the test that has the highest power among the

various alternatives. The power is compared when each test is set to have the same Type I error

rate, a. The Type I error rate is the probability that the null hypothesis will be rejected when it is

true. If the assumptions made about the data distribution are correct, the calculation of a forms

the basis for setting the critical value of the test statistic. If the assumptions are not valid, the

calculated value of a will differ from the true Type I error rate. The fewer assumptions that are

made, the more confidence can be placed in the calculation of the Type I error rate.

If a specific set of assumptions is made, the test results can be simulated using Monte Carlo

sampling techniques. Using a large number of simulations, the actual Type I and Type II error

rates for different tests can be compared. For each sample size, and specific set of assumptions, a

separate simulation must be performed. Although much can be learned about the relative

accuracy of statistical tests in this way, it is clearly not possible to explore every potential set of

assumptions.

An alternative is to look at large sample results. With very few exceptions, it can be proved that

the average of a large enough number of random data points tends to be normally distributed

(Central Limit Theorem). In the same way, the power of statistical tests can be examined when

the sample sizes are large enough. Note that what is meant by large enough is not precisely

specified. Depending on the situation, large enough might be 10, or it might be 1000. If the

samples size is allowed to grow large enough, the asymptotic (i.e., in the limit of arbitrarily large

sample size) behavior of tests can be compared. Better large sample test behavior may be taken

to imply that a test is better for all sample sizes. In reality, it can only be used as an indication of

which test might be preferred.

One measure commonly used to compare statistical tests is called the relative efficiency. This is

defined as the inverse of the ratio of sample sizes needed to achieve a given level of statistical

power. If a test has relative efficiency of two relative to another, it requires half the sample size

to achieve the same power. The asymptotic relative efficiency of one test to another, is the limit

of the relative efficiency when the sample size is arbitrarily large.

14-1 NUREG-1505
ALTERNATIVES AND MODIFICATIONS

Wilcoxon Signed Ranks Test (WSR test)

The asymptotic relative efficiency of the WSR test compared to the Sign test can be greater or

less than one. That is, either might be better, depending on the data distribution. The WSR test

tends to be better when the data distribution is symmetric, and the Sign test tends to be better

when the data distribution is skewed.

Student's t-Test

Student's t-test may be used if the data have a nornial distribution. This is a more restrictive

requirement than that of symmetry, since every normal distribution is symmetric, but there are

many other distributions that are also symmetric. The assumption of normality should be

checked before using this test. The Shapiro-Wilk test discussed in EPAIQA-G9 (1996) is one

such test. Others include the Kolmogorov-Smimov test, Lillifor's test, and the Chi-Squared test.

The asymptotic relative efficiency of the WSR test relative to the one-sample Student's t-test

ranges from 0.864 to infinity. As stated by Conover (1980): "the Wilcoxon test never can be too

bad, but it can be infinitely good ... .. The asymptotic relative efficiency of the WRS test

compared to the two-sample Student's t-test has the same range, from 0.864 to infinity.

Chen's Test

Chen's test (Chen, 1995) is a modification of the Student's t-test that has been suggested for use

when data are from a positively skewed distribution. Simulations show that it is generally more

powerful than other forms of the t-test. However, this test can only be used in Scenario B.

Lognormal Test

If the data are assumed to lognormal, the testing procedure of Land (1988) may be used. The

assumption of lognormality should be checked by testing the logarithms of the data for normality.

It is important to note that a test on the mean of a lognormal distribution cannot be performed by

using a Student's t-test on the mean of the logarithms of the data. This is because the mean of the

logarithms of the data is the logarithm of the median of the original data. The behavior of this

test relative to others when the assumption of lognormality is violated has not been studied.

Bootstrap Methods

The bootstrap is a simulation technique (Efron and Tibshirani, 1993). In essence, the distribution

of concentrations in a survey unit is approximated by the empirical distribution (e.g., histogram)

of the sample data taken. If n measurements are made, these n measurements are randomly

sampled n times with replacement. Each time this is done, the mean of the random sample is

calculated. After this has been done a specified number of times (generally between 50 and 200),

the standard deviation of all of the random sample means is calculated. This is then used as the

estimate of the standard error of the mean. There are, in addition, several methods for computing

bootstrap t-statistics. Usually 1000 or more replications are recommended for the bootstrap t.

Bootstrap methods generally have-good asymptotic properties, but can be sensitive to outliers,

and erratic when sample sizes are small.

NUREG-1505 14-2
ALTERNATIVES AND MODIFICATIONS

14.2 Retesting

It may happen that a survey unit fails the hypothesis test (i.e., the decision is made that the

survey unit does not meet the release criterion), yet the mean of the measured data is below the

release criterion. This is more likely to occur when the mean falls in the gray region than

otherwise. It analogous to the situation in which the mean is below the release criterion, but the

1- ca upper confidence level on the mean falls above the release criterion. It may be that the

survey unit does meet the release criterion, but the hypothesis test was not powerful enough to

detect that with the number of samples taken. Under some circumstances, one might like the

option to take additional random samples and re-perform the hypothesis test on the entire set of

data. The major difficulty with this is that the Type I error rate will now be greater than originally

specified in the DQOs.

Sequential testing is performed when data are collected and analyzed in stages. It differs from

hypothesis testing in that at each stage a third alternative is added to the decision of whether or

not to reject the null hypothesis, namely, to collect more data before deciding. The usual

motivation for sequential testing is to reduce the expected total number of samples from that

required when all the sample are taken at one stage. Sequential versions of the WSR and WRS

tests are discussed by Spurrier and Hewett (1976).

14.3 Composite Sampling

The number of measurements taken in a Class 1 survey unit may sometimes be driven more by

the need to locate small areas of elevated activity than by the need to achieve the specified

acceptable error rates for the statistical tests. When the scanning MDC is high, the sample size,

N, may need to be significantly increased, in order to decrease the area between samples on the

systematic grid. When this grid area, about A/N, is small enough, so that in turn the area factor is

sufficiently high, the result is a DCGLEMC that is detectable by scanning. If the sample size N is

much greater than that required for the statistical tests, some number, m, of neighboring samples

might be composited to reduce the total cost of analysis. Suppose there are N = mn

measurements. Each composite represents a contiguous area of approximately the same

proportion of the survey unit, m(A/N). The number of composite measurements, n, should be

equal to or greater than the number of measurements required by the statistical test. When the

elevated measurement comparison is performed against the composites measurement results, the

DCGLEMC should be divided by the number of samples included in each composite. If the

composite measurement is below DCGLEMc/m, no individual sample contributing to the

composite could exceed the DCGLEMC.

If a composite measurement is flagged by the EMC, it may be necessary to reanalyze each

sample included in that composite to determine which of them, if any, actually exceed the

DCGLEMC or, alternatively, the area of the survey unit represented by that composite

measurement should be reinvestigated.

14-3 NUREG-1505
.J
15 GLOSSARY

Amin: The smallest area of elevated activity identified using the DQO Process that it is important

to identify.

action level: A scanning measurement level for residual radioactivity that (1) is based on the

DCGL and (2) triggers a response, such as further investigation or cleanup, if exceeded. See

investigation level.

activity: See radioactivity.

affected area: This term was previously used for areas that would be designated Class 1 or

Class 2.

affected/non-uniform: This term was previously used for areas that would be designated

Class 1.

affected/uniform: This term was previously used for areas that would be designated Class 2.

ALARA (acronym for as low as is reasonably achievable): A basic concept of radiation

protection which specifies that exposure to ionizing radiation and releases of radioactive

materials should be reduced as far below regulatory limits as is reasonably achievable

considering economic, technological, and societal factors, among others. Reducing exposure at a

site to ALARA strikes a balance between what is possible through additional remediation and the

use of additional resources to achieve a lower level. A determination of ALARA is a site-specific

analysis that is open to interpretation, because it depends on approaches or circumstances that

may differ between regulatory agencies. An ALARA recommendation should not be interpreted

as a set limit or level.

alpha (a): The specified maximum probability of a Type I decision error, i.e., the maximum

probability of rejecting the null hypothesis when it is true. Alpha is also referred to as the size of

the test. Alpha reflects the amount of evidence the decision maker would like to see before

abandoning the null hypothesis.

alpha particle: A positively charged particle emitted by some radioactive materials undergoing

radioactive decay.

alternative hypothesis (Ha): See hypothesis.

area: A general term referring to any portion of a site, up to and including the entire site.

area of elevated activity: An area over which residual radioactivity exceeds a specified value

DCGLEMc.

area factor (A.): A factor used to adjust DCGLw to estimate DCGLEMc and the minimum

detectable concentration for scanning surveys in Class 1 survey units -

15-1 NUREG-1505
GLOSSARY

DCGLEMc = (DCGLw)(Am). Am is the magnitude by which the residual radioactivity in a small

area of elevated activity can exceed the DCGLw while maintaining compliance with the release

criterion.

arithmetic mean: The average value obtained when the sum of individual values is divided by

the number of values.

arithmetic standard deviation: A statistic used to quantify the variability of a set of data. It is

calculated in the following manner: (1) subtracting the arithmetic mean from each data value

individually; (2) squaring the differences; (3) summing the squares of the differences; (4)

dividing the sum of the squared differences by the total number of data values less one; and (5)

taking the square root of the quotient. The calculation process produces the Root Mean Square

Deviation (RMSD).

background radiation: Radiation from cosmic sources; naturally occurring radioactive

material, including radon (except as a decay product of source or special nuclear material); and

global fallout as it exists in the environment from the testing of nuclear explosive devices or

from nuclear accidents which contribute to background radiation and are not under the control of

the licensee. Background radiation does not include radiation from source, byproduct, or special

nuclear materials regulated by the NRC.

becquerel (Bq): The International System (SI) unit of activity equal to one nuclear

transformation (disintegration) per second. 1 Bq = 2.7x101" Curies (Ci) = 27.03 picocuries

(pCi).

beta (p): The probability of a Type 11 decision error, i.e., the probability of accepting the null

hypothesis when it is false. The complement of beta (1 -[3) is referred to as the power of the test.

beta particle: An electron emitted from the nucleus during radioactive decay.

bias: The systematic or persistent distortion of a measurement process which causes errors in

one direction.

biased sample or measurement: See judgment sample or measurement.

byproduct material: Any radioactive material (except special nuclear material) yielded in or

made radioactive by exposure to the radiation incident to the process of producing or utilizing

special nuclear material.

CEDE (committed effective dose equivalent): The effective dose equivalent is the summation

of the products of the dose equivalent received by specified tissues of the body and a tissuespecific weighting factor. It is a risk-equivalent value, expressed in Sv or rem, that can be used to

estimate the health effects on an exposed individual. See TEDE.

characterization survey: A type of survey that includes facility or site sampling, monitoring,

and analysis activities to determine the extent and nature Of contamination. Characterization

NUREG-1505 ..15-2
GLOSSARY

surveys provide the basis for acquiring necessary technical information to develop, analyze, and

select appropriate cleanup techniques.

Class 1 area: Areas containing locations in which, prior to remediation, the concentrations of

residual radioactivity may have exceeded the DCGLw.

Class 1 survey: A type of final status survey that applies to areas with the highest potential for

contamination. Class 1 surveys require (1) sufficient measurements on a systematic grid to meet

the desired error rates for the statistical hypothesis tests; (2) scanning over 100% of the survey

unit; (3) scanning MDC at or below the DCGLEMC.

Class 2 area: Areas containing no locations where, prior to remediation, the concentrations of

residual radioactivity may have exceeded the DCGLw.

Class 2 survey: A type of final status survey that require (1) sufficient measurements on a

systematic grid to meet the desired error rates for the statistical hypothesis tests; (2) judgmental

scanning of a portion, up to 100%, of the survey unit.

Class 3 area: Areas with a low probability of containing any locations with residual

radioactivity.

Class 3 survey: A type offinal status survey that require (1) sufficient measurements at

randomly chosen locations to meet the desired error rates for the statistical hypothesis tests; (2)

judgmental scanning over less than 10% of the survey unit.

classification: The act or result of separating areas or survey units into one of three designated

classes: Class I area, Class 2 area, or Class 3 area.

composite sample: A sample formed by collecting several samples and combining them (or

selected portions of them) into a new sample which is then thoroughly mixed.

confirmatory survey: A type of survey that includes limited independent (third-party)

measurements, sampling, and analyses to verify the findings of afinal status survey.

contamination: The presence of residual radioactivity in excess of levels which are acceptable

for release of a site or facility.

criterion: See release criterion.

critical group: The group of individuals reasonably expected to receive the greatest exposure to

residual radioactivity for any applicable set of circumstances.

critical level (Lv): A fixed value of the test statistic corresponding to a given probability level,

as determined from the sampling distribution of the test statistic. Lc is the level, in counts, at

which there is a statistical probability (with a predetermined confidence) of incorrectly

identifying a background value as greater than background.

15-3 NUREG-1505
GLOSSARY

curie (Ci): The customary unit of radioactivity. One curie (Ci) is equal to 37 billion

disintegrations per second (3.7 x 1010 dps = 3.7 x 1010 Bq), which is approximately equal to the

decay of one gram of 226

Ra. Fractions of a curie, e.g. picocurie (pCi) or 10". Ci and microcurie

(pCi) or 106 Ci, are levels typically encountered in decommissioning.

DQA (Data Quality Assessment): The scientific and statistical evaluation of data to determine

if the data are of the right type, quality, and quantity to support their intended use.

DQOs (Data Quality Objectives): Qualitative and quantitative statements that clarify study

objectives, define the appropriate type of data, and specify levels of potential decision errors that

will be used as the basis for establishing the quality and quantity of data needed to support

decisions.

DCGL (derived concentration guideline level): A derived, radionuclide-specific activity

concentration within a survey unit corresponding to the release criterion. The DCGL is based on

the spatial distribution of the contaminant and hence is derived differently for the Wilcoxon test

(DCGLw) and the Elevated Measurement Comparison (DCGLEMc). The DCGLW is derived

assuming that residual radioactivity is uniformly distributed over a wide area, i.e. the entire

survey unit. This can often be the default DCGL provided by an exposure pathway model. The

DCGLEMc is derived assuming that residual radioactivity is concentrated in a much smaller area,

i.e. in only a small percentage of the entire survey unit.

decay: See radioactive decay.

decision maker: The person, team, board, or committee responsible for the final decision

regarding disposition of the survey unit.

decommission: To remove a facility or site safely from service and reduce residual radioactivity

to a level that permits (1) release of the property for unrestricted use and termination of the

license or (2) release of the property under restricted conditions and termination of the license.

decommissioning: The process of removing a facility or site from operation, followed by

decontamination and license termination.

decontamination: The removal of radiological contaminants from, or their neutralization on, a

person, object or area to within established levels. Decontamination is sometimes used

interchangeably with remediation, remedial action, and cleanup.

delta (8): The amount that the distribution of measurements for a survey unit is shifted to the

right of the distribution of measurements of the reference area.

delta (A): The width of the gray region. A divided by the arithmetic standard deviation of the

measurements, a, is the relative shift, Ala, expressed in multiples of standard deviations. See

relative shift, gray region.

