Advanced Mass Calibration and 

Measurement Assurance Program for 

State Calibration Laboratories

(2005 Ed)

Fraley, Ken L. 

Harris, Georgia L.

NISTIR 5672

 

NISTIR 5672

Advanced Mass Calibration and 

Measurement Assurance Program for 

State Calibration Laboratories

(2005 Ed)

Fraley, Ken L.

Oklahoma Bureau of Standards

 Harris, Georgia L.

Weights and Measures Division

Technology Services

March 2005

U.S. DEPARTMENT OF COMMERCE

Carlos M. Gutierrez, Secretary

TECHNOLOGY ADMINISTRATION

Phillip J. Bond, Under Secretary of Commerce for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

Hratch G. Semerjian, Acting Director

March 16, 2005 

Page iii 

Preface

This publication was originally written by Ken Fraley, metrologist with the State of Oklahoma, and 

Georgia Harris, physical scientist with the NIST Weights and Measures Division. Ideas from final 

users regarding publication content were sought at the 1994 National Conference on Weights and 

Measures meeting held in San Diego, CA. The publication was written to provide guidance for 

calibration laboratories in their desire to provide improved precision mass calibrations and for the 

NIST Weights and Measures Division to ensure uniform evaluation of laboratories seeking to make 

mass measurements and be accredited at the advanced level of mass calibration. 

Since the first edition in 1995, numerous practical questions have been raised and additional input 

has been sought from mass calibration experts. This edition seeks to enhance the original 

publication and provide additional guidance. Copies of Standard Operating Procedures 5 and 28 (3-

1 Weighing Designs and Advanced Weighing Designs respectively) are included as an appendix. 

Additional weighing designs, equations for between-time standard deviations, and updates for 

uncertainty analysis are included as well. 

About the Authors

Ken Fraley is a metrologist with the State of Oklahoma. He has over 10 years of experience in 

making precision mass measurements using weighing designs as described in this publication. He 

has carried out extensive experimentation and implemented measurement assurance programs to 

ensure that measurements made in the Oklahoma laboratory are consistent and uniform with those at 

the national level. As background experience, Ken developed the first draft of this document based 

on discussions with NIST and early users of weighing designs in the State laboratories, and provided 

a critical review of Advanced Laboratory Auditing Program (LAP) problems submitted after the first 

Advanced Mass Metrology seminar in 1993. He has also coordinated and analyzed several 

interlaboratory comparisons among laboratories working at the advanced level described in this 

publication. He has analyzed data, prepared preliminary and final analyses, and presented results of 

many interlaboratory comparisons conducted at basic, intermediate and advanced levels. In this 

edition, Ken has clarified ideas that now have had over 10 years of refinement, has introduced 

extensive spreadsheet usage, and provided additional graphic content. 

Georgia Harris is the Group Leader of the Laboratory Metrology Group in the NIST Weights and 

Measures Division. She provided direction and encouragement to Ken in developing the ideas and 

provided editorial support for the initial publication. In this revision, she has provided updated 

copies of SOP 5, SOP 28, ideas and content for spreadsheet analysis, and enhancements based on 

answering many technical questions from laboratories working at this level, and from reviewing 

numerous annual submissions for laboratories seeking formal Recognition at this level. The 

Weights and Measures Division is responsible for providing technical support and guidance to the 

State legal metrology laboratories to ensure uniformity in the legal metrology measurement 

infrastructure; this publication is intended to provide support and guidance not only for State weights 
March 16, 2005 

Page iv 

and measures laboratories, but also for other calibration laboratories seeking to implement advanced 

weighing designs. 

The authors wish to express their thanks to M. Carroll Croarkin (NIST) for providing between-time 

standard deviation formulae and assistance regarding updating mass calibration uncertainties to meet 

the ISO Guide to the Expression of Uncertainty in Measurement, to Jerry L. Everhart (JTI Systems, 

formerly with EG&G Mound) for providing guidance in Process Measurement Assurance Programs, 

and to all of the metrologists who regularly participate in WMD training and regional meetings for 

their questions, comments, and desire to make precision mass measurements to the best of their 

capabilities. For the 2005 update, the authors greatly appreciate the efforts of Hung-kung Liu of the 

NIST Statistical Engineering Division in supplying missing factors for the between-time standard 

deviation equations and for his critical review of the document. 
March 16, 2005 

Page v 

Contents 

Preface...........................................................................................................................................iii 

Program Objective........................................................................................................................ 7 

Program Prerequisites.................................................................................................................. 8 

Training............................................................................................................................... 8 

Facilities.............................................................................................................................. 8 

Equipment........................................................................................................................... 9 

Standards........................................................................................................................... 10 

Advanced LAP Problems........................................................................................................... 11 

LAP Problem 1 ................................................................................................................. 12 

LAP Problem 2 ................................................................................................................. 12 

LAP Problem 3 ................................................................................................................. 12 

Follow up .......................................................................................................................... 12 

Establishing Measurement Controls......................................................................................... 15 

Process Evaluation............................................................................................................ 15 

Data Input ......................................................................................................................... 17 

Handling the Output.......................................................................................................... 18

Reviewing Mass Code Report.................................................................................................... 19 

Graphs and Control Charts....................................................................................................... 20 

Critical Graphs.................................................................................................................. 20 

Optional Graphs................................................................................................................ 21 

Proficiency Tests......................................................................................................................... 21 

Evaluation Criteria for Proficiency Tests................................................................................. 22 

Verification of Laboratory Values.................................................................................... 23 

Verification of the Laboratory Precision .......................................................................... 23 

File Management......................................................................................................................... 23 

Software Management................................................................................................................ 24 

Distribution ....................................................................................................................... 24 

Licensing and Software Quality Assurance...................................................................... 24 

Updating............................................................................................................................ 24 

Approved Weighing Designs............................................................................................ 24 

Documentation of Standard Operating Procedures................................................................ 25 

Traceability and Calibration Intervals..................................................................................... 25 

Formulae and Calculations........................................................................................................ 26 
March 16, 2005 

Page vi 
March 16, 2005 

Page 7 

Advanced Mass Calibration and Measurement Assurance Program 

 for State Calibration Laboratories

Program Objective

This publication provides guidelines for evaluating data from advanced mass calibrations and for 

establishing measurement assurance programs in precision mass calibration laboratories. The NIST 

Weights and Measures Division (WMD) will use these guidelines when evaluating advanced mass 

calibration data for State laboratories that request technical support, Recognition, and/or NVLAP 

accreditation. 

Advanced mass calibrations use weighing designs, such as those found in NBS Handbook 1451

(SOP 4, 5), NISTIR 6969, Selected Publications2, NBS Technical Note 9523, and the 

NIST/SEMATECH e-Handbook of Statistical Methods4

 that require the use of computer software 

(mass code) for the data reduction. These weighing designs are normally used when high precision 

(low uncertainty) mass measurement results are sought, although weighing designs can be used at 

any uncertainty level. The uncertainty reported using advanced weighing designs is based on the 

historically observed process of similar measurements and is very dependent upon correct 

procedures for defining these processes. 