NUREG-1505 15-4
GLOSSARY

derived guideline(s): A level or levels of radioactivity presented in terms of ambient radiation,

surface activity level(s), and soil activity concentration(s). Derived guidelines are derived from

activity/dose relationships through various exposure pathway scenarios. See DCGL.

detection sensitivity: The ability to identify the presence of radiation or radioactivity.

direct measurement: Radioactivity measurement obtained by placing the detector against the

surface or in the media being surveyed. The resulting radioactivity level is read out directly.

dose commitment: The dose that an organ or tissue would receive during a specified period of

time (e.g., 50 or 70 years) as a result of intake (as by ingestion or inhalation) of one or more

radionuclides from a given release.

dose equivalent (dose): A quantity that expresses all radiations on a common scale for

calculating the effective absorbed dose. This quantity is the product of absorbed dose (rads)

multiplied by a quality factor and any other modifying factors. Dose is measured in Sv or rem.

epsilon (e): A fraction of a survey unit that has not been remediated to the reference-based

cleanup standard. c is used in the Quantile test.

elevated area: See area of elevated activity.

elevated measurement: A measurement that exceeds a specified value, the DCGLEMc.

Elevated Measurement Comparison (EMC): This comparison is used in conjunction with the

Wilcoxon test to determine if there are any measurements that exceed a specified value

DCGLEMc.

exposure pathway: The route by which radioactivity travels through the environment to

eventually cause a radiation exposure to a person or group.

exposure rate: The amount of ionization produced per unit time in air by X-rays or gamma rays.

The unit of exposure rate is roentgens/hour (R/h); for decommissioning activities the typical

units are microroentgens per hour (VR/h), i.e. 10.6 R/h.

external radiation: Radiation from a source outside the body.

final status survey: Measurements and sampling to describe the radiological conditions of a

survey unit, following completion of decontamination activities (if any).

gamma (y) radiation: Penetrating high-energy, short-wavelength electromagnetic radiation

(similar to X-rays) emitted during radioactive decay. Gamma-rays are very penetrating and

require dense materials (such as lead or uranium) for shielding.

graded approach: An approach to data collection and interpretation that places the greatest

survey efforts on areas that have, or had, the highest potential for residual radioactivity.

15-5 NUREG-1505
GLOSSARY

gray region: A range of values below the appropriate DCGL for a site in which decision errors

carry relatively less risk or economic consequence.

grid: A network of intersecting and vertical lines used for the purpose of identification of exact

locations. See reference coordinate system, sampling grid.

grid area: See sampling grid area.

grid block: A square defined by two adjacent vertical and two adjacent horizontal reference

coordinate lines.

half-life (tl,2): The time required for one-half of the atoms present to disintegrate.

Historical Site Assessment (HSA): A detailed investigation to collect existing information,

primarily historical information, on a site and its surroundings.

hot measurement: See elevated measurement.

hot spot: See area of elevated activity.

hypothesis: An assumption about a property or characteristic of a set of data under study. The

goal of statistical inference is to decide which of two complementary hypotheses is likely to be

true. The null hypothesis describes what is assumed to be the true state of nature and the

alternative hypothesis describes the opposite situation.

impacted area: Areas with some potential for residual contamination.

indistinguishable from background: The detectable concentration distribution of a

radionuclide is not statistically different from the background concentration distribution of that

radionuclide in the vicinity of the site or, in the case of structures, in similar materials using

adequate measurement technology, survey, and statistical techniques.

inventory: Total residual quantity of formerly licensed radioactive material at a site.

investigation level: A radionuclide specific level of radioactivity that results in additional

investigation when it is exceeded, to determine if the survey unit was properly classified. See

action level, elevated measurement.

judgment sample or measurement: A sample or measurement taken at a location where

radiation levels or other site characteristics are expected to be unusual, based on the judgment

and/or experience of a skilled investigator. Also called authoritative or biased. Samples or

measurements that are not of this type are considered representative of the site being studied.

k: When conducting the Quantile test, k is the number of riieasurements from the survey unit that

are among the r largest measurements of the combined set of reference area and cleanup unit

measurements.

NUREG-1505 15-6
GLOSSARY

less-than data: Measurements that are less than the minimum detectable concentration.

license: A license issued under the regulations in Parts 30 through 35, 39, 40, 60, 61, 70 or Part

72 of 10 CFR Chapter I.

licensee: The holder of a license.

license termination: Discontinuation of a license, the eventual conclusion to decommissioning.

lower bound of the gray region (LBGR): The minimum value of-the gray region. The width of

the gray region (DCGL-LBGR) is also referred to as the shift, A.

lower limit of detection (LD): The smallest amount of radiation or radioactivity that statistically

yields a net result above the method background. The critical level, Lc, is the value set for

deciding that radioactivity is detected with a specified Type I error rate. The detection limit, LD,

is the level at which the power to detect net radioactivity is set.

m: The number of measurements required from the reference area to conduct a statistical test

with specified Type I and Type H error rates.

minimum detectable concentration (MDC): The a priori activity level that a specific

instrument and technique can be expected to detect a specified percentage of the time. When

stating the detection capability of an instrument, this value should be used. The MDC is the

detection limit, LD, multiplied by an appropriate conversion factor to give units of activity.

missing or unusable data: Data (measurements) that are mislabeled, lost, or do not meet quality

control standards. Less-than data are not considered to be missing or unusable data.

N: N = m + n, is the total number of measurements required from the reference area and a survey

unit being compared with the reference area. See.m and n.

n: Number of measurements required from a survey unit to conduct a statistical test that has

specified Tyfie I and Type 11 error rates.

naturally occurring radionuclides: Radionuclides and their associated progeny produced

during the formation of the earth or by interactions of terrestrial matter with cosmic rays.

non-detect: A measurement below the critical level, Lc.

non-impacted area: Areas in which there is no reasonable possibility of residual contamination.

nonparametric test: A test based on relatively few assumptions about the exact form of the

underlying probability distributions of the measurements. As a consequence, nonparametric tests

are generally valid for a fairly broad class of distributions. The Wilcoxon Rank Sum test and the

Sign test are examples of nonparametric tests.

15-7 NUREG-1505
GLOSSARY

normal (Gaussian) distribution: A family of bell-shaped distributions described by the mean

and variance.

outlier: Measurements that are unusually large relative to the bulk of the measurements in the

data set.

Pitman efficiency: A measure of performance for statistical tests. It is equal to the reciprocal of

the ratio of the sample sizes required by each of two tests to achieve the same power, as these

sample sizes become large.

power (1-p): The probability of rejecting the null hypothesis when it is false. The power is

equal to one minus the Type 11 error rate, i.e. (1- P3).

precision: A measure of mutual agreement among individual measurements, usually under

prescribed similar conditions, expressed generally in terms of the arithmetic standard deviation.

quality assurance (QA): An integrated system of management activities involving planning,

implementation, assessment, reporting, and quality improvement to ensure that a process, item,

or service is of the type and quality needed and expected by the client.

Quality Assurance Project Plan (QAPP): A formal document describing in comprehensive

detail the necessary QA, QC, and other technical activities that must be implemented to ensure

that the results of the work performed satisfies the stated performance criteria.

quality control (QC): The overall system of technical activities that measure the attributes and

performance of a process, item, or service against defined standards to verify that they meet the

stated requirements established by the client. QC includes operational techniques and activities

that are used to fulfill requirements for quality.

Quantile Test: A nonparametric test that looks at only the r largest measurements of the N

combined reference area and survey unit measurements. If a sufficiently large number of these r

measurements are from the survey unit, then the test indicates the survey unit has not attained the

reference-based cleanup standard.

RA: The acceptable level of risk associated with not detecting an area of elevated activity of area

Amn

radiation (ionizing radiation): Alpha particles, beta particles, gamma rays, x-rays, neutrons,

high-speed electrons, high-speed protons, and other particles capable of producing ions.

Radiation, as used in this document, does not include non-ionizing radiation, such as radio- or

microwaves, or visible, infrared, or ultraviolet light.

radioactive decay: The spontaneous transformation of an unstable atom into one or more

different nuclides accompanied by either the emission of energy and/or particles from the

nucleus, nuclear capture or ejection of orbital electrons, or fission. Unstable atoms decay into a

more stable state, eventually reaching a form that does not decay further or has a very long halflife.

NUREG-1505 15-8
GLOSSARY

radioactivity: The mean number of nuclear transformations occurring in a given quantity of

radioactive material per unit time. The International System (SI) unit of radioactivity is the

Becquerel (Bq). The customary unit is the Curie (Ci).

radiological survey: Measurements of radiation levels associated with a site together with

appropriate documentation and data evaluation.

radionuclide: An unstable nuclide that undergoes radioactive decay.

random error: The deviation of an observed value from the true value is called the error of

observation. If the error of observation behaves like a random variable-i.e., its value occurs as

though chosen at random from a probability distribution of such errors-it is called a random

error. See systematic error.

readily removable: A qualitative statement of the extent to which a radionuclide can be

removed from a surface or medium using non-destructive, common, housekeeping techniques

(e.g., washing with moderate amounfs of detergent and water) that do not generate large volumes

of radioactive waste requiring subsequent disposal or produce chemical wastes that are expected

to adversely affect public health or the environment.

reference area: Geographical area from which representative reference measurements are

performed for comparison with measurements performed in specific survey units at remediation

site. A site radiological reference area (background area) is defined as an area that has similar

physical, chemical, radiological, and biological characteristics as the site area being remediated,

but which has not been contaminated by site activities. The distribution and concentration of

background radiation in the reference area should be the same as that which would be expected

on the site if that site had never been contaminated. More than one reference area may be

necessary may be necessary for valid comparisons if a site exhibits considerable physical,

chemical, radiological, or biological variability.

reference coordinate system: A grid of intersecting lines referenced to a fixed site location or

benchmark. Typically the lines are arranged in a perpendicular pattern dividing the survey

location into squares or blocks of equal areas. Other patterns include three-dimensional and

polar coordinate systems.

reference grid: A network of parallel horizontal and vertical lines forming squares on a map

that may be overlaid on a property parcel for the purpose of identification of exact locations. See

reference coordinate system.

reference region: The geographical region from which reference areas will be selected for

comparison with survey units.

relative shift (A/o): A divided by a, the standard deviation of the measurements, is the relative

shift expressed in multiples of standard deviations. See delta.

15-9 15-9 NUREG-1505
GLOSSARY

release criteria: The criteria for license termination given in 10 CFR 20.1402 and 10 CFR

20.1403, expressed in terms of total effective dose equivalent (TEDE).

release criterion: A regulatory limit expressed in terms of dose or risk.

rem (roentgen equivalent man): The conventional unit of dose equivalent. The corresponding

International System (SI) unit is the Sievert (Sv): 1 Sv = 100 rem.

remediation: The process and associated activities 'resulting in removal of contamination from a

site. Remediation is sometimes used interchangeably with the terms remedial action, response

action, or decontamination.

remediation control survey: A type of survey that includes monitoring the progress of remedial

action by real time measurement of areas being decontaminated to determine whether or not

efforts are effective and to guide further decontamination activities.

removable activity: Surface activity that can be readily removed and collected for measurement

by wiping the surface with moderate pressure.

residual radioactivity: Radioactivity in structures, materials, soils, groundwater, and other

media at a site resulting from activities under the licensee's control. This includes radioactivity

from all licensed and unlicensed sources used by the licensee, but excludes background

radioactivity. It also includes radioactive materials remaining at the site as a result of routine or

accidental releases of radioactive material at the site and previous burials at the site, even if those

burials were made in accordance with the provisions of 10 CFR Part 20.

restricted use: A designation following remediation requiring radiological controls at a

formerly licensed site.

sample: A part or selection from a medium located in a survey unit or reference area that

represents the quality or quantity of a given parameter or nature of the whole area or unit; a

portion serving as a specimen.

sampling grid: The pattern of points formed by the locations chosen for systematic sampling.

sampling grid area: The area bounded by adjacent sampling locations on a systematic sampling

grid. If the linear distance between samples is L, then the grid area is L2 for a square sampling

grid and 0.866L2 for a triangular sampling grid.

scanning: An evaluation technique performed by moving a detection device over a surface at a

specified speed and distance above the surface to detect elevated levels of radiation.

scoping survey: A type of survey that is conducted to identify (1) radionuclide contaminants, (2)

relative radionuclide ratios, and (3) general levels and extent of contamination.

NUREG-1505 15-10
GLOSSARY

shape parameter (S): For an elliptical area of elevated activity, the ratio of the semi-minor axis

length to the semi-major axis length is the shape parameter. For a circle, the shape parameter is

one. A small shape parameter corresponds to a flat ellipse.

Sievert (Sv): The special name for the International System (SI) unit of dose equivalent.

1 Sv = 100 rem = 1 Joule per kilogram.

Sign test: A nonparametric statistical test used to demonstrate compliance with the release

criterion when the radionuclide of interest is not present in background and the distribution of

data is not symmetric. See also Wilcoxon Rank-Sum test.

site: Any installation, facility, or discrete, physically separate parcel of land, or any building or

structure or portion thereof, that is being considered for release.

size (of-a test): See alpha.

soil: The top layer of the earth's surface, consisting of rock and mineral particles mixed with

organic matter. A particular kind of earth or ground-e.g., sandy soil.

soil activity (soil concentration): The level of radioactivity present in soil and expressed in

units of activity per soil mass (typically Bq/kg or pCi/g).

source material: Uranium and/or thorium other than that classified as special nuclear material.

source term: All residual radioactivity remaining at the site, including material released during

normal operations, during inadvertent releases or accidents, and includes radioactive materials

which may have been buried at the site in accordance with 10 CFR Part 20.

special nuclear material: Plutonium, 233

U, and uranium enriched in ..5U; material capable of

undergoing a fission reaction.

square sampling grid: A systematic grid of sampling locations that is arranged in a square

pattern. See sampling grid.

standard normal distribution: A normal (Gaussian) distribution with mean zero and variance

one.

standard operating procedure (SOP): A written document that details the method for an

operation, analysis, or action with thoroughly prescribed techniques and steps, and that is

officially approved as the method for performing certain routine or repetitive tasks.

subsurface soil sample: A soil sample taken deeper than 15 cm below the soil surface.

surface contamination: Residual radioactivity found on building or equipment surfaces and

expressed in units of activity per surface area (Bq/m2 or dpm/100 cm2).

15-11 NUREG-1505
GLOSSARY

surface soil sample: A soil sample taken from the first 15 cm of surface soil.

surrogate: Radionuclide A is a surrogate for radionuclide B if there is an established ratio

between their concentrations in a survey unit. The concentration of radionuclide B can then be

inferred from the measured concentration of radionuclide A.

survey: A systematic evaluation and documentation of radiological measurements with a

correctly calibrated instrument or instruments that meet the sensitivity required by the objective

of the evaluation.

survey plan: A plan for determining the radiological characteristics of a site.

survey unit: A geographical area of specified size and shape at a remediated site for which a

separate decision will be made whether the unit attains the site-specific reference-based cleanup

standard for the designated pollution parameter. Survey units are generally formed by grouping

contiguous site areas with a similar use history and the same classification of contamination

potential. Survey units are established to facilitate the survey process and the statistical analysis

of survey data.

/

systematic error: An error of observation based on system faults which are biased in one or

more ways, e.g., tending to be on one side of the true value more than the other.

systematic sampling: Taking measurements or samples at locations according to a spatial

pattern. See triangular sampling grid, square sampling grid.

tandem testing: Two or more statistical tests conducted using the same data set.

TEDE (total effective dose equivalent): The effective dose equivalent is the summation of the

products of the dose equivalent received by specified tissues of the body and a tissute-specific

weighting factor. It is a risk-equivalent Vialue, expressed in Sv or rem, that can be used to

estimate the health-effects on an exposed individual. When calculating TEDE, the licensee shall

base estimates on the greatest annual TEDE dose expected within the first 1000 years after

decommissioning. Estimates must be substantiated using actual measurements to the maximum

extent practical. See CEDE.

test statistic: A function of the measurements (or their ranks) that has a known distribution if

the null hypothesis is true. This is compared to the critical level to determine if the null

hypothesis should be rejected.

tied measurements: Two or more measurements that have the same value.

triangular sampling grid: A grid of sampling locations that is arranged in a triangular pattern.

See sampling grid.

NUREG-1505 15-12
GLOSSARY

two-sample t-test: A parametric statistical test used in place of the Wilcoxon Rank Sum (WRS)

test if the reference area and survey unit measurements are known to be normally (Gaussian)

distributed and there are no less-than measurements in either data set.

Type I decision error: A decision error that occurs when the null hypothesis is rejected when it

is true. The probability of making a Type I decision error is called alpha (a).

Type II decision error: A decision error that occurs when the null hypothesis is accepted when

it is false. The probability of making a Type H decision error is called beta (P3).

unaffected area: An impacted area that is expected to contain little, if any, residual radioactivity,

based on a knowledge of site history and previous survey information. This term was previously

used for areas designated Class 3.

unity rule (mixture rule): A rule applied when more than one radionuclide is present at a

concentration that is distinguishable from background and where a single concentration

comparison does not apply. In this case the mixture of radionuclides is compared against default

concentrations by applying the unity rule. This is accomplished by determining (1) the ratio

between the concentration of each radionuclide in the mixture and (2) the concentration for that

radionuclide in an appropriate listing of default values. The sum of the ratios for all

radionuclides in the mixture should not exceed 1.

unrestricted area: Any area where access is not controlled by a licensee for purposes of

protection of individuals from exposure to radiation and radioactive materials-including areas

used for residential purposes.

unrestricted release: Release of a site from regulatory control without requirements for future

radiological restrictions. Also known as unrestricted use.

Wilcoxon Rank Sum (WRS) test: A nonparametric statistical test used to determine

compliance with the release criterion when the radionuclide of concern is present in background.

See also Sign test.

z•: the 4ith quantile of a standard normal distribution. Also called the 100 percentile. The

probability of observing a value less than zý is equal to 4P. See standard normal distribution.

15-13 NUREG-1505
'I
16 BIBLIOGRAPHY

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Society. 1984.

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BIBLIOGRAPHY

Ling, R.F., and J.W. Pratt. "The Accuracy of Peizer Approximations to the Hypergeometric

Distribution, With Comparisons to Some Other Approximations," Journal of the American

Statistical Association 79 (385) 49-60. 1984.

Microsoft Excel for Windows, Version 5.0. Redmond, WA: Microsoft Corporation. 1993.

Neptune, 0., E.P. Brantley, M.J. Messner, and D.I. Michael. "Quantitative Decision Making in

Superfund: A Data Quality Objectives Case Study," Hazardous Materials Control 3(3):18-27.

1990.

Noether, G.E. "Sample Size Determination for Some Common Nonparametric Tests,"

Journal of the American Statistical Association 82:645-647. 1987.

Norton, R.M. "Pocket-Calculator Approximation for Areas Under the Standard Normal Curve,"

American Statistician 43:24-26. 1989.

Nuclear Regulatory Commission. "Draft Branch Technical Position on Site Characterization for

Decommissioning." Washington, D.C. November 1994.

Nuclear Regulatory Commission. "Guidelines for Decontamination of Facilities and Equipment

Prior to Release for Unrestricted Use or Termination of Licenses for Byproduct, Source, or

Special Nuclear Materials." Washington, D.C. May 1987.

NUREG-0586. "Final Generic Environmental Impact Statement on Decommissioning of

Nuclear Facilities." Washington, D.C.: Nuclear Regulatory Commission. 1988.

NUREG-1496 [a]. "Generic Environmental Impact Statement in Support of Rulemaking on

Radiological Criteria for Decommissioning of NRC-Licensed Facilities," Volume 1- Main

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NUREG-1496 [b]. "Generic Environmental Impact Statement in Support of Rulemaking on

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NUREG-1500. "Working Draft Regulatory Guide on Release Criteria for Decommissioning:

NRC Staff's Draft for Comment." Washington, D.C.: Nuclear Regulatory Commission.- August

1994.

NUREG-1501. "Background as a Residual Radioactivity Criterion for Decommissioning"

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NUREG-1505 Nuclear Regulatory Commission (NRC). 1997a. "A Nonparametric Statistical•

Methodology for the Design and Analysis of Final Status Decommissioning Surveys" (Draft

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BIBLIOGRAPHY

NUREG- 1506. "Measurement Methods for Radiological Surveys in Support of New

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Commission. 1995.

NUREG/CR-1507. "Minimum Detectable Concentrations with Typical Radiation Survey

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NUREG/CR-5512. "Residual Radioactive Contamination From Decommissioning," Volume I

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16-5 NUREG-1505
BIBLIOGRAPHY

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BIBLIOGRAPHY

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16-7 NUREG-1505

APPENDIX: STATISTICAL TABLES

Table A.1 Cumulative Normal Distribution Function dO(z)

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.00 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.10 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5674 0.5714 0.5753

0.20 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

0.30 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.40 0.6554 0.659 1 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.50 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.60 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.70 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.80 10.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133

0.90 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.00 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.10 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

1.20 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.30 0.9032 0.9049 0.9066 10.9082 0.9099 0.9115 10.9131 0.9147 0.9162 0.9177

1.40 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.50 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

1.60 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.95 15 0.9525 0.9535 0.9545

1.70 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633

1.80 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706

1.90 10.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

2.00 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 10.9812 0.9817

2.10 0.982 1 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857

2.20 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890

2.30 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.99 13 0.99 16

2.40 0.9918 0.9920 0.9922 10.9925 0.9927 0.9929 10.9931 0.9932 0.9934 10.993 6

2.50 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 1 0.9952

2.60 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 10.9964

2.70 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.997 1 0.9972 0.9973 10.9974

2.80 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 10.9981

2.90 0.998 1 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 10.9986

3.00 10.9987 0.9987 0.9987 10.9988 0.9988 0.9989 10.9989 0.9989 0.9990 0.9990

3.10 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 10.9993

3.20 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 _0.9995 0.9995

3.30 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 (0.9996 0.9996 0.99961 0.9997

3.40 0.9997 10.9997 0.9997 0.9997 10.9997 0.99971 0.9997 0.9997 0.97% 0.999

Negative values of z can be obtained from the relationship c(D(-z) = I - cID(z).

A-1 A-i NUREG-1505
STATISTICAL TABLES

Table A.2a Sample Sizes for the Sign Test

(Number of measurements to be performed in each survey unit)

(uf) or (0,a)

0.01 0.01 0.01 0.01 0.01 0.025 0.025 0.025 0.025 0.05 0.05 0.05 0.1 0.1 0.25

a 0.01 0.025 0.05 0.1 0.25 0.025 0.05 0.1 0.25 0.05 0.1 0.25 0.1 0.25 0.25

0.10 4095 3476 2984 2463 1704 2907 2459 1989 1313 2048 1620 1018 1244 725 345

0.20 1035 879 754 623 431 735 622 503 333 518 410 258 315 184 88

0.30 468 398 341 282 195 333 281 227 150 234 185 117 143 83 40

0.40 270 230 197 162 113 192 162 131 87 136 107 68 82 48 23

0.50 178 152 130 107 75 126 107 87 58 89 71 45 54 33 16

0.60 129 110 94 77 54 92 77 63 42 65 52 33 40 23 11

0.70 99 83 72 59 41 70 59 48 33 50 40 26 30 18 9

0.80 80 68 58 48 34 57 48 39 26 40 32 21 24 15 8

0.90 66 57 48 40 28 47 40 33 22 34 27 17 21 12 6

1.00 57 48 41 34 24 40 34 28 18 29 23 15 18 11 5

1.10 50 42 36 30 21 35 30 24 17 26 21 14 16 10 5

1.20 45 38 33 27 20 32 27 22 15 23 18 12 15 9 5

1.30 41 35 .30 26 17 29 24 21 14 21 17 11 14 8 4

1.40 38 33 28 23 16 27 23 18 12 20 16 10 12 8 4

1.50 35 30 27 22 15 26 22 17 12 18 15 10 11 8 4

1.60 34 29 24 21 15 24 21 17 11 17 14 9 11 6 4

1.70 33 28 24 20 14 23 20' 16 11 17 14 9 10 6 4

1.80 32 27 23 20 14 22 20 16 11 16 12 9 10 6 .4

1.90 30 26 22 18 14 22 18 15 10 16 12 9 ,10 6 4

2.00 29 26 22 18 12 21 18 15 10 15 12 8 10 6 3

2.50 28 23 21 17 12 20 17 14 10 15 11 8 9 5 3

3.00 27 23 20 17 12 20 17 14 9 14 11 8 9 5 3

NUREG-1505 A-2
STATISTICAL TABLES

Table A.2b Sample Sizes for the Wilcoxon Rank Sum Test

(Number of measurements to be performed in the reference area and in each survey unit)

(aAD) or (0,a)

0.01 0.01 0.01 0.01 0.01 0.025 0.025 0.025 0.025 0.05 0.05 0.05 0.1 0.1 0.25

A/a 0.01 0.025 0.05 0.1 0.25 0.025 0.05 0.1 0.25 0.05 0.1 0.25 0.1 0.25 0.25

0.10 5452 4627 3972 3278 2268 3870 3273 2646 1748 2726 2157 1355 1655 964 459

0.20.1370 1163 998 824 570 973 823 665 440 685 542 341 416 243 116

0.30 614 521 448 370 256 436 369 298 197 307 243 153 187 109 52

0.40 350 297 255 211 146 248 210 170 112 175 139 87 106 62 30

0.50 227 193 166 137 95 162 137 111 73 114 90 57 69 41 20

0.60, 161 137 117 97 67 114 97 78 52 81 64 40 49 29 14

0.70 121 103 88 73 51 86 73 59 39 61 48 30 37 22 11

0.80 95 81 69 57 40 68 57 46 31 48 38 24 29 17 8

0.90 77 66 56 47 32 55 46 38 25 39 31 20 24 14 7

1.00 64 55 47 39 27 46 39 32 21 32 26 16 20 12 6

1i.o0 55 47 40 33 23 39 33 27 18 28 22 14 17 10 5

1.20 48 41 35 29 20 34 29 24 16 24 19 12 15 9 4

1.30 43 36 31 26 18 30 26 21 14 22 17 11 13 8 4

1.40 38 32 28 23 16 27 23 19 13 19 15 10 12 7 4

1.50 35 30 25 21 15 25 21 17 11 18 14 9 11 7 3

1.60 32 27 23 19 14 23 19 16 11 16 13 8 10 6 3

1.70 30 25 22 18 13 21 18 15 10 15 12 8 9 6 3

1.80 28 24 20 17 12 20 17 14 9 14 11 7 9 5 3

1.90 26 22 19 16 11 19 16 13 9 13 11 7 8 5 3

2.00 25 21 18 15 11 18 15 12 8 13 10 7 8 5 3

2.25 22 19 16 14 10 16 14 11 8 11 9 6 7 4 2

2.50 21 18 15 13 9 15 13 10 7 11 9 6 7 4 2

2.75 20 17 15 12 9 14 12 10 7 10 8 5 6 4 2

3.00 19 16 14 12 8 14 12 10 6 10 8 5 6 4 2

3.50 18 16 13 11 8 13 11 9 6 9 8 5 6 4 2

4.00 18 15 13 11 8 13 11 9 6 1.9 7 5 6 4 2

A-3 NUREG-1505
STATISTICAL TABLES

Table A.3 Critical Values for the Sign Test Statistic S+

N 0.005 0.01 0.025 0.05 0.1 0.2 0.3 0.4 0.5

4 4 .4 4 4 3 3 3 2 2

.5 5 5 5 4 4 3 3 3 2

6 6 6 5 5 5 4 4 3 3

.7 7 6 6 6 5 5 4 4 3

8 7 7 7 6 6 5 5 4 4

9 8 8 7 7 6 6 5 5 4

10 9 9 18 8 7 6 6 5 5

11 10 9 9 8 8 7 6 6 5

12 10 10 9 9 8 7 7 6 6

13 11 11 10 9 9 8 7 7 6

14 12 11 11 10 9 9 8 7 7

15 12 12 11 11 10 9 9 8 7

16 13 13 12 11 11 10 9 9 8

17 14 13 12 12 11 10 10 9 8

18 14 14 13 12 12 11 10 10 9

19 15 14 14 13 12 11 11 10 9

20 16 15 14 14 113 12 11 11 10

21 16 16 15 14 13 12 12 11 10

22 17 16 16 15 14 13 12 12 11

23 18 17 16 15 15 14 13 12 11

24 18 18 17 16 15 14 13 13 12

25 19 18 17 17 16 15 14 13 12

26 19 19 18 17 J16 15 14 14 13

27 20 19 19 18 17 16 15 14 13

28 21 .20 19 18 .17 16 15 15 14

29 21 21 20 19 18 17 16 15 14

30 22 21 20 19 19 17 16 16 15

NUREG- 1505A4 A-4
STATISTICAL TABLES

Table A.3 Critical Values for the Sign Test Statistic S+ (continued)

____ ___ _ __ __ ___ _ __ __ _ ___

N 0.005 0.01 0.025 0.05 0.1 0.2 0.3 0.4 0.5

31 23 22 21 20 19 18 17 16 15

32 23 23 22 21 20 18 17 17 16

33 24 23 22 21 20 19 18 17 16

34 24 *24 23 22 21 19 19 18 17

35 25 24 23 22 21 20 19 18 17

36 26 25 24 23 22 21 20 19 18

37 26 26 24 23 22 21 20 19 18

38 27 26 25 24 23 22 21 20 19

39 27 27 26 .25 23 22 21 20 19

40 28 27 26 25 24 23 22 21 20

41 29 28 27 26 25 23 22 21 20

42 29 28 27 26 25 24 23 22 21

43 30 29 28 27 26 24 23 22 21

44 30 30 28 27 26 25 24 23 22

45 31 30 29 28 27 25 24 23 22

46 32 31 30 29 27 26 25 24 23

47 32 31 30 29 28 26 25 24 23

48 33 32 31 30 28 27 26 25 24

49 33 33 31 30 29 27 26 25 2

50 34 33 32 31 30 28 27 26 2

For N larger than 50, the critical value, CV, may be obtained from the expression:

CV = 0.5 [N + zV FN], where z is the (1- a) percentile of a standard normal distribution,

which can be found in the following table:

IC 0I.005 0.01 I0.025 0.05 0.1 0.2 1 0.3 0.4 I0.51

z 12.575 2.326 1.96 1.645 1.282 0.842 10.524 0.253 0.000

Other values can be found in Table A. 1.