NIST calibrations provide traceable standards at one point in time. The major advantage in this 

program is the ability to evaluate reference/working standards and the measurement process over 

time, providing ongoing assurance regarding accuracy and traceability of the mass standards for both 

the laboratory and its customers. Ongoing evaluation of the measurement process provides the 

laboratory with data that can be used to establish or adjust calibration intervals for reference/working 

standards. The measurement assurance program is also critical for defining and reporting realistic 

uncertainties. 

 

1

 Taylor, John K. And Henry V. Oppermann, NIST [NBS] Handbook 145, Handbook for the Quality 

Assurance of Metrological Measurements, November 1986. 

2

 Selected Laboratory and Measurement Practices, and Procedures, to Support Basic Mass Calibrations 

3

 Cameron, J. M., M. C. Croarkin, and R. C. Raybold, Technical Note 952, Designs for the Calibration of 

Standards of Mass, June 1977. 

4

 NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, 2005. 
March 16, 2005 

Page 8 

Program Prerequisites

The following items are listed as general guidelines for a laboratory conducting an internal 

evaluation of its program for suitability in the advanced mass calibration program for States. These 

guidelines have been established based on good measurement practices, good laboratory practices, 

and a similar fee-funded program (NIST Mass MAP) operated by the NIST Mass and Force Group. 

Many specific technical recommendations are taken from NIST Handbook 143, State Weights and 

Measures Laboratory Program Handbook, G. Harris, Editor, March 2003, and NIST Handbook 150-

2G: NVLAP Calibration Laboratories, Technical Guide for Mechanical Measurements, C. Douglas 

Faison and Carroll S. Brickenkamp, Editors, March 2004. Deviations from recommendations are 

occasionally made when data is available to support it; however, judgments should be made 

carefully when evaluating data, since some deviations from these practices will inadvertently 

increase measurement uncertainties and may contribute to measurement errors. 

Training 

< Satisfactory completion of an Intermediate Metrology Training course and Laboratory 

Auditing Program (LAP) problems within the last five years are expected before attendance 

at the Advanced Mass Training course which is traditionally taught in odd-numbered years. 

< Although required for NIST Recognition, attendance at the regional measurement assurance 

program meetings is not required to perform advanced mass calibrations. However, regular 

updates on precision mass procedures and issues are often provided at the regional meetings. 

< Satisfactory completion of the Advanced Mass Training course and Advanced LAP 

problems and successful application of these guidelines are expected before Recognition or 

accreditation at the Echelon I level as described in NIST Handbooks 143 and 150-2G. 

Facilities 

< Environment -- The temperature for the laboratory where mass measurements are made 

should be selected at a point between 20 o

C and 23 oC, with allowable variation of ± 1 oC 

(e.g., 22 o

C ± 1 oC), with a maximum change of 0.5 oC per hour. Air flow must be low 

enough so that it does not interfere with balance or mass comparator operation. Humidity 

should be set between 40 % and 60 % relative humidity (e.g., 45 % ± 5 %). Environmental 

conditions must be monitored according to technical criteria and measurements should not 

be made when prescribed conditions are not maintained. Deviation from stated 

environmental parameters requires a thorough evaluation of the impact of the deviation on 

measurement errors and uncertainties. 

< Vibration -- The laboratory location and design should be such as to avoid or minimize 

potential sources of vibration that will interfere with precision mass calibration. 
March 16, 2005 

Page 9 

< Cleanliness -- Good housekeeping practices and cleanliness specifications are found in 

NIST Handbook 145, 143 and NIST/NVLAP Handbook 150-2G. Contamination from dust, 

hair, paper, shipping/packaging/storage materials like felt and velvet, and other air 

contaminants has been found to be a critical concern. 

Equipment 

< Computer and printer -- A computer of sufficient memory and processing ability is 

essential. In the laboratory, every effort must be made to minimize the impact of introducing 

temperature gradients in the measuring areas. Laptops or other wireless or panel monitors 

are preferred to large heat-producing systems. The printer should be capable of routinely 

producing the 40-page reports generated by the mass code, and it should also be capable of 

printing graphics. The printer should not be located in the precision mass laboratory. 

< Balances -- A list of the laboratory's balances and process control chart data should be 

submitted to WMD for evaluation. Control data for each balance should consist of the 

following: 

- Balance manufacturer, model number and serial number; 

- Capacity and resolution; and 

- Pooled standard deviations (accepted within-process standard deviations, sw, and 

accepted long term standard deviation of the check standard, st), showing number of 

degrees of freedom, loads, and specific weighing designs. 

Laboratory balances will be evaluated to determine their suitability for this program. Minimum 

balance performance specifications recommended for this program are as follows: 

Loads *Standard Deviation should be <: 

20 kg 1.5 mg 

10 kg 0.20 mg 

1 kg 0.050 mg 

600/500/400/200 g 0.025 mg 

60/50/40/20 g 0.0050 mg 

6/5/4/2 g 0.0025 mg 

600/500/400/200 mg 0.0013 mg 

60/50/40/20 mg 0.00050 mg 

6/5/4/2 mg 0.00030 mg 

* The standard deviations in this table were developed by calculating 1/3 of OIML Class E1

tolerances, accounting for typical uncertainties associated with the standard and other factors, 

and developed by working backwards from a target expanded uncertainty. Many 

laboratories will not have balances or processes that can achieve these results on a routine 

basis. 
March 16, 2005 

Page 10 

< Barometer -- A barometer having documented accuracy of ± 65 Pa (0.5 mm Hg) with 

evidence of traceable measurement results (typically from an accredited laboratory) must be 

available. 

< Thermometers -- Thermometers to measure air temperature having accuracy of ± 0.1 oC 

with evidence of traceable measurement results (typically from an accredited laboratory) are 

required. Temperature measurements made at this level are generally made within the 

balance chambers. 

< Hygrometer/Psychrometer -- Percent relative humidity should be measured with an 

accuracy of ± 5 % and have evidence of traceable measurement results (typically from an 

accredited laboratory). 

< An environmental recording device is critical for monitoring laboratory conditions. Even 

though a number of environmental corrections are made in the mass measurement process, 

ensuring environmental stability during the 24-hour period preceding a calibration 

(particularly for temperature) is important to ensure proper thermal and environmental 

equilibration of mass standards. 

Standards 

< Reference (formerly called Primary) Standards -- A minimum of two 1-kg reference 

standards (four 1-kg reference standards are recommended) with NIST calibration and 

density determination are needed. Calibration values should be less than two years old. 

Standards should be calibrated at least every two years unless a measurement assurance 

program that monitors the reference kilogram standards is in place and demonstrates ongoing 

stability and validity of the mass values. If only two reference standards are available, and if 

they are used with equal frequency, the measurement assurance program will not be 

considered adequate without some type of verification using standards from outside the 

laboratory that have recent NIST-calibration. GMP 11, Good Measurement Practice for 

Assignment and Adjustment of Calibration Intervals for Laboratory Standards, and GMP 13, 

Good Measurement Practice for Ensuring Traceability (or other equivalent procedure), 

should be implemented in the laboratory to document the traceability hierarchy and 

calibration intervals that the laboratory will follow. (See discussions on Proficiency Tests 

and Graphs and Control Charts.) 