A-5 A-S NUREG-1 505
STATISTICAL TABLES

Table A.4 Critical Values for the WRS Test

NOTE: When using this table under Scenario A, m is the number of reference area samples and

n is the number of survey unit samples. When using this table for Scenario B, the roles of m and

n in this table are reversed.

n= 2

m=2 a-=0.001 7

a=0.005 7

a--0.01 7

a=0.025 7

N a--0.05 7

a=-0.1 7

n= 2

m=3 a-=0.001 12

a=0.005 12

a=0.01 12

a=0.025 12

a=0.05 12

a=0.1 11

n= 2

rn=4 a--0.001 18

a=0.005 18

t--0.0O 18

a=0.025 18

a--0.05 18

a--0.1 17

n= 2

m=5 a--0.001 25

a--0.005 25

a=0.01 25

a--0.025 25

a-=0.05 24

a--0.1 23

n= 2

m=6 a--0.001 33

a--0.005 33

a-=0.01 33

a--0.025 33

a--0.05 32

a--0.1 31

n= 2

m=7 a=0.001 42

a--0.005 42

a=0.01 42

a=0.025 42

a=0.05 41

&--0.1 40

n= 2

m= 8 a--0.001 52

a--0.005 52

a=0.01 52

a--0.025 51

a=0.05 50

a-=0.1 49

NUREG-1505

3

9

9

9

9

9

8

3

15

15

15

15

14

13

3

22

22

22

22

21

20

3

30

30

30

29

28

27

3

39

39

39

37

36

35

5

13

13

13

13

12

11

5

21

21

21

20

19

18

5

30

30

29

28

27

25

5

40

39

38

37

35

34

5

51

49

48

47

45

43

5

63

61

59

57

56

54

5

75

73

71

69

67

65

7

17

17

17

17

16

15

7

27

27

26

25

24

22

7

38

37

36

34

33

31

7

50

48

46

44

43

41

7

63

59

58

56

54

51

7

75

72

70

68

65

63

7

89

85

84

81

78

75

8

19

19

19

18

17

16

8

30

30

29

27

26

24

8

42

40

39

37

36

34

8

54

52

50

48

46

44

8

67

64

62

60

58

55

8

81

77

76

73

70

67

8

95

92

90

86

84

80

10

23

23

23

22

21

19

10

36

35

34

32

31

29

10

49

47

46

44

42

39

10

63

60

58

56

53

51

10

77

74

72

69

66

63

10

92

88

86

83

80

76

11

25

25

25

23

23

21

iII

39

38

37

35

33

31

11

53

51

49

47

45

42

11

67

64

62

60

57

54

11

82

79

77

73

70

67

12

27

27

27

25

24

22

12

42

40

39

37

36

33

12

57

54

52

50

48

45

12

72

68

66

63

61

57

12

88

83

81

78

75

71

13 14

29 31

29 31

28 30

27 29

26 27

24 26

13 14

45 48

43 46

42 45

40 42

38 40

35 37

13 14

60 64

58 61

56 59

53 56

51 54

48 50

13 14

76 81

72 77

70 74

67 71

64 68

61 64

13 14

93 98

88 93

86 91

82 87

79 83

75 79

15

33

33

32

31

29

27

15

51

48

47

45

43

40

15

68

64

62

59

57

53

15

85

81

78

75

71

67

16

35

35

34

33

31

29

16

54

51

50

47

45

42

16

71

68

66

62

59

56

16

89

85

82

79

75

71

17

37

37

36

34

33

30

17

56

54

52

50

47

44

17

75

71

69

66

62

59

17

94

89

86

82

79

74

18

39

39

38

36

34

32

18

59

57

55

52

50

46

18

78

75

72

69

65

61

18

98

93

90

86

82

77

19

41

40

39

38

36

33

20

43

42

41

40

38

35

4

19 20

62 65

59 62

58 60

55 57

52 54

48 50

19 20

82 86

78 81

76 79

72 75

68 71

64 67

19 20

102 107

97 101

94 98

90 94

86 89

81 84

4

45

44

43

42

41

39

3 4

49 56

49 55

.48 54

47 52

46 51

44 49

3 4

60 68

60 66

59 65

57 63

56 62

54 60

6

57

54

53

51

49

47

6

69

66

65

63

61

58

6

82

79

77

75

73

70

9

72

69

67

64

62

59

9

87

83

81

78

75

72

15 16 17 18 19 20

103 108 113 118 123 128

98 103 107 112 117 122

95 100 104 109 114 118

91 95 100 104 109 113

87 91 96 100 104 108

83 87 91 94 98 102

11 12 13 14 15 16 17 18 19 20

98 104 110 116 122 128 133 139 145 151

94 99 105 110 116 121 127 132 138 143

92 97 102 108 113 118 123 129 134 139

88.93 98 103 108 113 118 123 128 133

85 90 94 99 104 109 113 118 123 128

81 85 90 94 99 103 108 112 117 121

9 10 11 12 13 14 15 16 17 18 19 20

102 109 115 122 128 135 141 148 154 161 167 174

98 104 110 116 122 129 135 141 147 153 159 165

96 102 108 114 120 125 131 137 143 149 155 161

92 98 104 109 115 121 126 132 137 143 149 154

89 95 100 105 111 116 122 127 132 138 143 148

85 91 96 101 106 111 116 121 126 131 136 141

. A-6
STATISTICAL TABLES

Table A.4 Critical Values for the WRS Test (continued)

n =

m=9 a--0.001

a--0.005

a=O.01

a--0.025

a--0.05

a--O.!

2

63

63

63

62

61

60

3

72

71

70

69

67

66

.3

'85

84

83

81

80

78

4

81

79

77

76

74

71

4

94

92

91

89

87

84

5

88

86

84

82

80

77

5

103

100

98

96

93

91

6

96

93

91

88

86

83

7 8

104 111

100 107

98 105

95 101

92 98

89 94

9

118

114

III

108

104

100

m= 10 a--0.001

a--0.005

a--0.01

a--0.025

a=0.05

a--0.1

nI=

m=11 a--0.001

c--0.005

a=0.01

a=0.025

a=0.05

a--0. I

2

75

75

75

74

73

71

2

88

88

88

87

86

84

6"7 8 9

111 119 128 136

108 115 123 131

106 113 121 128

103 110 117 124

100 107 114 120

97 103 110 116

10

126

121

118

114

110

106

10

144

138

135

131

127

122

10

163

157

153

149

144

139

10

183

176

173

168

165

158

11

133

127

125

120

116

112

11

152

146

142

138

133

128

11

171

165

161

156

152

146

12

140

134

131

126

122

117

12

160

153

150

145

140

135

12

180

173

169

164

159

153

13 14

147 155

141 148

138 144

133 139

128 134

123 129

15

162

155

151

145

140

134

16

169

161

157

151

146

140

16

191

183

178

172

166

160

13 14 15

167 175 183

160 168 175

157 164 171

151 158 165

147 153 160

141 147 153

17

176

168

164

158

152

145

18

183

175

170

164

158

151

17 18

199 207

190 197

186 193

179 186

173 179

166 172

19 20

190 198

182 188

177 184

170 176

164 170

157 162

19 20

215 222

205 212

200 207

192 199

186 192

178 184

3 4 5 6 7 8 9

99 109 118 127 136 145 154

98 107 115 124 132 140 148

97 105 113 122 130 138 146

95 103 111 118 126 134 141

93 101 108 115 123 130 137

91 98 105 112 119 126 133

n = •2

m = 12 a--0.001 102

a--0.005 102

a--0.01 102

a--0.025 100

a--0.05 99

a--O.l 97

3 4 5 6

114 125 135 145

112 122 131 140

111 120 129 138

109 118 126 135

108 116 124 132

105 113 120 128

7 8 9

154164 173

149 158 167

147 156 164

143 151 159

140 147 155

135 143 150

11 12

192 202

185 194

181 190

176 184

171 179

165 172

11 12

213 223

206215

202 211

196 205

191 199

185 193

13 14 15 16 17 18 19 20

188 197 206 214 223 231 240 248

181 189 197 205 213 221 229 237

177 185 193 200 208 216 224 232

171 179 186 194 201 208 216 223

166 173 180 187 195 202 209 216

160 167 173 180 187 194 201 207

13 14 15 16 17 18 19 20

210 220 230 238 247 256 266 275

202 211 220 228 237 246 254 263

198 207 215 223 232 240 249 257

192200208 216 224 232 240 248

186 194 202 209 217 225 233 240

180 187 194 202 209 216 224 231

13 14 15 16 17 18 19 20

233 243 253 263 273 282 292 302

225 234 243 253 262 271 280 290

220 229 238 247 256 265 274 283

214 222 231 239 248 257 265 274

208 216 224 233 241 249 257 266

201 209217 224 232 240 248 256

n--=

m = 13 a--0.001

a--0.005

ix=0.01

a--0.025

a=0.05

a=0. I

2

117

117

116

115

114

112

3 4 5

130 141 152

128 139 148

127 137 146

125 134 143

123 132 140

120 129 137

3 4 5

147 159 171

145 156 167

144 154 164

141 151 161

139 149 158

136 145 154

6

163

158

156

152

149

145

7 8 9 .10

173 183 193 203

168 177 187 196

165 174 184 193

161 170 179 187

157 166 174 183

153 161 169 177

n= 2

m= 14 a--0.001 133

a=0.005 133

a--0.01 132

d-=0.025 131

a=0.05 129

a--0.1 128

6 7 8 9 10 11 12 13 14 15 16 17 18 "19 20

182 193 204 215 225 236 247 257 268 278 289 299 310 320 330

177 187 198 208 218 228 238 248 258 268 278 288 298 307 317

175 185 194 204 214 224 234 243 253 263 272 282 291 301 311

171 180 190 199 208 218 227 236 245 255 264 273 282 292 301

167 176 185 194 203 212 221 230 239 248 257 265 274 283 292

163 171 180 189 197 206214 223 231 240 248 257 265 273 282

m= 15 a--0.001

a--0.005

a=0.01

a--0.025

a--0.05

a--0. I

2 3 4 5

150 165 178 190

150 162 174 186

149 161 172 183

148 159 169 180

146 157 167 176

144 154 163 172

6 7

202 212

197 208

194 205

190 200

186 196

182 191

8 9 10 11 12

225 237 248 260 271

219 230 240 251 262

215 226 236 247 257

210 220 230 240 250

206 215 225 234 244

200 209 218 227 236

13 14 15

282 293 304

272 283 293

267 278 288

260 270 280

253 263 272

246 255 264

16 17

316 327

304 314

298 308

289 299

282 291

273 282

18

338

325

319

309

301

291

19 20

349 360

335 346

329 339

319 329

310 319

300 309

A-7 NUREG-1505
STATISTICAL TABLES

Table A.4 Critical Values for the WRS Test (continued)

m = 16 a--0.001

a--0.005

a=0.01

a--0.025

a--0.05

a--O.

n--

m =17 a--0.001

a--0.005

a--0.01

a--0.025

a--0.05

a---O.l

M =18 a--0.001

a--0.005

a--0.01

a--0.025

a=0.05

a--O.l

m =19 a--0.001

a--0.005

a--0.01

a--0.025

a--0.05

a---O.l

n =

rn=20 x--0.001

a--0.005

a--0.01

a=0.025

a--0.05

ca-0.1

2

168

168

167

166

164

162

2

187

187

186

184

183

180

2

207

207

206

204

202

200

2

228

227

226

225

223

220

3 4 5 6 7 8 9 10

184 197 210 223 236 248 260 272

181 194 206 218 229 241 252 264

180 192 203 215 226 237 248 259

177 188 200 210 221 232 242 253

175 185 196 206 217 227 237 247

172 182 192 202 211 221 231 241

11 12

284 296

275 286

270 281

264 274

257 267

250 260

13 14 15 16 17 18 19 20

308 320 332 343 355 367 379 390

298 309 320 331 342 353 365 376

292 303 314 325 336 347 357 368

284 295 305 316 326 337 347 357

278 288 298 308 318 328 338 348

269 279 289 298 308 317 327 336

3 4 5 6 7 8 9 10 11 12

203 218 232 245 258 271 284 297 310 322

201 214 227 239 252 264 276 288 300312

199 212 224 236 248 260 272 284 295 307

197 209 220 232 243 254 266 277 288 299

194 205 217 228 238 249 260 271 282 292

191 202212 223 233 243 253 264 274 284

3 4 5 6 7 8 9 10 11 12

224 239 254 268 282 296 309 323 336 349

222 236 249 262 275 288 301 313 326 339

220 233 246 259 272 284 296 309 321 333

217 230 242 254 266 278 290 302 313 325

215 226 238 250 261 273 284 295 307 318

211 222 233 244 255 266 277 288 299 309

13 14 15

335 347 360

324 336 347

318 330 341

310321 332

303 313 324

294305 315

16 17 18 19 20

372 384 397 409 422

359 371 383 394 406

353 364 376 387 399

343 354 365 376 387

335 345 356 366 377

325 335 345 355 365

13 14 15 16 17 18 19 20

362 376 389 402 415 428 441 454

351 364 376 388 401 413 425 438

345 357 370 382 394 406 418 430

337 348 360 372 383 395 406418-

329 340 352 363 374 385 396 407

320 331 342 352 363 374 384 395

3 4 " 5

246 262 277

243 258 272

242 256 269

239 252 265

236 248 261

232 244 256

6 7 8 9 10 11 12

292 307 321 335 350 364 377

286 300 313 327 340 353 366

283 296 309 322 335 348 361

278 290 303 315 327 340 352

273 285 297 309 321 333 345

267 279 290 302 313 325 336

6 7 8 9 10 11 12

317 333 348 363 377 392 407

311 325 339 353 367 381 395

307 321 335 349 362 376 389

302 315 329 341 354 367 380

297 310 322 335 347 360 372

291 303 315 327 339 351 363

13 14 15

391 405 419

379 392 405

373 386 399

364377 389

356 368 380

347 358 370

13 14 15

421 435 450

409422 436

402 416 429

393 406 419

385 397 409

375 387 399

16 17 18 19 20

433 446 460 473 487

419 431 444 457 470

411 424 437 449 462

401 413 425 437 450

392 403 415 427 439

381 392 403 415 426

16 17 18 19 20

464 479 493 507 521

450 463 477 490 504

442 456 469 482 495

431 444 457 470 482

422 434 446 459 471

410 422 434 446 458

2 3 4 5

250 269 286 302

249 266 281 296

248 264 279 293

247 261 275 289

245 258 271 284

242 254 267 279

Reject the null hypothesis if the test statistic is greater than the table (critical) value.