< Check standards -- Check standards (sometimes called control standards) are not required 

to be calibrated by NIST. However, having the check standards calibrated by NIST or a 

competent external source provides an effective mechanism to identify and evaluate biases 

that may be occurring in the measurement processes that would otherwise go undetected; 

having an external calibration is essential. 
March 16, 2005 

Page 11 

The check standards must be one-piece design (to provide the necessary stability and act as 

surrogates to the reference standards) with known densities and have been assessed to 

comply with limits on magnetic susceptibility as required in ASTM and OIML standards, in 

the following decade denominations: 1 kg, 100 g, 10 g, 1 g, 100 mg, 10 mg, 1 mg. “ASTM 

Class 1, Type 1, Grade S” or “OIML Class E2” verbiage may be used if specifying weights 

for purchase. Additional check standards above 1 kilogram are needed to handle the entire 

range of calibration services, e.g., 10 kg. 

Measurement Assurance (Control) 

< The laboratory needs to have a measurement assurance (control) system already in place 

before trying to perform advanced mass calibrations. Practical, hands-on experience in the 

laboratory is essential to making good mass measurements. A current measurement 

assurance system and data are essential for demonstrating measurement proficiency and/or 

justifying any deviations from these recommendations. 

Advanced LAP Problems

See Figure 1 for a graphic view of the components required in a complete analysis of the Advanced 

LAP problems. 

Laboratory Auditing Program (LAP) problems are used for establishing a baseline for the 

initialization of the check standards, to provide initial data to assess the between-time component of 

the measurement process, for providing validation on uncertainty statements, and for evaluating the 

proficiency level at which the laboratory uses the mass code. The data collected in the LAP 

problems is reduced by each laboratory using the mass code. Each qualified metrologist must 

complete training and the Advanced LAP problems and be able to reduce and analyze their own 

data. Each laboratory is responsible for graphing and analyzing data when determining the "in 

control" or "out of control" condition of their standards (see sections on Establishing Measurement 

Control and Graphs and Control Charts). Observed surveillance values must be compared to the 

reported NIST values to determine the level of control. A copy of all data, data files, reports, graphs, 

and final analysis are to be sent to WMD for evaluation along with the most recent NIST calibration 

report for the standards used. The final written analysis, demonstrating a thorough understanding of 

the measurement assurance system and uncertainty analysis at this level of work will be considered 

as the most critical component of the completed Advanced LAP problems. 

NOTE: Standards should not be cleaned using solvents during the initial data-collection period as 

these tests will provide data for determining the total and between-time standard deviations for each 

series. Cleaning standards changes their mass values and may invalidate the calibration. Cleaning 

plans and procedures must be documented as a part of the laboratory procedures. At least 7 to 10 

days are required for environmental equilibration on standards that have been cleaned with solvents 

prior to recalibration. 
March 16, 2005 

Page 12 

LAP Problem 1: Ten (10) complete runs on reference/working State standards from 1 kg to 

1 mg. (Initial data from the first one or two runs may be submitted to be sure 

the laboratory is on the right track.) 

LAP Problem 2: Two (2) complete runs on standards from 30 kg to 2 kg. 

The series selected when ascending from 1-kg may be the same as those used when descending; 

however, a single restraint is usually used and the between-time standard deviation formulae must be 

derived. 

If the laboratory maintains both metric and avoirdupois standards with NIST traceability, one 

additional LAP problem should be conducted. The avoirdupois standards start at 1 pound and 

usually will require a “cross over” from the kilogram. Special tare weights (92.815 g) can be 

obtained to facilitate this process. 

LAP Problem 3: Two (2) complete runs on standards from 50 lb to 1 μlb. 

Follow up: Note measurement assurance guidelines and traceability for using the initial 

LAP problems in continuing measurement assurance.

Note: Two complete runs is not adequate to provide data for establishing initial limits nor for 

establishing a baseline for acceptable measurement assurance nor for validating uncertainties at this 

level. If the laboratory plans to provide internal calibration results or extended service to customers, 

additional data must be obtained and analyzed. Process statistics determined with limited data must 

have uncertainties that reflect the actual degrees of freedom. The laboratory may limit their 

application of this process to the smaller mass levels (e.g., 1 kg to 1 mg, or 100 g to 1 mg). LAP 

problems 1 and 2 may be combined if a limited range will be used for this level in the laboratory. 

For example, ten runs from 10 kg to 1 mg, would be another acceptable approach. If the laboratory 

does not plan to use the mass code for avoirdupois standards, LAP problem 3 is not required. 
March 16, 2005 

Page 13 
March 16, 2005 

Page 14 

Restraint for 

Series #2

Restraint for 

Series #3 

Restraint for Series #4 

Restraint for Series #5 

Restraint for Series #6 

Restraint for Series #7 
March 16, 2005 

Page 15 

Series 1 

P1.kg P1:kg 

C1kg 

∑1kg 

1st Double Sub + - 1 kg load

2nd Double Sub + - 1 kg load

3rd Double Sub + - 1 kg load

4th Double Sub + - 1 kg load

5th Double Sub + - 1 kg load

6th Double Sub + - 1 kg load

Series 2 

P500g P300g P200g P100g C100g 

∑100g 

Series 3 

P50g P30g P20g P10g C10g 

∑10g 

1st Double Sub + - - + - 600 g load 1st Double Sub + - - + - 60 g load

2nd Double Sub + - - + - 600 g load 2nd Double Sub + - - + - 60 g load

3rd Double Sub + - - - + 600 g load 3rd Double Sub + - - - + 60 g load

4th Double Sub + - - 500 g load 4th Double Sub + - - 50 g load

5th Double Sub + - - - - 500 g load 5th Double Sub + - - - - 50 g load

6th Double Sub + - + - - 400 g load 6th Double Sub + - + - - 40 g load

7th Double Sub + - - + - 400 g load 7th Double Sub + - - + - 40 g load

8th Double Sub + - - - + 400 g load 8th Double Sub + - - - + 40 g load

9th Double Sub + - - 200 g load 9th Double Sub + - - 20 g load

10th Double Sub + - - 200 g load 10th Double Sub + - - 20 g load

11th Double Sub + - - 200 g load 11th Double Sub + - - 20 g load

Series 4 

P5g P3g P2g P1g C1g 

∑1g 

Series 5 

P500mg P300mg P200mg P100mg C100mg 

∑100mg 

1st Double Sub + - - + - 6 g load 1st Double Sub + - - + - 600mg load

2nd Double Sub + - - + - 6 g load 2nd Double Sub + - - + - 600mg load

3rd Double Sub + - - - + 6 g load 3rd Double Sub + - - - + 600mg load

4th Double Sub + - - 5 g load 4th Double Sub + - - 500mg load

5th Double Sub + - - - - 5 g load 5th Double Sub + - - - - 500mg load

6th Double Sub + - + - - 4 g load 6th Double Sub + - + - - 400mg load

7th Double Sub + - - + - 4 g load 7th Double Sub + - - + - 400mg load

8th Double Sub + - - - + 4 g load 8th Double Sub + - - - + 400mg load

9th Double Sub + - - 2 g load 9th Double Sub + - - 200mg load

10th Double Sub + - - 2 g load 10th Double Sub + - - 200mg load

11th Double Sub + - - 2 g load 11th Double Sub + - - 200mg load

Series 6 

P50mg P30mg P20mg P10mg C10mg 

∑10mg 

Series 7 

P5mg P3mg P2mg 

P:1mg P.1mg 

C1mg 

1st Double Sub + - - + - 60 mg load 1st Double Sub + - - + - 6 mg load

2nd Double Sub + - - + - 60 mg load 2nd Double Sub + - - + - 6 mg load

3rd Double Sub + - - - + 60 mg load 3rd Double Sub + - - - + 6 mg load

4th Double Sub + - - 50 mg load 4th Double Sub + - - 5 mg load

5th Double Sub + - - - - 50 mg load 5th Double Sub + - - - - 5 mg load

6th Double Sub + - + - - 40 mg load 6th Double Sub + - + - - 4 mg load

7th Double Sub + - - + - 40 mg load 7th Double Sub + - - + - 4 mg load

8th Double Sub + - - - + 40 mg load 8th Double Sub + - - - + 4 mg load

9th Double Sub + - - 20 mg load 9th Double Sub + - - 2 mg load

10th Double Sub + - - 20 mg load 10th Double Sub + - - 2 mg load

11th Double Sub + - - 20 mg load 11th Double Sub + - - 2 mg load

Measurement Matrix 

1 kg through 1 mg 

Establishing Measurement Controls

Process Evaluation 

For each combination of a weighing design, specific load, and specific balance (e.g., a 4-1 design, at 

a 1-kg load, on an AT 1005 balance), a measurement control process should be defined and data 

collected to characterize both the standard uncertainty of the process, sw, and the standard 
March 16, 2005 

Page 16 

uncertainty for the standard over time, st. These values are essential for correctly reducing 

measurement data and calculating the uncertainty assigned to each mass value in a report. This 

process is more than simple statistical process control because the assigned values for the standards 

and check standards are verified for accuracy with each run. 

In a surveillance test of reference/working standards from 1-kg to 1-mg, there are seven series as 

characterized below: 

Series 

Weighing Design 

(Tech Note 952) Decade Load 

Balance 

(examples only) 

1 A.1.2 (1, 1, 1, 1) (4-1) 1 kg AT 1005 

1A (option) A.1.4 (1, 1, 1, 1, 1) (5-1) 1 kg AT 1005 

2 C.2 (5, 3, 2, 1, 1, 1) 600, 500, 400, 200 g AT 1005 

(optional) A.1.2 or A.1.4 100 g AT 106, CC111 

3 C.2 (5, 3, 2, 1, 1, 1) 60, 50, 40, 20 g AT 106, CC111 

4 C.2 (5, 3, 2, 1, 1, 1) 6, 5, 4, 2 g UMT-6, CC6 

5 C.2 (5, 3, 2, 1, 1, 1) 600, 500, 400, 200 mg UMT-6, CC6, S5 

6 C.2 (5, 3, 2, 1, 1, 1) 60, 50, 40, 20 mg UMT-6, CC6, S5 

7 C.1 (5, 3, 2, 1, 1) or C.2 6, 5, 3, 2, 1 mg UMT-6, CC6, S5 

NOTE: When data is reduced using the mass code, switching balances within a series may result in 

artificially low process standard deviations and will result in errors in between-time calculations. 

As noted earlier, the Advanced LAP problems are the minimum recommendations for collecting data 

that will begin to characterize the measurement process. Ten complete runs on the 

reference/working standards, 1-kg through 1-mg (all seven series reduced using the mass code), 

must be made. An additional series at a 100-g load is essential to establish a secondary starting 

point for the calibration of 100-g kits. This means that initial data will be collected for at least these 

eight series. 

The section on Calculations in SOP No. 28 shows how to calculate each of the standard uncertainty 

values that should be entered in the mass code data file once data from the initial ten runs is 

available. Meanwhile, because statistical data for the new process and check standards may be 

unavailable, simulated values, based on knowledge of each measurement process, should be used in 

order for the mass code to reduce data. For the first ten runs, process standard deviation values, 

based on previous three-ones (also called three in one, abbreviated 3-1) weighing designs or multiple 

double substitutions for each balance are satisfactory. Using this process, only the size of the 
March 16, 2005 

Page 17 

uncertainties and the statistical tests will be affected when data is reduced; the mass values are not 

affected. However, once actual data is available, the simulated data in the data file must be replaced, 

and final reports generated. All reports should be considered as “draft” during the process of 

gathering the initial data; the statistics and uncertainty values must be updated prior to performing 

the final data analysis. 

Proper characterization of the measurement process is more critical when using advanced weighing 

designs in decade series with the mass code than when using routine mass comparisons with 

one-to-one standards. This is primarily because the statistics used by the mass code distribute the 

uncertainty from the starting restraint (reference standard) proportionally among all the weights in 

each series. Also, the standard uncertainty of the process (previously called random error) is 

distributed among all the weights in each particular series. This type of data reduction allows the 

mass code to assign smaller uncertainties; however, the validity of these uncertainties is very 

dependent upon a well-characterized measurement process. A major difference between advanced 

weighing designs using the mass code, and routine calibrations such as the 3-1 weighing design and 

the double substitution, is that the mass code uses the uncertainty of the starting restraint only. The 

3-1 design and the double substitution use the uncertainty of a single standard at each denomination. 

 In a 1-kg to 1-mg mass code calibration, the starting restraint portion of the uncertainty is 

distributed among all other denominations and soon becomes negligible around five grams. 

Therefore, weights below five grams are primarily dependent upon the standard uncertainty of the 

process when assigning an uncertainty to a test weight. 

Data Input 

The [1993] Fortran version of the mass code requires input for statistical measurement control 

parameters in four locations: 

1) Line eight: "ran err" or "random error" for the standard is entered first; this hasn't 

been used and zero must be entered; 

2) Line eight: "sys err" or "systematic error" for the standard, sr, is entered second; this 

value is taken from the [NIST] calibration report and should be divided by the 

k-factor that was used in reporting the uncertainty (see calculations in SOP No. 28 

for formula to be used when using more than one restraint); 

3) Line thirteen: the within-process standard deviation is the same as the standard 

uncertainty of the process, sw. This value is used for the F-test and is entered at the 

beginning of line thirteen on the first series (at the beginning of the line that contains 

the sensitivity weight data for subsequent series). The value is based on pooled data 

for observed standard deviations for the process (see calculations in SOP No. 28 for 

formula to be used); and 

4) Line thirteen: the between-time standard deviation, sb, is entered at the end of line 

thirteen on the first series (at the end of the line that contains sensitivity weight data 

for subsequent series) and is the calculated value that measures the variation of the 

value for the check standard over time, st, less the contribution from the standard 
March 16, 2005 

Page 18 

uncertainty of the process (see the calculations in SOP No. 28 for formula to be 

used.) 

In Lab Wizard 1.0, the C++, Windows version of the mass code, these four parameters are identified 

in the data entry area called "Description of Weights." They are identified in Restraint 

Specifications and Statistical Parameters sections as: "Random error," "Systematic error," "Standard 

deviation," and "Between std. dev.". These items correspond exactly to the Fortran version of the 

mass code and are entered in the same location in the data file. 