For n or m greater than 20, the table (critical) value can be calculated for Scenario A from

m(n+m+l)12 + zlnm(n+m+1)ll2

if there are few or no ties, and from

m(n+m+l)/2 + z ]!i (n+m+l1)-E iI1) ] 1j-l (n+m)(n+m-I)

if there are many ties, where g is the number of groups of tied measurements and tj is the number

of tied measurements in thejth group. For Scenario B, the roles of n and m are reversed in these

equations. z is the (1 -a) percentile of a standard normal distribution (see Table A. 1).

NUREG-1505 A-8
STATISTICAL TABLES

Table A.5a 0.025 and 0.975 Percentiles of the Chi-Squared Distribution

DOF 0.025 0.975 DOF 0.025 0.975 DOF 0.025 0.975 DOF 0.025 0.975

____46 29.2 66.6 91 66.5 119.3 280 235.5 328.2

2 0.05 7.4 47 30 67.8 92 67.4 120.4 285 240.1 333.7

3 0.2 9.3 48 30.8 69.0 93 68.2 121.6 290 244.7 339.1

4 0.5 11.1 49 31.6 .70.2 94 69.1 122.7 295 249.3 344.5

5 0.8 12.8 50 32.4 71.4 95 69.9 123.9 300 253.9 349.9

6 1.2 14.4 51 33.2 72.6 96 70.8 125.0 305 258.5 355.3

7 1.7 16.0 52 34.0 73.8 97 71.6 126.1 310 263.1 360.7

8 2.2 17.5 53 34.8 75.0 98 72.5 127.3 315 267.7 366.1

9 2.7 19.0 54 35.6 76.2 99 73.4 128.4* 320 272.3 371.4

10 3.2 20.5 55 36.4 77.4 100 74.2 129.6 325 277.0 1376.8

11 3.8 21.9 56 37.2 78.6 105 78.5 135.2 330 281.6 382.2

12 4.4 23.3 57 38.0 79.8 110 82.9 140.9 335 286.2 387.6

13 5.0 24.7 58 38.8 80.9 115 87.2 146.6 340 290.8 393.0

14 5.6 26.1 59 39.7 82.1 120 91.6 152.2 345 295.4 398.4

15 6.3 27.5 60 40.5 83.3 125 95.9 157.8 350 300.1 403.7

16 6.9 28.8 61 41.3 84.5 130 100.3 163.5 355 304.7 409.1

17 7.6 30.2 62 42.1 85.7 135 104.7 169.1 360 309.3 414.5

18 8.2 31.5 63 43.0 86.8 140 109.1 174.6 365 314.0 419.8

19 8.9 32.9 64 43.8 88.0 145 113.6 180.2 370 318.6 425.2

20 9.6 34.2 65 44.6 89.2 150 118.0 185.8 375 323.2 430.5

21 10.3 35.5 66 45.4 90.3 155 122.4 191.4 380 327.9 435.9

22 11.0 36.8 67 46.3 91.5 160 126.9 196.9 385 332.5 441.3

23 11.7 38.1 68 47.1 92.7 165 131.3 202.5 390 337.2 446.6

24 12.4 39.4 69 47.9 93.9 170 135.8 208.0 395 341.8 452.0

25 13.1 40.6 70 48.8 95.0 175 140.3 213.5 400 346.5 457.3

26 13.8 41.9 71 49.6 96.2 180 144.7 219.0 405 351.1 462.7

27 14.6 43.2 72 50.4 97.4 185 149.2 224.6 410 355.8 468.0

28 15.3 44.5 73 51.3 98.5 190 153.7 230.1 415 360.5. 473.3

29 16.0 45.7 74 52.1 99.7 195 158.2 235.6 420 365.1 478.7

30 16.8 47.0 75 52.9 100.8 200 162.7 241.1 425 369.8 484.0

31 17.5 48.2 76 53.8 102.0 205 167.2 246.5 430 374.4 489.3

32 18.3 49.5 77 54.6 103.2 210 171.8 252.0 435 379.1 '494.7

33 19.0 50.7 78 55.5 104.3 215 176.3 257.5 440 383.8 500.0

34 19.8 52.0 79 56.3 105.5 220 180.8 263.0 445 388.4 505.3

35 20.6 53.2 80 57.2 106.6 225 185.3 268.4 450 393.1 510.7

36 21.3 54.4 81 58.0 107.8 230 189.9 273.9 455 397.8 516.0

37 22.1 55.7 82 58.8 108.9 235 194.4 279.4 460 402.5 521.3

'38 22.9 56.9 83 59.7 110.1 240 199.0 284.8 465 407.1 526.6

39 23.7 58.1 84 60.5 111.2 245 203.5 290.2 470 411.8 532.0

40 24.4 59.3 85 61.4 112.4 250 208.1 295.7 475 416.5 537.3

41 25.2 60.6 86 62.2 113.5 255 212.7 301.1 480 421.2 542.6

42 26.0 61.8 87 63.1 114.7 260 217.2 306.6 485 425.9 547.9

43 26.8 63.0 88 63.9 -115.8 265 221.8 312.0 490 430.6 553.2

44 27.6 64.2 89 64.8 117.0 270 226.4 317.4 495 435.2 558.5

45 28.4 .65.4 90 65.6 118.1 275 231.0 1322.8 500 _[43-9.9 1563.9

A-9 A-9 NUREG- 1505
STATISTIAL TABLES

Table A.5b 0.05 and 0.95 Percentiles of the Chi-Squared Distribution

DOF 0.05 0.95 DOF 0.05 0.95 DOF 0.05 0.95 DOF 0.05 0.95

_______46 31.4 62.8 91 70.0 114.3 280 242.2 320.0

2 0.1 6.0 47 32.3 64.0 92 70.9 115.4 285 246.9 325.4

3 0.4 7.8 48 33.1 65.2 93 71.8 116.5 290 251.6 330.7

4 0.7 9.5 49 33.9 66.3 94 .72.6 ,117.6 295 256.2 336.1

5 1.1 11.1 50 34.8 67.5 95 73.5 118.8 300 260.9 341.4

6 1.6 12.6 51 35.6 68.7 96 74.4 119.9 3505 265.5 346.7

7 .2.2 14.1 52 36.4 69.8 97 75.3 121.0 310 270.2 352.1

8 2.7 15.5 53 37.3 71.0 98 76.2 122.1 315 274.9 357.4

9 3.3 16.9 54 38.1 72.2 99 77.0 123.2 320 279.6 362.7

10 3.9 18.3 55 39.0 73.3 100 77.9 124.3 325 284.2 368.0

11 4.6 19.7 56 39.8 74.5 105 82.4 129.9 330 288.9 373.4

12 5.2 21.0 57 40.6 75.6 110 86.8 135.5 335 293.6 378.7

13 5.9 22.4 58 41.5 76.8 115 91.2 141.0 340 298.3 384.0

14 6.6 23.7 59 42.3 77.9 120 95.7 146.6 345 303.0 389.3

15 7.3 25.0 60 43.2 79.1 125 100.2 152.1 350 307.6 394.6

16 8.0 26.3 61 44.0 80.2 130 104.7 157.6 355 312.3 399.9

17 8.7 27.6 62 44.9 81.4 135 109.2 163.1 360 317.0 405.2

18 9.4 28.9 63 45.7 82.5 140 113.7 168.6 365 321.7 410.5

19 10.1 30.1 64 46.6 83.7 145 118.2 174.1 370 326.4 415.9

20 10.9 31.4 65 47.4 84.8 150 122.7 179.6 375 331.1 421.2

21 11.6 32.7. 66 48.3 86.0 155 127.2 185.1 380 335.8 426.5

22 12.3 33.9 67 49.2 87.1 160 131.8 190.5 385 340.5 431.8

23 13.1 35.2 68 50.0 88.3 165 136.3 196.0 390 345.2 437

24 13.8 36.4 69 50.9 89.4 170 140.8 201.4 395 349.9 442.3

25 14.6 37.7 70 51.7 90.5 175 145.4 206.9 400 354.6 447.6

26 15.4 38.9 71 52.6 91.7 180 150.0 212.3 405 359.4 452.9

27 16.2 40.1 72 53.5 92.8 185 154.5 217.7 410 364.1 458.2

28 16.9 41.3 73 54.3 93.9 190 159.1 223.2 415 368.8 463.5

29 17.7 42.6 74 55.2 95.1 195 163.7 228.6 420 373.5 468.8

30 18.5 43.8 75 56.1 96.2 200 168.3 234.0 425 378.2 474.1

31 19.3 45.0 76 56.9 97.4 205 172.9 239.4 430 382.9 479.3

32 20.1 46.2 77 57.8 98.5 210 177.5 244.8 435 387.6 484.6

33 20.9 47.4 78 58.7 99.6 215 182.1 250.2 440 392.4 489.9

34 21.7 48.6 79 59.5 100.7 220 186.7 255.6 445 397.1 495.2

35 22.5 49.8 80 60.4 101.9 225 191.3 261.0 450 401.8 500.5

36 23.3 51.0 81 61.3 103.0 230 195.9 266.4 455 406.5 505.7

37 24.1 52.2 82 62.1 104.1 235 200.5 271.8 460 411.3 511.0

38 24.9 53.4 83 63.0 105.3 240 205.1 277.1 465 416.0 516.3

39 25.7 54.6 84 63.9 106.4 245 209.8 282.5 470 420.7 521.5

40 26.5 55.8 85 64.7 107.5 250 214.4 287.9 475 425.5 526.8

41 27.3 56.9 86 65.6 108.6 255 219.0 293.2 480 430.2 532.1

42 28.1 58.1 87 66.5 109.8 260 223.7 298.6 485 434.9 537.3

43 29.0 59.3 88 67.4 110.9 265 228.3 304.0 490 439.7 542.6

44 29.8 60.5 89 68.2 112.0 270 232.9 309.3 495 444.4 547.9

45 30.6 61.7 90 69.1 .113.1 275 237.6 314.7 500 449.1 53.

NUREG-1505A-1 A-10
STATISTICAL TABLES

Table A.6 1000 Random Numbers Uniformly Distributed Between Zero and One

0.382000 0.100681 0.596484 0.899106 0.884610 0.958464 0.014496 0.407422 0.863247 0.138585

0.245033 0.045473 0.032380 0.164129 0.219611 0.017090 0.285043 0.343089 0.553636 0.357372

0.371838 0.355602 0.910306 0.466018 0.426160 0.303903 0.975707 0.806665 0.991241 0.256264

0.951689 0.053438 0.705039 0.816523 0.972503 0.466323 0.300211 0.750206 0.351482 0.775658

0.074343 0.198431 0.064058 0.358348 0.487045 0.511216 0.373455 0.985900 0.040712 0.230720

0.004975 0.926145 0.100314 0.256691 0.775689 0.679647 0.809107 0.724326 0.085055 0.132267

0.756157 0.626514 0.173650 0.404798 0.552324 0.711509 0.555162 0.181158 0.970275 0.686941

0.528794 0.796686 0.805658 0.262215 0.177953 0.866756 0.114841 0.059511 0.761559 0.738395

0.986297 0.925596 0.903867 0.544969 0.500778 0.674978 0.489822 0.145787 0.037965 0.796258

0.671560 0.731681 0.584521 0.152226 0.892178 0.377819 0.200476 0.205786 0.333964 0.325144

0.300211 10.802179 0.696097 0.271493 0.904050 0.039 125 0.709037 0.4537 19 0.5 16648 0.256539

0.291299 0.802149 0.789026 0.675954 0.755333 0.948515 0.619404 0.722068 0.968047 0.368603

0.850429 0.557054 0.873074 0.441053 0.217750 0.859035 0.280343 0.703299 0.707389 0.375835

0.329691 0.085971 0.976867 0.285531 0.534318 0.407392 0.997711 0.894711 0.810816 0.908597

0.574511 0.706076 0.401440 0.111026 0.897366 0.386334 0.095798 0.777642 0.783563 0.66573 1

0.656850 0.258461 0.765191 0.700308 0.858821. 0.002808 0.678610 0.928831 0.042482 0.5181431

0.912137 0.954314 0.594317 0.557665 0.968169 0.483016 0.255623 0.817896 0.496048 0.850642

0.668111 0.926939 0.451765 0.168096 0.061953 0.005158 0.541093 0.617603 0.492904 0.579455

0.601886 0.930052 0.533982 0.132054 0.082278 0.575915 0.829218 0.065676 0.270943 0.699698

0.414197 0.365581 0.435072 0.330088 0.211097 0.740471 0.523453 0.896786 0.603412 0.522782

0.589770 0.584918 0.497024 0.110508 0.593036 0.558916 0.774102 0.230811 0.731193 0.586718

0.545518 0.807337 0.964293 0.095370 0.108554 0.712271 0.883114 0.189917 0.015961 0.184545

0.580035 0.930620 0.162328 0.195105 0.677847 0.548387 0:'294565 0.540819 0.173223 0.185308

0.852351 0.029842 0.250771 0.431745 0.544877 0.967467 0.724082 0.951018 0.570269 0.940641

0.259865 0.863244 0.883755 0.820887 0.041871 0.898312 0.420820 0.128208 0.030122 0.204718

0.682028 0.921291 0.472304 0.478530 0.584979 0.362407 0.823328 0.310556 0.990448 0.771569

0.729270 0.163274 0.808313 0.926084 0.232276 0.381664 0.090182 0.911985 0.852016 0.573138

0.014222 0.693411 0.700644 0.888028 0.168981 0.886013 0.287088 0.231574 0.042299 0.915372

0.167180 0.938932 0.121555 0.341258 0.095462 0.944060 0.511490 0.629475 0.835566 0.974364

0.475723 0.151585 0.284555 0.801660 0.808496 0.695242 0.068636 0.081332 0.442824 0.264687,

0.265145 0.810297 0.200934 0.454268 0.408216 0.935545 0.093844 0.174780 0.433546 0.144322

0.075777 0.015259 0.719321 0.367351 0.660054 0.020234 0.878536 0.025666 0.302469 0. 164434

0.459975 0.553606 0.958678 0.306223 0.213416 0.227607 0.721305 0.900418 0.075594 0.833796

0.943815 0.252419 0.533128 0.203742 0.756645 0.594531 0.518601 0.151555 0.382244 0.765648

0.496139 0.841884 0.155309 0.775964 0.892819 0.121097 0.654134 0.037446 0.531602 0.842860

0.849757 0.541856 0.223121 0.718528 0.678793 0.766930 0.171300 0.724631 0.946379 0.592608

0.350017 0.598224 0.965484 0.008179 0.506699 0.451033 0.838801 0.891781 0.949034 0.034455

0.789300 0.643147 0.771020 0.685781 0.446150 0.951476 0.675222 0.487564 0.491775 0.479080

0.046205 0.671194 0.576434 0.742393 0.432936 0.795495 0.906827 0.97 1435 0.095004 0.732383

0.414594 0.229041 0.770135 0.990143 0.91 1405 0.571306 0.318003 0.405896 0.136082 0.529832

0.397961 0.738731 0.101321 0.698172 0.835597 0.193823 0.644856 0.702261 0.835353 0.410321

0.592151 0.337565 0.695029 0.942595' 0.436842 0.153020 0.178442 0.767235 0.944792 0.263741

0.202582 0.846797 0.588214 0.673330 0.995331 0.281198 0.216346 0.887326 0.380352 0.078341

0.868526 0.098819 0.345164 0.075137 0.136021 0.727836 0.548448 0.040132 0.979186 0.594958

0.923246 0.489242 0.405713 0.511582 '0.032014 0.245125 0.916623 0.566973 0.330088 0.272591

0.125889 0.894131 0935 0.820124 0.460524 0.19516 0.466231 0.399457 0 .346446 0.808039

0.388287 10.634632 0.787317 0.595080 0.051607 0.510758 0.742241 10.4355301 0.049532 10.055055

0..Q208197.1 0.602466 10.270211 10,9804389 0.41-4136 1 .065 082422 0.323283 10.032533 1 .860

A-11 A-li NUREG- 1505
STATISTICAL TABLES

Table A.6 1000 Random Numbers Uniformly Distributed Between Zero and One (continued)

0.539445 0.606800 0.814722 0.019013 0.247505 0.560747 0.556291 0.470656 0.404859 0.929167

0.798975 0.091525 0.317698 0.397290 0.174932 0.421186 0.559557 0.168889 0.800104 0.430860

0.759972 0.225105 0.622150 0.103946 0.988800 0.796197 0.130375 0.932188 0.392621 0.077609

0.332774 0.608875 0.239357 0.721244 0.406964 0.701254 0.027192 0.133396 0.931364 0.582018

0.814356 0.110660 0.102451 0.402142 0.425642 0.845058 0.845119 0.385937 0.692495 0.829066

0.220954 10.229682 0.382153 0.693106 0.042573 0.884823 10.972777 0.505203 0.098636 0.507340

0.930357 0.565050 0.896664 0.854518 0.643971 0.223579 0.329630 0.909757 0.675314 0.759026

0.172857 0.017762 0.479110 0.840846 0.816828 0.310343 0.250435 0.612934 0.142796 0.704489

0.055086 0.927061 0.947508 0.254158 0.899716 0.758171 0.822321 0.136662 0.897244 0.332255

0.894345 0.055300 0.184576 0.998718 0.334727 0.338786 0.748619 0.377758 0.040986 0.519761

0.816065 0.814844 0.399274 0.672750 0.964843 0.672597 0.533036 0.293008 0.611682 0.492172

0.944273 0.998169 0.033418 0.554918 0.953581 0.832606 0.577563 0.875576 0.845332 0.457472

0.218421 0.158483 0.124149 0.005921 0.1 15513 0.747948 0.788903 0.886196 0.329905 0.103885

0.030519 0.312815 0.355022 0.834254 0.260231 0.847896 0.618427 0.815149 0.873440 0.733177

0.610187 0.430342 0.656911 0.294473 0.794671 0.460952 0.244270 0.199683 0.961852 0.656941

0.526933 0.851344 0.173956 0.521256 0.426038 0.695456 0.964629 0.338450 0.932218 0.727439

0.945738 0.978454 0.819025 0.976653 0.478957 0.506973 0.936644 0.250526 0.568072 0.133976

0.491592 0.920164 0.628040 0.531144 0.219459 0.027345 0.287332 0.049684 0.856014 0.496231

0.879940 0.870052 0.813654 0.728660 0.782586 0.614124 0.469192 0.366253 0.254555 0.452071

0.969054 0.499466 0.935179 0.234046 0.958464 0.105197 0.655873 0.678976 0.140599 0.994232

0.723197 0.150853 0.072329 0.049440 0.328562 0.054598 0.101932 0.036836 0.241157 0.381909

0.856441 0.769402 0.440474 0.438948 0.183264 0.906125 0.844386 0.286966 0.954039 0.167058

0.789361 0.926969 0.634968 0.245521 0.418592 0.759667 0.032044 0.198798 0.635029 0.352428

0.115635 0.598682 0.774468 0.043733 0.331706 0.222968 0.656362 0.417005 0.305551 0.569323

0.167089 0.174657 0.344340 0.972533 0.033326 0.444044 0.644856 0.606586 0.940641 0.045930

0.612262 0.152745 0.654134 0.757500 0.196844 0.415021 0.297617 0.100406 0.654347 0.663869

0.446913 0.685781 0.636311 0.989959 0.193823 0.646870 0.017029 0.448683 0.837245 0.190832

0.362743 0.277322 0.357433 0.801294 0.449965 0.965270 0.329722 0.121860 0.738639 0.459212

0.433882 0.053316 0.206793 0.284555 0.215369 0.094302 0.452803 0.863735 0.255135 0.457717

0.471084 0.813227 0.403394 0.993774 0.008576 0.500290 0.234413 0.450667 0.906888 0.640950

0.378277 0.104862 0.337260 0.219703 0.885647 0.291208 0 .334605 0.513443 0.282022 0.865230,

0.536180 0.499435 0.467940 0.127750 0.531877 0.429914 0.120792 0.244270 0.732139 0.853603

0.755211 0.689261 0.159276 0.865780 0.097385 0.747276 0.069582 0.868282 0.930692 0.601917

0.811060 0.620289 0.067843 0.044069 0.824061 0.861599 0.606433 0.330332 0.190893 0.289193

0.123814 0.223182 0.640248 0.168859 0.967864 0.744072. 0.412915 0.362438 0.299387 0.670553

0.895444 0.052797 0.043214 0.308939 0.427168 0.946837 10.403211 0.573534 0.187719 0.238624

0.098300 0.903592 0.570605 0.113498 0.975494 0.049745 0.949370 0.07422 1 0.155797 0.260659

0.242744 0.870907 0.370037 0.901120 0.184881 0.315409 0.639912 0.033387 0.706595 0.762139

0.961303 0.992523 0.334574 0.535844 0.145756 0.486953 0.014252 0.087832 0.476363 0.255989

0.262673 0.162389 0.334208 0.814905 0.057863 0.753716 0.452040 0.319407 0.370617 0.135319

0.951018 0.080447 0.745293 0.743187 0.816828 0.760552 0.835383 0.193213 0.971679 0.783074

0.300851 0.215155 0.598010 0.985809 0.972686 0.783410 0.870754 0.049684 0.691305 0.226569

0.753624 0.734336 0.725364 0.596881 0.016511 0.451338 0.176397 0.080142 0.129551 0.206793

0.276589 0.728629 0.190497 0.444288 0.984863 0.296884 0.541551 0.819300 0.788263 0.462722

0.231910 0.436689 0.114872 0.213904 0.982208 0.582293 0.264016 0.682455 0.807215 0.479843

0.260292 0.207038 0.953032 0.104526 0.537889 0.905576 0.304727 0.584613 0.861049 0.338408

0.921445 0.221931 0.387249 0.944273 0.854060 0.662954 0.279092 0.829157 0.902921 0.251473

0.771294 0.190527 -0.470717 0.0941 19 0.099124 0.157109 0.167852 0.870144 0.194250 0.526231

0.325571 0.916288 0.554521- 0.779473 0.073763 0.618763 0.193701 0.858242 0.139470 0.642750

0.077822 0.480300 0.022187- 0.03 1037 0.953795 0.278542 0.702658 0.703482 0.976897 0.543352

0.540727 0.387371 0.043519 10.849574 0.519364 0.511490 0.235298 0.349712 10.490188 10.093661

0.280221 0.014191 0.577441 10.154180 0.438429 0.600330 0.396466 10.685476 10.7174601 0.630940

NUREG- 1505 A1 A-12
STATISTICAL TABLES

Table A.7a Values of r and k for the Quantile Test When a Is Approximately 0.01 (

Number of Survey Unit Measurements, n

m 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

5 r,k 11,11 13,13 16,16 19,19 22,22 25,25 28,21 r,k

a 0.008 0.015 0.014 0.013 0.013 0.013 0.011 a

10 6,6 7,7 9,9 11,11 13,13 14,14 16,16 18,18 19,19 21,21 23,22 25,2f 26,2( 28,21 30,30

0.005 0.013 0.012 0.011 0.01 0.014 0.013 0.012 0.01 0.014 0.013 0.012 0.015 0.01 0.01

15 3,3 7,6 6,6 7,7 8,8 10,10 11,11 12,12 13,13 15,15 16,1( 17,1 18,11 19,1 21,2 22,2, 23,2 24,2, 26,2 27,2

0.009 0.007 0.008 0.011 0.014 0.00 0.011 0.013 0.014 0.01 0.01 0.01 0.01 0.01 0.010.01 0.01, 0.01.1 0.01 0.01

20 6,4 4,4 5,5 6,6 7,7 8,8 9,9 10,10 11,11 12,12 13,13 14,1, 15,1! 16,1( 17,17 18,1 19,11 19,19 20,20 21,2

0.005 0.008 0.009 0.01 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.01: 0.01 0.01, 0.01: 0.01: 0.01 0.01.1 0.01 0.01

25 4,3 7,5 4,4 5,5 6,6 7,7 8,8 9,9 9,9 10,10 11,11 12,1: 12,12 13,1 14,1z 15,1! 16,1 16,1( 17,17 18,1

0.009 0.012 0.015 0.013 0.011 0.01 0.009 0.009 0.01 0.012 0.011 0.01 0.01 0.01 0.01 0.01 : 0.01 0.01, 0.01, 0.01

30 4,3 3,3 4,4 5,5 6,6 6,6 7,7 8,8 8,8 9,9 10,1 10,1 11,11 1211 12,1: 13,12 14,1 14,1 15,1 15,1

0.006 0.012 0.009 0.007 0.006 0.012 0.01 0.008 0.013 0.011 0.00 0.01 0.011 0.01 0.01: 0.01 0.011 0.01. 0.01 0.01

35 2,2 3,3 4,4 4,4 5,5 6,6 6,6 7,7 7,7 8,8 9,9 9,9 10,1C 10,1( 11,11 11,1 12,1 13,1 13,1' -14,1

0.013 0.008 0.006 0.014 0.01 0.007 0.012 0.009 0.014 0.011 0.00c. 0.01 0.01 0.01' 0.01 0.01 0.01, 0.01 0.01 0.01

40 2,2 3,3 7,5 4,4 5,5 5,5 6,6 6,6 7,7 7,7 8,8 8,8 9,9 9,9 10,1 10,1 11,11 11,11 12,1: 12,1"

0.01 0.006 0.013 0.01 0.006 0.012 0.008 0.013 0.00ý 0.013 0.01 0.014 0.011 0.014 0.011 0.01, 0.01, 0.011 0.01: 0.01

45 2,2 6,4 3,3 4,4 4,4 5,5 5,5 6,6 6,6 7,7 7,7 8,8 8,8 9,9 9,9 10,1 10,1 10,1 11,1 11,I

0.008 0.008 0.013 0.007 0.014 0.008 0.014 0.009 0.013 0.009 0.01 0.00S 0.01 0.00 0.01, 0.00W 0.01: 0.011 0.01 0.01

50 4,3 3,3 4,4 4,4 5,5 5,5 5,5 6,6 6,6 7,7 7,7 8,8 8,8 8,8 9,9 9,9 10,1 10,1 10,1

0.013 0.01 0.005 0.01 0.006 0.01 0.015 0.009 0.013 0.00 0.01 0.00 0.01 0.01 0.01 0.01! 0.01 0.013 0.01

55 4,3 3,3 7,5 4,4 4,4 5,5 5,5 6,6 6,6 6,6 7,7 7,7 8,8 8,8 8,8 9,9 9,9 9,9 10,1

0.01 0.008 0.013 0.008 0.014 0.007 0.011 0.007 0.01 0.01 0.00 0.01: 0.00E 0.01 0.012 0.00 0.01" 0.01' 0.01

60 4,3 3,3 3,3 4,4 4,4 5,5 5,5 5,5 6,6 6,6 6,6 7,7 7,7 7,7 8,8 8,8 8,8 9,9 9,9

0.0080.007 0.014 0.006 0.011 0.006 0.00 0.013 0.007 0.01 0.014 0.00. 0.01 0.01, 0.01 0.01: 0.01! 0.01 0.01ý

65 4,3 3,3 3,3 6,5 4,4 4,4 5,5 5,5 5,5 6,6 6,6 6,6 7,7 7,7 7,7 8,8 8,8 8,8 9,9

0.007 0.006 0.012 0.006 0.009 0.013 0.007 0.01 0.014 0.00 0.011 0.014 0.00S, 0.01 0.01 0.00 0.01, 0.01 0.01

70 2,2 6,4 3,3 7,5 4,4 4,4 5,5 5,5 5,5 5,5 6,6 6,6 6,6 7,7 7,7 7,7 8,8 8,8 8,8

0.014 0.008 0.01 0.013 0.007 0.011 0.005 0.008 0.011 0.01! 0.00E 0.011 0.01 0.00S 0.01 0.01 0.00,. 0.01 0.01.