NOTE: Both the Fortran and Lab Wizard 1.0 versions of the mass calibration software have been 

modified to conform to the NIST policy on uncertainty (see NIST Technical Note 1297) as far as 

possible. References to "systematic error" have been changed to "type B uncertainty"; references to 

"random error" have been changed to "type A uncertainty"; and references to "uncertainty" have 

been changed to "expanded uncertainty." The type B uncertainties are calculated as one standard 

deviation estimates for systematic error, and the type A uncertainties are calculated as one standard 

deviation estimates for random error. The expanded uncertainties are calculated as the root sum 

squares of the type A and type B uncertainties multiplied by a coverage factor of two. 

However, to preserve the integrity of the statistical control procedure for mass calibrations, the 

operation of the code deviates from the NIST policy in the evaluation of type A and type B 

uncertainties. The policy defines type A (random) uncertainties in a global manner; i.e., as a 

function of both local phenomena (balance precision and long-term measurement precision) and 

hierarchical phenomena (uncertainties associated with previously assigned values of reference 

standards). The test for statistical control for each series requires a standard deviation based only on 

local phenomena. The code does not distinguish between these two requirements, and it will 

produce an improper t-test if the hierarchical uncertainties are treated as random components. The 

solution to this conflict is to distinguish between local and hierarchical uncertainties and to define 

hierarchical uncertainties as type B uncertainties.5

 

Handling the Output 

To establish measurement controls once the mass code has been run, certain data must be extracted 

from each series and placed in a spreadsheet or database for storage, analysis, and measurement 

assurance: 

Test Number; 

Operator ID; 

Date of Test; 

Balance ID; 

Restraint ID; 

Check Standard ID; 

 5

 Croarkin, M. C., Internal NIST Correspondence, July & August 1993. 
March 16, 2005 

Page 19 

Check Standard Nominal Value; 

Starting Restraint Number; 

Calibration Design ID; 

Average Corrected Temperature in Degrees C; 

Average Corrected Pressure in Pascals; 

Average Corrected Humidity in Percent; 

Average Computed Air Density in mg/cm3

; 

Observed Standard Deviation of the Process; 

Accepted Standard Deviation of the Process; 

Degrees of Freedom; 

F-Ratio; 

Observed Correction of the Check Standard; 

Accepted Mass Correction of the Check Standard; 

t-Value; and 

Expanded Uncertainty assigned to the Check Standard. 

When either version of the mass code is run, it produces two files, one called "control" and one 

called "statis." The "control" file contains a string of data for each series that contains: the month, 

day, year, check standard identification, observed value for the check standard, balance 

identification, process standard deviation, degrees of freedom, weighing design identification, 

average temperature, range of temperature during the measurement, average pressure, range of 

pressure during the measurement, average humidity, range of humidity during the measurement, air 

density, range of air density during the measurement, operator identification, and a flag for process 

control results (0 = ok; a number 1, 2, or 3 (depending on the version of the mass code used) flags 

that the check standard failed, the observed standard deviation failed, or both failed). This file can 

be imported to a spreadsheet using a parse function so that each item is entered into a separate cell 

and saved. The "statis" file contains F-test and t-test data for each series run in the mass code. 

Unless an assignable blunder is detected in the data, all out of control series must be saved in the 

control chart file or the statistical limits will gradually become artificially small resulting in an 

increased number of failed tests. 

Each of these files should be saved by another name and/or in another directory immediately after 

each run of the mass code because the data is not cumulatively saved. These files contain 

information for the preceding mass code reduction only, and are written over with each run of the 

mass code.

Reviewing Mass Code Report

Several sections within the mass code report should be reviewed for adequacy. Key areas are 

observation values, the F-test, and the t-test. If unusually high t-test or F-test values are observed, 

one should check for data entry errors first. 
March 16, 2005 

Page 20 

Evaluate the F-test to make sure that the observed standard deviation agrees statistically with the 

accepted standard deviation of the process. An F-value is quoted and immediately below this value 

in the report is a statement that shows whether the F-test passed or failed. 

Evaluate the t-test to verify that the observed mass correction of the check standard agrees 

statistically with the accepted mass correction of the check standard. A t-value is quoted and 

immediately below the t-value in the report is a statement that says whether the check standard is 

in-control or out-of-control. 

Should either test fail and no data-entry errors are found, the series should be rerun. If the process is 

gradually changing, the t-values or F-values will usually fall in a range from two to nine. If 

statistical data is graphed properly, trends can be identified before they become critical. If many 

tests fail, it could suggest a change in the measurement process in which case the data is combined 

with the other data to define the new measurement parameters. If failed data with no attributable 

errors is routinely and incorrectly discarded, statistical limits will be artificially reduced. 

Uncertainty values based on such data will be invalid. 

Graphs and Control Charts

Critical Graphs 

Standard Control Charts – When advanced mass calibrations are used for surveillance testing, each 

weight (500 g to 1 mg) involved may be graphed separately. These data and charts can be used to 

verify calibration values and to determine appropriate calibration intervals. When appropriate for 

reference/working standards, a new accepted value may be calculated as either the mean of all 

values, or the predicted value from the linear fit of the data at a time six months in the future. 

When the measurement process has been sufficiently characterized and advanced mass calibrations 

are used for routine calibrations, a graph must be prepared for the check standard at each decade to 

properly characterize each measurement process. Analysis of the data provides the statistics for 

calculating the between-time standard deviations and can be used to verify standard and check 

standard values. The standard deviation over time is calculated from either the standard deviation 

about the mean, or the residual standard deviation from the linear fit. The standard deviation is later 

compared with the process deviation to detect if there is a between-time component of error in the 

measurement process. As noted for surveillance testing, the accepted values can be calculated in one 

of two ways. 

Measurements and control charts for two external or "monitoring" 1-kg check standards are 

recommended to verify the calibration values for the two 1-kg reference standards. Measurements 

are made between these two external standards and the reference standards using a 4-1 weighing 

design. The external kilogram standards may be part of a circulating mass package with recent 

NIST calibration (as used in proficiency testing) or may be maintained in the laboratory but must 

receive less frequent (less than 25 % as often) use than the reference standards. Analysis of 
March 16, 2005 

Page 21 

calibration results over time as recorded on control charts can provide a realistic estimate of 

calibration uncertainty and allow the investigation of drift for standard values due to time and use. 

Historical analysis can also help set calibration intervals. 

Process charts -- Control charts for the process standard deviation (for each series/balance) are used 

to establish process variability and an accepted within-process standard deviation, sw; each point can 

note the degrees of freedom differentiating between series. Standard deviations for each balance are 

plotted versus time. Plots are critiqued for outliers and degradation. A new accepted value is 

calculated by pooling the standard deviation for each balance. 

Optional Graphs 

Summation Graphs -- With a measurement control program in place for 3-1 weighing designs that 

shows the values for summations of standards, data from the mass code reports may be plotted with 

those values (when the 3-1 design uses the summation as the check standard). A number of items 

should be plotted versus time: the mean for the 3-1 summation values, the upper and lower 

control/warning limits, the mass code summation values and uncertainties, and a line showing the 

NIST calibration value. This graph may look cluttered, but it provides enormous insight to the 

relationships between the NIST value, the mass code values, and the 3-1 values for the same group 

of weights. It also provides a comparison between the separate measurement processes. The 3-1 

values may show greater precision than the mass code values. This is because in some cases the 3-1 

values may be assigned using a balance with a lower standard deviation of the process than the 

balance used for the mass code. 