75 2,2 4,3 3,3 3,3 4,4 4,4 4,4 5,5 5,5 5,5 6,6 6,6 6,6 6,6 7,7 7,7 7,7 8,8. 8,8

0.013 0.014 0.008 0.014 0.006 0.00 0.013 0.00 0.009 0.012 0.00 0.00S 0.01 0.01 0.00 0.01 0.01 0.00 0.0

80 2,2 4,3 3,3 3,3 6,5 4,4 4,4 5,5 5,5 5,5 5,5 6,6 6,6 6,6 6,6 7,7 7,7 7,7 7,7

0.011 0.012 0.007 0.012 0.006 0.008 0.011 0.005 0.00ý 0.01 0.013 0.00 0.00S 0.01 0.01, 0.00 0.01 0.01, 0.01

85 2,2 4,3 3,3 3,3 7,5 4,4 4,4 4,4 5,5 5,5 5,5 5,5 6,6 6,6 6,6 6,6 7,7 7,7 7,7

0.01 0.01 0.006 0.011 0.013 0.006 0.009 0.013 0.00 0.00 0.011 0.01 0.00- 0.01 0.01' 0.014 0.00 0.01 0.01

90 4,3 3,3 3,3 3,3 4,4 4,4 4,4 5,5 5,5 5,5 5,5 5,5 6,6 6,6 6,6 6,6 7,7 7,7

0.009 0.005 0.009 0.014 0.005 0.00' 0.011 0.005 0.00 0.00€. 0.01 0.01 0.00 0.01 0.01 0.01, 0.00 0.01

95 4,3 6,4 3,3 3,3 6,5 4,4 4,4 4,4 5,5 5,5 5,5 5,5 6,6 6,6 6,6 6,6 6,6 7,7

0.008 0.008 0.008 0.013 0.005 0.007 0.01 0.013 0.006 0.00a 0.01 0.01" 0.00j 0.001 0.01 0.011 0.01' 0.00

100 rk 4,3 4,3 3,3 3,3 7,5 4,4 4,4 4,4 4,4 5,5 5,5 5,5 5,5 6,6 6,6 6,6 6,6 6,6

___ a 0.007 0.014 0.007 0.011 0.01 0.006 0.00 0.011 0.01 0.00 0.00 0.010.01 0.00 0.00 0.01 0.01 0.01

") Values of the parametersr and k needed for the Quantile test calculated by Gilbert and Simpson (PNL-7409,

1992) for certain combinations of m (the number of measurements in the reference area) and n (the number of

measurements in the survey unit) The value of a listed is that obtained from simulation studies.

A-13 NUREG-1505
STATISTICAL TABLES

Table A.7b Values of r and k for the Quantile Test When a Is Approximately 0.025

Number of Survey Unit Measurements, n

m 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

5 rk 9,9 12,12 15,15 17,17 20,20 22,22 25,25 r,k

0.03 0.024 0.021 0.026 0.024 0.028 0.025

10 7,6 6,6 8,8 9,9 11,11 12,12 14,1 17,17 18,18 20,2( 21,21 23,22 24,2ý 26,2( 27,27

0.029 0.028 0.022 0.029 0.024 0.029 0.02! 0.025 0.029 0.02f 0.02S 0.02( 0.02! 0.02( 0.02!

15 11,5 6,5 5,5 6,6 7,7 8,8 9,9 10,1 11,11 13,13 15,15 14,11 16,1( 17,1", 18,11 19,1! 21,2 21,2 22,2 23,2

0.03 0.023 0.021 0.024 0.026 0.02 0.028 0.02 0.03 0.02; 0.02 0.021 0.024 0.021 0.02! 0.020 0.02 0.02 0.02 0.02

20 8,4 3,3 4,4 5,5 6,6 7,7 12,11 13,11 9,9 10,1 11,11 12,1- 13,1. 13,11 14,11 15,1 16 1 17,1 17,17 18,1

0.023 0.03 0.026 0.024 0.022 0.02 0.021 0.02 0.028 0.02f 0.02, 0.02. 0.02: 0.02 0.02' 0.026 0.02 0.02, 0.02 0.0

25 2,2 8,5 6,5 7,6 5,5 6,6 10,9 7,7 8,8 13,12 9,9 10,1 11,1 11,1 12,1, 13,1 13,1. 14,1- 15,1 15,1

0.023 0.027 0.021 6.023 0.025 0.02 0.026 0.027 0.023 0.027 0.027 0.024 0.02: 0.021 0.02! 0.82 0.624 0.02 0.02 0.01

30 6,3 6,4 9,6 4,4 7,6 5,5 9,8 6,6 7,7 12,11 8,8 9,9 9,9 10,1( 10,1( 11,1 11,1 12,1 13,1 13,1

0.026 0.026 0.026 0.021 0.029 0.02J 0.024 0.029, 0.021 0.021 0.025 0.021 0.021 0.02: 0.02! 0.02! 0.03 0.02 2 0.0

35 7,3 4,3 3,3 6,5 4,4 10,8 5,5 9,8 6,6 7,7 7,7 8,8 8,8 9,9 9,9 10,1C 10,1 11,1 11,11 12,1

0.03 0.03 0.023 0.02 0.026 0.02 0.027 0.024 0.027 0.02 0.02" 0.021 0.024 0.021 0.02" 0.02 0.027 0.02 0.027 0.02

40 3,2 4,3 8,5 11,7 6,5 4,4 10,8 5,5 9,8 6,6 10,9 7,7 12,1 8,8 8,8 9,9 9,9 10,1 10,1 I1,1

0.029 0.022 0.028 0.025 0.028 0.03 0.026 0.027 0.023 0.02f 0.02E 0.024 0.02 0.02 0.02( 0.02 0.02 0.02 0.02 0.02

45 3,2 8,4 6,4 3,3 8,6 4,4 7,6 5,5 5,5 9,8 6,6 10,9 7,7 7,7 8,8 8,8 8,8 9,9 9,9 10,1

0.023 0.029 0.036 0.020 0.021 0.023 0.025 0.02 0.024 0.023 0.024 0.02 0.02: 0.02' 0.02 0.02 0.03 0.02 0.02 0.0

50 2,2 6,4 3,3 11,7 6,5 4,4 7,6 5,5 5,5 9,8 6,6 6,6 7,7 7,7 12,1 8,8 8,8 13,1 9,9

0.025 0.022 0.021 0.077 6.02 0.026 0.028 0.021 0.02E 0.02A 0.02- 0.02S 0.02 0.02! 0.02 0.02: 0.024 0.02 0.02

55 2,2 4,3 8,5 3,3 8,6 4,4 4,4 10,8 5,5 5,5 9,8 6,6 6,6 10,9 7,7 7,7 12,11 8,8 8,8

0.022 0.029 0.028 0.028 0.021 0.02 0.029 0.021 0.022 0.02E 0.02 0.09A 0.02 0.02 0.02. 0.027 0.02 0.02 0.02

60 14,5 4,3 8,5 3,3 11,7 6,5 4,4 7,6 10,8 5,5 5,5 9,8 6,6 6,6 10,9 7,7 7,7 7,7 8,8

0.022 0.024 0.021 0.023 0.029 0.024 0.02A 0.02A 0.024 0.023 0.02S 0.02, 0.02, 0.02" 0.027 0.021 0.02 0.03 0.02

65 6,3 7,4 6,4 10,6 3,3 8,6 6,5 4,4 7,6 10,8 5,5 5,5 9,8 6,6 6,6 10,9 7,7 7,7 7,7

0.028 0.021 0.025 0.025 0.02 0.021 0.029 0.026 0.02( 0.02f 0.022 0.02S 0.02 0.02. 0.02( 0.02 0.024 0.024 0.0

70 6,3 2,2 6,4 8,5 3,3 13,8 6,5 4,4 4,4 7,6 10,8 5,5 5,5 9,8 6,6 6,6 6,6 10,9 7,7

0.024 0.029 0.021 0.028 0.025 0.026 0.023 0.022 0.024 0.02E 0.02 0.02, 0.02J 0.02 0.021 0.02! 0.024 0.03 0.02

75 11,4 2,2 4,3 8,5 3,3 9,6 8,6 6,5 4,4 7,6 7,6 10,8 5,5 5,5 9,8 6,6 6,6 6,6 10,9

0.022 0.026 0.028 0.022 0.022 0.02 0.021 0.02 0.024 0.023 0.03 0.02S 0.02ý 0.02S 0.02 0.02 0.02, 0.021 0.02

80 7,3 2,2 4,3 6,4 10,6 3,3 13,8 6,5 4,4 4,4 7,6 10,8 5,5 5,5 5,5 9,8 6,6 6,6 6,6

0.028 0.024 0.024 0.028 0.024 0.02 0.027 0.023 0.02 0.02C 0.024 0.02 0.07 0.02! 0.02S 0.02 0.02 0.02ý 0.02

85 3,2 2,2 4,3 6,4 8,5 3,3 9,6 8,6 6,5 4,4 4,4 7,6 10,8 5,5 5,5 5,5 9,8 6,6 6,6

0.029 0.021 0.021 0.023 0.028 0.023 0.03 0.02 0.026 0.02 0.02E 0.02 0.02 0.021 0.02! 0.02 0.02 0.02 0.02

90 5,3 11,5 9,5 8,5 3,3 3,3 13,8 6,5 6,5 4,4 4,4 7,6 10,8 5,5 5,5 5,5 9,8 9,8

__0.02 0.027 0.023 0.023 0.021 0.028 0.028 0.022 0.029 0.02• 0.02 0.02[ 0.02( 0.02: 0.02• 0.03 0.02, 0.02

95 10,4 2,2 4,3 6,4 10,6 3,3 11,7 8,6 6,5 4,4 4,4 7,6 7,6 10,8 5,5 5,5 5,5 9,8

__0.029 0.029 0.028 0.029 0.023 0.025 0.02( 0.02 0.025 0.02' 0.02( 0.024 0.02• 0.02' 0.02, 0.021 0.03 0.02

100 r~k 6,3 2,2 4,3 6,4 8,5 3,3 3,3 13,8 6,5 6,5 4,4 4,4 7,6 10,8 10,8 5,5 5,5 5,5

___ 0.029 0.027 0.025 0.025 0.028 0.022 0.02€ 0.02• 0.02• 0.021 0.02! 0.02• 0.02: 0.02' 0.021 0.02: 0.02e 0.0

NUREG-1505 A-14
STATISTICAL TABLES

Table A.7c Values of r and k for the Quantile Test When ca Is Approximately 0.05

Number of Survey Unit Measurements, n

m 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

5 rk 8,8 10,10 13 13 15 15 17,17 19,19 21,21 r,k

a 0.051 0.057 0.043 0.048 0.051 0.054 0.05( a

10 4,4 5,5 14,12 8,8 9,9 10,10 12,12 13,13 14,14 15,15 17,1' 18,1E 19,1S 20,2( 21,2 23,2.

0.043 0.057 0.045 0.046 0.052 0.058 0.046 0.05 0.054 0.057 0.04ý 0.052 0.05f 0.05, 0.05S 0.05i

15 2,2 3,3 4,4 5,5 6,6 7,7 8,8 9,9 9,9 10,10 11,11 12,1, 13,11 14,11 15,1f 16,11 16,1i 17,1' 18,11 19,1

0.053 0.052 0.05 0.048 0.046 0.045 0.052 0.043 0.06 0.057 0.055 0.05ý 0.05' 0.051 0.05 0.04c 0.054 0.05" 0.054 0.05

20 9,4 8,5 6,5 4,4 5,5 9,8 6,6 7,7 8,8 8,8 9,9 10,1 10,1( 11,11 12,1, 12,12 13,1 14,1' 14,1, 15,1

0.04 0.056 0.04 0.053 0.043 0.052 0.056 48 0.043 0.05 0.051 0.041 0.05' 0.052 0.04 0.05, 0.05 0.04! 0.05" 0.05

25 6,3 6,4 3,3 6,5 4,4 5,5 5,5 6,6 11,1 7,7 8,8 8,8 9,9 9,9 10,1( 11,1 11,11 11,1 12,1 12,1

0.04110.043 0.046 0.052 0.055 0.041 0.059 0.046 0.042 0.05 0.042 0.053 0.04f 0.05 0.0 0.041 0.05 0.05 0.05: 0.0

30 3,2 2,2 10,6 3,3 11,8 4,4 8,7 5,5 6,6 6,6 7,7 7,7 8,8 8,8 9,9 9,9 9,9 10,1( 10,19 11,1

0.047 0.058 0.052 0.058 0.045 0.05 0.044 0.054 0.04 0.053 0.041 0.05 0.042 0.051 0.04 0.05 0.05 0.04 0.05' 0.04

35 8,3 2,2 6,4 3,3 6,5 4,4 4,4 8,7 5,5 9,8 6,6 6,6 7,7 7,7 8,8 8,8 8,8 9,9 9,9 10,1

0.046 0.045 0.058 0.043 0.041 0.04 0.057 0.043 0.051 0.052 0.047 0.051 0.042 0.051 0.041 0.04 0.051 0.045 0.05: 0.04

40 4,2 5,3 4,3 10,6 3,3 6,5 4,4 4,4 8,7 5,5 9,8 6,6 6,6 11,1 7,7 7,7 8,8 8,8 8,8 9,9

0.055 0.048 0.057 0.059 0.053 0.048 0.043 0.058 0.042 0.048 0.047 0.042 0.051 0.04, 0.04! 0.05ý 0.04 0.04 0.0 0.04

45 4,2 9.4 2,2 8,5 3,3 8,6 6,5 4,4 4,4 8,7 5,5 5,5 9,8 6,6 6,6 11,1C 7,7 7,7 8,8 8,8

0.045 0.047 0.059 0.052 0.042 0.041 0.054 0.045 0.058 0.041 0.04( 0.05 0.05( 0.041 0.05 0.04( 0.041 0.05 0.04 0.04

50 6,3 2,2 6,4 12,7 3,3 8,6 6,5 4,4 4,4 8,7 5,5 5,5 9,8 6,6 6,6 6,6 7,7 7,7 7,7

0.051 0.05 0.051 0.05 0.049 0.04 0.059 0.047 0.059 0.041 0.04f 0.05z 0.05 0.04' 0.05 0.051 0.041 0.041 0.05

55 3,2 2,2 4,3 8,5 3,3 5,4 6,5 9,7 4,4 4,4 8,7 5,5 5,5 9,8 6,6 6,6 6,6 11,1 7,7

10.059 0.043 0.056&0.058 0.041 0.041 0.046 0.04 0.04 0.05 0.04 0.04. 0.052 0.04E 0.04 0.041 0.05ý 0.04 0.04

60 3,2 5,3 4,3 6,4 3,3 3,3 8,6 6,5 9,7 4,4 4,4 13,10 5,5 5,5 5,5 9,8 6,6 6,6 6,6

0.052 0.052 0.046 0.059 0.035 0.047 0.043 51 0.046 0.049 0.055 0.052 0.04 0.05 0.051 0.05, 0.04, 0.05 0.05

65 .3,2 5,3 2,2 6,4 10,6 3,3 3,3 6,5 6,5 4,4 4,4 4,4 13,19 5,5 5,5 5,5 9,8 6,6 6,6

0.045 0.043 0.053 0.048 0.05 0.04 0.052 0.041 0.055 0.042 0.05 0.06 0.05 0.041 0.04E 0.05! 0.05, 0.041 0.04

70 8,3 9,4 2,2 4,3 8,5 5,4 3,3 3,3 6,5 6,5 4,4 4,4 4,4 13,10 5,5 5,5 5,5 9,8 9,8

0.057 0.048 0.047 0.055 0.05 0.041 0.046 0.057 0.04 0.058 0.04 0.051 0.06 0.05q 0.04. 0.04" 0.05 0.04 0.05

75 8,3 6,3 2,2 4,3 6,4 10,6 3,3 3,3 8,6 6,5 9,7 4,4 4,4 5,5 13,1( 8,7 5,5 5,5 5,5

0.049 0.056 0.043 0.047 0.054 0.053 0.04 0.051 0.044 0.049 0.041 0.044 0.052 0.06 0.05 0.04. 0.04( 0.05 0.05

80 4,2 6,3 5,3 2,2 6,4 8,5 5,4 3,3 3,3 6,5 6,5 9,7 4,4 4,4 7,6 13,1C 87 5,5 5,5

.0.059 0.048 0.053 0.055 0.046 0.055 0.041 0.04 0.055 0.041 0.05- 0.04. 0.04f 0.052 0.05 0.05 0.04 0.04 0.05

85 4,2 3,2 5,3 2,2 4,3 4,3 10,6 5,4 3,3 3,3 6,5 6,5 9,7 4,4 4,4 7,6 10,8 8,7 5,5

0.054 0.058 0.047 0.05 0.054 0.048 0.056 0.049 0.04 0.05 0.044 0.05f 0.04( 0.04( 0.05' 0.05 0.06 0.04 0.04

90 3,2 5,3 2,2 6,4 6,4 8,5 5,4 3,3 3,3 8,6 6,5 6,5 4,4 4,4 4,4 7,6 10,8 8,7

0.053 0.041 0.046 0.059 0.051 0.058 0.042 0.044 0.053 0.045 0.04 0.05E 0.04 0.04, 0.05ý 0.05! 0.06 0.04

95 3,2 9,4 2,2 2,2 4,3 8,5 10,6 5,4 3,3 3,3 6,5 6,5 9,7 4,4 4,4 4,4 7,6 10,

0.048 0.048 0.042 0.056 0.059 0.05 0.058 0.048 0.048 0.056 0.041 0.05 0.04( 0.04d 0.04J 0.05, 0.59 0.05

100 r,k 3,2 6,3 5,3 2,2 4,3 6,4 10,6 5,4 3,3 3,3 3,3 6,5 6,5 9,7 4,4 4,4 4,4 7,6

0.044 0.057 0.054 0.052 0.053 0.05 0.04 0.04 0.04 0.05 0.05 0.05 0.04 0.04 0.04 0.05 0.05

A-15 NUREG-1505
STATISTICAL TABLES

Table A.7d Values of r and k for the Quantile Test When a Is Approximately 0.10

Number of Survey Unit Measurements, n

m 5 10 15 20 25 30 35 40 45 50 55 60. 65 70 75 80 85 90 95 100

5 rk 7,7 8,8 10,10 12,12 14,14 15,1 17,17 r,k

a 0.083 0.116 0.109 0.104 0.1 0.11 0.11 a

10 3,3 4,4 5,5 6,6 7,7 8,8 9,9 10,1( 11,11 12,12 13,1 14,14 15,1f 16,1( 1712 18,1

0.105 0.108 0.109 0.109 0.109 0.109 0.10 0.10 0.10S 0.10S 0.10S 0.10S 0.10S 0.10S 0.10! 0.10(

15 9,4 10,6 3,3 4,4 5,5 5,5 6,6 7,7 7,7 8,8 9,9 9,9 10,10 11,1 11,11 12,1: 13,1" 13,1 14,1 15,1

0.098 0.106 0.112 0.093 0.081 0.11 0.102 0.092 0.11 0.10( 0.09 0.11 0.10! 0.101 0.11 0.11 0.1 0.11 0.11 0.1C

20 3,2 2,2 5,4 3,3 4,4 4,4 5,5 10,9 6,6 7,7 7,7 8,8 8,8 9,9 9,9 10,1C 10,1 11,11 11,1 12,1

0.091 0.103 0.093 0.115 0.085 0.119 0.093 0.084 0.09 0.083 0.10: 0.081 0.10 0.09: 0.10' 0.09 0.10 0.09 0.11 0.1

25 4,2 7,4 8,5 3,3 3,3 4,4 4,4 8,7 5,5 10,9 6,6 6,6 7,7 7,7 8,8 8,8 8,8 9,9 9,9 10,1

0.119 0.084 0.112 0.08 0.117 0.08 0.107 0.10E 0.101 0.08E 0.09 0.114 0.093 0.104 0.09 0.1 0.11' 0.1 0. fi1: 0.09

30 4,2. 5,3 2,2 14,8 3,3 3,3 9,7 4,4 8,7 5,5 5,5 6,6 6,6 6,6 7,7 7,7 7,7 8,8 8,8 8,8

0.089 0.089 0.106 0.111 0.088 0.11 0.116 0.1 0.093 0.08E 0.10( 0.08 0.09 0.11 0.08' 0.1 0.11: 0.09 0.10 0.11

35 5,2 3,2 2,2 6,4 5,4 3,3 3,3 9,7 4,4 4,4 8,7 5,5 5,5 6,6 6,6 6,6 6,6 7,7 7,7 7,7

0.109 0.119 0.086 0.12 0.091 0.093 0.12 0.11 0.094 0.114 0.10' 0.09 0.11 0.08 0.09z 0.10 0.12 0.09, 0.10 0.11

40 5,2 3,2 5,3 2,2 12,7 5,4 3,3 6,5 9,7 4,4 4,4 8,7 5,5 5,5 5,5 6,6 6,6 6,6 6,6 7,7

0.087 0.098 0.119 0.107 0.109 0.102 0.097 0.10C 0.10ý 0.09 0.10• 0.09- 0.08ý 0.09S 0.11, 0.08 0.09 0.10 0.11 0.0

45 6,2 3,2 5,3 2,2 6,4 7,5 5,4 3,3 6,5 9,7 4,4 4,4 4,4 8,7 5,5 5,5 5,5 6,6 6,6 6,6

0.103 0.082 0.094 0.091 0.115 0.086 0.112 0.1 0.101 0.101 0.08 0.10: 0.11, 0.10, 0.09 0.10: 0.115 0.08 0.09 0.1

50 7,3 9,4 7,4 2,2 10,6 5,4 3,3 3,3 6,5 9,7 4,4 4,4 4,4 8,7 5,5 5,5 5,5 5,5 6,6

0.083 0.115 0.097 0.108 0.112 0.09 0.084 0.103 0.102 0.10f 0.08' 0.09 0.11, 0.09S 0.08 0.95 0.10 0.11 0.08

55 4,2 3,2 5,3 2,2 6,4 14,8 5,4 3,3 3,3 6,5 9,7 4,4 4,4 4,4 4,4 8,7 5,5 5,5 5,5

0.109 0.114 0.114 0.095 0.112 0.111 0.098 0.088 0.104 0.103 0.10z 0.08: 0.091 0.10" 0.12 0.10 0.08 0.09 0.10

60 4,2 3,2 5,3 2,2 2,2 8,5 5,4 5,4 3,3 3,3 6,5 9,7 4,4 4,4 4,4 4,4 8,7 5,5 5,5

0.095 0.1 0.097 0.084 0.109 0.119 0.081 0.105 0.091 0.10( 0.10. 0.10: 0.08 0.09', 0.10' 0.11. 0.1 0.08 0.09

65 4,2 3,2 5,3 7,4 2,2 6,4 12,7 5,4 5,4 3,3 3,3 6,5 9,7 7,6 4,4 4,4 4,4 8,7 8,7

0.084 0.089 0.082 0.090 0.097 0.11 0.113 0.089 0.111 0.092 0.10 0.10A 0.10 0.08, 0.09 0.1 0.11 0.09, 0.10

70 5,2 7,3 9,4 5,3 2,2 2,2 8,5 7,5 5,4 3,3 3,3 3,3 6,5 9,7 7,6 4,4 4,4 4,4 4,4

0.115 0.101 0.106 0.112 0.088 0.109 0.114 0.081 0.09( 0.08! 0.09e 0.105 0.10 0.19 0.08, 0.081 0.09 0.10 0.11

75 5,2 7,3 3,2 5,3 7,4 2,2 2,2 10,6 5,4 5,4 3,3 3,3 3,3 6,5 9,7 7,6 4,4 4,4 4,4

103 0.088 0.111 0.098 0.101 0.09 0.119 0.11 0.083 0.10 0.085 0.094 0.11 0.10! 0.1 0.081 0.086 0.09 0.10

80 5,2 4,2 3,2 5,3 7,4 2,2 2,2 8,5 14,8 5,4 5,4 3,3 3,3 3,3 6,5 6,5 9,7 4,4 4,4

0.093 0.116 0.101 0.086 0.086 0.09! 0.109 0.111 0.11 0.08 0.10j 0.081 0.09, 0.11 0.10 0.12 0.11 0.08 0.09

85 5,2 4,2 3,2 9,4 5,3 2,2 2,2 2,2 10,6 7,5 5,4 5,4 3,3 3,3 3,3 6,5 6,5 9,7 4,4

0.084 0.106 0.092 117 0.111 0.083 0.101 0.11 0.112 0.08 0.09A 0.11 0.09 0.10 0.11 0.10 0.11 0.11 0.08

90 4,2 3,2 3,2 5,3 7,4 2,2 2,2 8,5 12,7 5,4 5,4 3,3 3,3 3,3 3,3 6,5 6,5 9,7

0.097 0.085 0.119 0.099 0.095 0.09 0.109 3 0.108 0.11 0.08 4 0.09 2 0.08 0.09 0.10 0.11 0.10 . 0.11 1 0.11

95 4,2 7,3 3,2 5,3 7,4 2,2 2,2 2,2 10,6 14,8 5,4 5,4 3,3 3,3 3,3 3,3 6,5 6,5

0.089 100 0.11 0.089 0.084 0.086 0.102 0.11) 0.08 0.11 0.08E 0.10 0.08 0.09 0.10" 0.111 0.106 0.11

100 rk 4,2 7,3 3,2 5,3 5,3 2,2 2,2 2,2 6,4 12,7 7,5 5,4 5,4 3,3 3,3 3,3 3,3 6,5

- 02 I I 0.080 . 0.1 0.11 0.1

NUREG-1505 A-16
NRC FORM 335 U.S. NUCLEAR REGULATORY COMMISSION 1. REPORT NUMBER

(2-89) (Assigned by NRC, Add Vol., Supp.. Rev.,

NRCM 1102. and Addendum Numbers, it any.)

3201.3202 BIBLIOGRAPHIC DATA SHEET

(See Instructions on the reverse)

2. TITLE AND SUBTITLE NUREG-1505 Rev. 1

A Nonparametric Statistical Methodology for the Design and Analysis of Final Status

Decommissioning Surveys 3. DATE REPORT PUBLISHED

Interim Report for Use and Comment MONTH YEAR

June 1998

4. FIN OR GRANT NUMBER

5. AUTHOR(S) 6. TYPE OF REPORT

C.V. Gogolak*, Environmental Measurements Laboratory G.E. Powers, U.S. Nuclear Regulatory Commission Technical

A.M. Huffert, U.S. Nuclear Regulatory Commission 7. PERIOD COVERED (Inclusive Dates)

9/1/95 to 4/30/98

8. PERFORMING ORGANIZATION - NAME AND ADDRESS (If NRC, provide Division, Office or Region, U.S. Nuclear Regulatory Commission, and mailing address: if contractor,

provide name and mailing address.)

*U.S. Department of Energy Division of Regulatory Applications

Environmental Measurements Laboratory Office of Nuclear Regulatory Research

201 Varick Street, 5th Floor U.S. Nuclear Regulatory Commission

New York, NY 10014 Washington, DC 20555-0001

9. SPONSORING ORGANIZATION - NAME AND ADDRESS (If NRC, type "Same as above' If contractor, provide NRC Division, Office or Region, U.S. Nuclear Regulatory Commission,

and mailing address.)

Division of Regulatory Applications

Office of Nuclear Regulatory Research

U.S. Nuclear Regulatory Commission

Washington, DC 20555-0001

10. SUPPLEMENTARY NOTES

George E. Powers, NRC Project Manager

11. ABSTRACT (200 words or less)

.This report describes a nonparametric statistical methodology for the design and analysis of final status decommissioning

surveys in support of the final rulemaking on Radiological Criteria for License Termination published by the Nuclear Regulatory

Commission in the Federal Register on July 21, 1997. The techniques described are expected to be applicable to a broad

range of circumstances, but do not preclude the use of alternative methods as particular situations may warrant.

Nonparametric statistical methods for testing compliance with decommissioning criteria are provided both for the case in which

the radionuclides of concern occur in background and also for the case in which they do not occur in background. The tests

described are the Sign test, the Wilcoxon Rank Sum test, and a Quantile test. These tests are performed in conjunction with an

Elevated Measurement Comparison to provide confidence that the radiological criteria specified for license termination are

met. The Data Quality Objectives process is used for the planning of final site surveys. This includes methods for determining

the number of samples needed to obtain statistically valid comparisons with decommissioning criteria and the methods for

conducting the statistical tests with the resulting sample data.

12. KEY WORDS/DESCRIPTORS (List words or phrases that will assist researchers in locating the report.) 13. AVAILABILITY STATEMENT

Statistics, Decommissioning, Survey(s), Quality Assurance, Optimization, Nonparametric Statistics, unlimited

Sign test, Wilcoxon Rank Sum test, Quantile test, Elevated Measurement Comparison 14. SECURITY CLASSIFICATION

(This Page)

unclassified

(This Report)

unclassified

15. NUMBER OF PAGES

16. PRICE

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