F-values -- Graphing the F-values for a particular series can show trends in the process and can 

evaluate the appropriateness of the assigned process parameters. The F-values are plotted 

chronologically with the mean. The graph should be analyzed for trends and for uniform 

distribution above and below the F-value of 1.00. 

t-values -- Graphing the t-values of each series allows visualization of the extent of agreement 

between the observed and accepted value of the check standard. It also permits a comparison with 

the current measurement process. The t-values are plotted chronologically with the mean. The 

graph should be analyzed for trends and for uniform distribution above and below the t-value of 

1.00. 

Proficiency Tests

Interlaboratory comparisons (proficiency tests) should be conducted at least once every four years at 

the advanced level and may consist of two kits with an entire set of standards: 1 kg through 1 mg. 

Each regional measurement assurance group will have a schedule that addresses compliance with the 

WMD PT policy. Charts should be prepared with an "uncertainty bar" format with lines showing the 

lab average, lab median, the NIST value and NIST upper and lower uncertainties. There should be 

two charts for each denomination (one for each kit). A third chart should be a scatter plot with a 
March 16, 2005 

Page 22 

"Youden" analysis, with the center of the circle at the two lab medians and the diameter of the circle 

based upon the "residual standard deviation" of the participating laboratories. This type of analysis 

will provide useful information about potential errors in the laboratory, about the uncertainty 

reported, and about drift of artifacts during the intercomparisons. This information provides 

opportunity for evaluating a laboratory's capability to meet stated uncertainties. 

Evaluation Criteria for Proficiency Tests

Verification of the NIST Value 

The NIST value will be verified by comparison with the median of all accepted laboratory results. A 

standard deviation among laboratories and an overall median are calculated using all participating 

lab values. Using only those values that fall within the two standard deviation limits, an adjusted 

standard deviation (a measure of reproducibility between laboratories) and an adjusted round robin 

median are calculated. These new statistical values are used in the evaluation. Verification of the 

NIST value (VE) is based on the following formula: 

The normalized error (Enormal) concept (which is used internationally) is used to verify the accepted 

value for each artifact by using the Verification Error (VE). The normalized error is simply a ratio 

of the difference between the observed and accepted values and the combined uncertainty (at k=1) in 

the process combined by the root sum square method. The VE value must be less than one to verify 

the NIST value. When VE is equal to or greater than one, the standards should be calibrated by 

NIST and the new NIST values and uncertainties should be used to evaluate the proficiency test. 

The limits on the verification test require the adjusted median value to agree with the reference value 

within very tight limits. These tight limits are essential for evaluation of proficiency when many 

laboratories are working at state-of-the art levels. 

(1sd + NIST )

( RR - NIST ) (VE)= 2

unc k

2

RR

median value

( =1)
March 16, 2005 

Page 23 

Verification of Laboratory Values 

After verification of the NIST value, the following formula is used to evaluate the acceptance of 

each laboratory value (NVLAP Handbook 150, Procedures and General Requirements, July 2001.): 

Enormal calculated for a laboratory must be less than one to pass this test. The laboratory uncertainty 

and the reference value uncertainty are calculated at a 95 % confidence level. 

Verification of the Laboratory Precision 

This criterion evaluates and validates the reported uncertainty of the laboratory for its suitability to 

the level against which the laboratory is being evaluated. At the highest level of mass calibration, 

the uncertainty assigned by the participating laboratory should be less than the tolerance. There are 

several perspectives regarding the use of tolerance and uncertainty ratios; this is only a general 

guideline and is not intended to be a requirement. However, if the laboratory will determine 

compliance to OIML or ASTM standards, the expanded uncertainty at k=2, must be less than 1/3 of 

the applicable tolerance. 

Ulab<Tolerance OR Ulab<Tolerance/ 3

NOTE: Criteria for determining satisfactory/unsatisfactory compliance for proficiency tests may be 

revised in the future to handle uncertainties based on 95 percent confidence intervals rather than 99.7 

percent confidence intervals. 

File Management

Mass Code reports that have been generated and printed do not need to be saved on a hard disk; they 

can take up valuable space on the computer. Instead, a directory may be created to store each data 

file. Data files take up little disk space and the complete report can be generated as needed. 

Once the ten basic runs are made on the standards (LAP Problem 1) and the process parameters are 

defined, the original data files can be updated with the actual process parameters (replace simulated 

data with observed data), and final reports can be generated. These new reports will contain the 

same mass values, but the quoted uncertainties will reflect the true process more closely. 

( Lab + NIST )

( Lab - NIST ) E = 2

unc 2unc

value value

normal
March 16, 2005 

Page 24 

Software Management

Distribution 

Mass code software will be distributed by the NIST Weights and Measures Division upon acceptable 

completion of the Advanced Mass Metrology seminar. Lab Wizard software is “personalized” for 

each laboratory: the name of the laboratory and of the metrologist are embedded in the software and 

reports are generated containing the name of the laboratory and the metrologist. 

Licensing and Software Quality Assurance 

Each copy of mass code software distributed by WMD will be serialized, and a list of trained 

metrologists and their laboratories is maintained. All practical steps have been taken at NIST to 

ensure correct results when the software is used with proper data files. However, each laboratory 

must verify this independently and must document the verification. Each licensee must agree to 

refrain from copying or transferring software to others who have not participated in Advanced Mass 

training unless WMD gives permission in writing. WMD will only provide technical support to 

metrologists who have participated in the Advanced Mass training from the NIST Weights and 

Measures Division. 

Updating 

The mass code will be periodically updated and new versions will be released to trained metrologists 

when available. 

Approved Weighing Designs 

The Weights and Measures Division recognizes a variety of weighing designs such as those found in 

NBS Technical Notes 844 and 952, and the NIST/SEMATECH e-Handbook. However, weighing 

designs are used throughout the world with variations from those presented in NIST publications. 

Metrologists should use good judgment in developing unusual weighing designs and may submit 

them to NIST for review or validation whenever appropriate. Any designs submitted to NIST for 

review should be accompanied by sufficient experimental data to provide adequate evaluation. 

Metrologists should consider, and be able to justify, variations in weighing designs, the selection and 

use of check standards, length of designs and time restraints particularly with respect to drift, 

selection of balance, use of sensitivity weights, etc., to suit particular calibration applications. With 

the use of electronic mass comparators, length of time during a design and fatigue of the operator 

(which affect design selection) are of less concern than with the older mechanical balances. When 

developing new designs, another consideration should be that weights of equal nominal values 

should have the same uncertainty. NIST/WMD strongly recommends designs that incorporate check 

standards for process and standard verification. 
March 16, 2005 

Page 25 

Any unusual weighing designs not submitted to NIST for review will be subject to critical review 

during on-site assessments. 

Documentation of Standard Operating Procedures

To help with laboratory compliance to NVLAP Handbook 150, ANSI/NCSL Z540-1-1994, and 

ISO/IEC 17025, SOP No. 28 “Recommended Standard Operating Procedures for Using Advanced 

Weighing Designs” was developed and is included with this update as an appendix. 

Traceability and Calibration Intervals

Ensuring traceability and providing documentation of how traceability is maintained is a critical 

concern for customers and for accreditation bodies when evaluating a laboratory. Measurement 

traceability for mass measurements can be maintained through two reference kilograms that have 

been calibrated at NIST as long as appropriate mass calibration and measurement assurance 

procedures are used (and documented). The process described in this document provides guidance 

for laboratories on how to maintain adequate traceability and uncertainty needs. See GMP 11 and 

GMP 13 for additional examples of traceability hierarchies and calibration interval requirements. 

Each laboratory must have an implementation policy that covers the requirements in GMP 11 and 

GMP 13 to ensure traceability and appropriate calibration intervals. Both of these GMPs are posted 

on the NIST website. 

The assigned LAP problems can be used to initiate an appropriate measurement assurance program 

and prepare graphs. The data from the analysis and preparation of the graphs must be evaluated 

against original NIST values for the reference/working mass standards and check standards as 

appropriate. Additional data collected periodically is added to the original graphs. Data should be 

updated periodically and evaluated. How often data is updated will depend on the laboratory 

workload. 

Each laboratory must document which standards are used at each level in their traceability hierarchy 

process, what specific measurement assurance is in place at each step, and how often 

intercomparisons are conducted. The measurement assurance program, as described, is fully 

integrated into the actual calibration process. Therefore, this is not an exercise used to provide data 

for an accreditation body, but actually provides checks on the system that the laboratory can use to 

ensure that each measurement performed for a customer is accurate and traceable, with validated 

uncertainties. 

It is critical for laboratories to participate in interlaboratory comparisons that provide periodic 

checks on the measurement process. The data must be correlated with the measurement assurance 

program to be meaningful. Any discrepancies indicate the need for further investigation and 

possible need for calibration of the reference standards. Laboratories must participate in this level of 

intercomparison on a frequency no greater than four years (per WMD policy published in NISTIR 
March 16, 2005 

Page 26 

7082, Proficiency Test Policy and Plan (for State Weights & Measures Laboratories), G. Harris and 

J. Gust, January 2004.) 

Formulae and Calculations

The following items are calculated using formulae located in SOP No. 28 “Recommended Standard 

Operating Procedure for Using Advanced Weighing Designs” which is included in this edition as an 

appendix. 

• sr – The standard uncertainty of the starting restraint in the first series. 

• sw – The within-process standard deviation. 

• sb – The between-time standard deviation for each particular series . 

• Effective densities for summation standards. 

• Effective cubical coefficients of expansion for summation standards. 

SOP 5 is inserted here as Appendix A for printing only; it is kept separate from the document for website downloads. 

SOP 28 is inserted here as Appendix B for printing only; it is kept separate from the document for website downloads. 
March 16, 2005

SOP 5 Page 1 of 15

 SOP 5 

 

 Recommended Standard Operations Procedure 

 for 

 Using a 3-1 Weighing Design 

1. Introduction 

 1.1. Purpose 

 The 3-1 weighing design is a combination of three double substitution comparisons of three 

weights of equal nominal value; a standard, an unknown weight, and a second standard 

called a check standard. (The check standard may be made up of a summation of weights.) 

The weights are compared using an equal-arm, single-pan mechanical, full electronic, or a 

combination balance utilizing built-in weights and a digital indication. The specific SOP for 

the double substitution procedure for each balance is to be followed. The 3-1 weighing 

design provides two methods of checking the validity of the measurement using an F-test to 

check the measurement process and a t-test to evaluate the stability of the standard and check 

standard. Hence, the procedure is especially useful for high accuracy calibrations in which it 

is critical to assure that the measurements are valid and well documented. This procedure is 

recommended as a minimum for precision calibration of laboratory working standards that 

are subsequently used for lower level calibrations and for routine calibration of precision 

mass standards used for balance calibration. For surveillance of reference and working mass 

standards and calibration of precision mass standards used to calibrate other mass standards, 

see SOP 28 for the use of higher level weighing designs. 

 

 1.2 Prerequisites 

 1.2.1 Calibrated mass standards, traceable to NIST, with valid calibration 

certificates must be available with sufficiently small standard uncertainties 

for the intended level of calibration. Reference standards should only be 

used to calibrate the next lower level of working standards in the laboratory 

and should not be used to routinely calibrate customer standards. 

 

 1.2.2 The balance used must be in good operating condition with sufficiently small 

process standard deviation as verified by F-test values, pooled short term 

standard deviations, and by a valid control chart for check standards or 

preliminary experiments to ascertain its performance quality when new 

balances are put into service. 

 

 1.2.3 The operator must be experienced in precision weighing techniques. The 

operator must have specific training in SOP 2, SOP 4, SOP 5, SOP 29, and 

be familiar with the concepts in GMP 10. 
March 16, 2005

SOP 5 Page 2 of 15

 1.2.4 The laboratory facilities must meet the following minimum conditions to 

meet the expected uncertainty possible with this procedure: 

Table 1. Environmental conditions 

Echelon 

Temperature Relative 

Humidity 

(percent) 

I 20 °C to 23 °C, allowable variation of ± 1 °C 

Maximum change of 0.5 °C/h 40 to 60 ± 5 

II 20 °C to 23 °C, allowable variation of ± 2 °C 

Maximum change of 1.0 °C/h 40 to 60 ± 10 

 

2 Methodology 

 2.1 Scope, Precision, Accuracy 

 

 This method can be performed on any type of balance using the appropriate double 

substitution SOP for the particular balance. Because considerable effort is involved, this 

weighing design is most useful for calibrations of the highest accuracy. The weighing design 

utilizes three double substitutions to calibrate a single unknown weight. This introduces 

redundancy into the measurement process and permits two checks on the validity of the 

measurement; one on accuracy and stability of the standard and the other on process 

repeatability. A least-squares best fit analysis is done on the measurements to assign a value 

to the unknown weight. The standard deviation of the process depends upon the resolution 

of the balance and the care exercised to make the required weighings. The accuracy will 

depend upon the accuracy and uncertainty of the calibration of the standard weights and the 

precision of the comparison. 

 2.2 Summary 

 

 A standard weight, S, an unknown weight, X, and a check standard, Sc are intercompared in 

a specific order using the double substitution procedure. The balance and the weights must 

be prepared according to the appropriate double substitution SOP for the particular balance 

being used. Once the balance and weights have been prepared, all readings must be taken 

from the reading scale of the balance without adjusting the balance or weights in any way. 

Within a double substitution all weighings are made at regularly spaced time intervals to 

average out any effects due to instrument drift. Because of the amount of effort required to 

perform the 3-1 weighing design, the procedure includes the air buoyancy correction. 
March 16, 2005

SOP 5 Page 3 of 15

 2.3 Apparatus/Equipment Required 

 2.3.1 Precision analytical balance or mass comparator with sufficient capacity and 

resolution for the calibrations planned. 

 2.3.2 Working standard weights and sensitivity weights with valid calibrations 

traceable to NIST. 

 2.3.3 Small working standards with valid calibrations traceable to NIST to be used 

as tare weights. 

 

 2.3.4 Uncalibrated weights to be used to adjust the balance to the desired reading 

range or adequate optical or electronic range for the intended load and range. 

 2.3.5 Forceps to handle the weights or gloves to be worn if the weights are moved 

by hand. 

 2.3.6 Stop watch or other timing device to observe the time of each measurement 

or the operator is experienced with determining a stable indication. If an 

electronic balance is used that has a means for indicating a stable reading, the 

operator may continue to time readings to ensure consistent timing that can 

minimize errors due to linear drift. 

 2.3.7 Thermometer accurate to 0.10 °C to determine air temperature.1

 

 2.3.8 Barometer accurate to 0.5 mm of mercury (66.5 Pa) to determine air pressure. 

 2.3.9 Hygrometer accurate to 10 percent to determine relative humidity. 

 2.4 Procedure 

 2.4.1 Place the test weight and standards in the balance chamber or near the 

balance overnight to permit the weights and the balance to attain thermal 

equilibrium. The equilibration time is particularly important with weights 

larger than 1 gram. Conduct preliminary measurements to determine the tare 

weights that may be required, the size of the sensitivity weight required, 

adjust the balance to the appropriate reading range of the balance indications, 

and to exercise the balance. Refer to the appropriate double substitution SOP 

 

 1The thermometer, barometer, and hygrometer are used to determine the air density at the time of 

the measurement. The air density is used to make an air buoyancy correction. The accuracies 

specified are recommended for high precision calibration. Less accurate equipment can be used 

with only a small degradation in the overall accuracy of the measurement (See SOP 2). 
March 16, 2005

SOP 5 Page 4 of 15

for details. 

 2.4.2 Weighing Design Matrix 

 The following table shows the intercomparisons to be made in the 3-1 design, in a 

matrix format as shown in NBS Technical Note 952, Designs for the Calibration of 

Standards of Mass, J. M. Cameron, M. C. Croarkin, and R. C. Raybold, 1977.: 

Weight ID 

Comparison 

S X Sc 

a1 + - 

a2 + - 

a3 + - 

Standard + 

Check Standard + 

 This design is represented as design ID “A.1.1”in Technical Note 952, with the 

exception that the design order is reversed and Restraint B is used. The restraint is 

another name for the “standard” used in the comparison that may be found in NBS 

Technical Note 952. This matrix may be useful for anyone using the NIST Mass 

Code for data reduction. When creating a data file for this design, the design matrix 

will appear as follows: 

restraint 1 0 0 

Check 0 0 1 

following series sum 0 0 0 

Report 0 1 1 

1st double sub 1 -1 0 

2nd double sub 1 0 -1 

3rd double sub 0 1 -1 

 2.4.3 Measurement Procedure 

 Record the pertinent information for the standard, S, unknown, X, and check 

standard, Sc, as indicated on a suitable data sheet such as the one in the 

Appendix of this SOP. Record the laboratory ambient temperature, 
March 16, 2005

SOP 5 Page 5 of 15

barometric pressure, and relative humidity. Perform the measurements in the 

order shown in the following table. 

Double 

Substitution 

Measurement 

Number 

Weights 

on Pan 

Observation 

 

a1: S vs X 1 S + ts O1

 2 X + tx O2

 3 X + tx + sw O3

 4 S + ts + sw O4

a2: S vs Sc 5 S + ts O5

 6 Sc + tsc O6

 7 Sc + tsc + sw O7

 8 S + ts + sw O8

a3: X vs Sc 9 X + tx O9

 10 Sc + tsc O10

 11 Sc + tsc + sw O11

 12 X + tx + sw O12

 

 where: 

Variable Description 

ts calibrated tare weights carried with S 

tx calibrated tare weights carried with X 

tsc calibrated tare weights carried with Sc

sw calibrated sensitivity weight 

3 Calculations 

 3.1 Calculate the air density, ρA, as described in the Appendix to SOP No. 2. 

 3.2 Calculate the measured differences, a1, a2, and a3, for the weights used in each double 

substitution using the following formula (note: do not confuse this formula with the 

calculations used in SOP 4; the signs will be opposite from Option A of SOP 4): 
March 16, 2005

SOP 5 Page 6 of 15

where: 

 Variable Description 

Msw mass of the sensitivity weight 

ρsw density of the sensitivity weight 

 

 3.3 Calculate the short term within process standard deviation, sw, for the 3-1 weighing 

design. This standard deviation has one degree of freedom. 

 

 sw = 0.577(a1 - a2 + a3) 

 

 3.4 Compute the F statistic which compares the short term within process standard 

deviation, sw, to the pooled within process standard deviation. (See chapter 8.4 and 

8.5 for a discussion of the statistics used in weighing designs.) 

 

The F-statistic so computed must be less than the F-value obtained from an F-table at 

99 % confidence level (Table 9.5) to be acceptable. The F-value is obtained from the 

F-table for numerator degrees of freedom equal one, and denominator degrees of 

freedom equal to the number of degrees of freedom in the pooled within process 

standard deviation. If the data fails the F-test and the source of the error cannot be 

determined conclusively, the measurement must be repeated. 

 3.5 Compute the observed mass value of the check standard. 

 

 Compute the least-squares measured difference dsc for Sc. 

( Pooled s )

s F -statistic= 2

w

w

2

O - O

M 1-

 2

( O - O +O - O )

a = 3 2

sw

A

sw

1 2 4 3

x

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

ρ

ρ

3

- a - 2 a - a d = 1 2 3

sc
March 16, 2005

SOP 5 Page 7 of 15

 

 3.6 Compute the observed mass of Sc, Msc. 

3.7 Evaluation of the observed mass of Sc, Msc . 

 The mass determined for the check standard should be plotted on the control chart 

and must lie within the control limits. If it does not, and the source of error cannot be 

found, the measurement must be repeated. The ‘Accepted MSc’ is the mean of the 

historically observed mass values for the check standard. 

|Observed MSc – Accepted MSc| > 3sd Status: Out of Control 

2sd < |Observed MSc – Accepted MSc| < 3sd Status: In Control*Warning 

|Observed MSc – Accepted MSc| < 2sd Status: In Control 

Perform an Enormal test to compare the mean value of the Msc value from the 3-1 

design to a calibration value that has demonstrated measurement traceability for the 

check standard. 

2 2

Msc CalibratedMsc

Sc Sc

n

U U

M CalibratedM

E

+

− =

The Enormal value must be less than one to pass. 

 3.8 Compute the least-squares measured difference, dx, for X. 

3

- 2 a - a +a d = 1 2 3

x

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

1-

M 1- +d + M 1- - M 1-

M =

sc

A

t

A

t

t

A sc t

s

A s

sc

sc

sc

s

s

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ
March 16, 2005

SOP 5 Page 8 of 15

3.9 Compute the mass of X, Mx. 

where: 

Variable Description 

ρA air density 

Mi mass for weight I 

ρi reference density for weight I 

 3.10 Calculate the conventional mass of X versus the desired reference density of 8.0 

g/cm3 or apparent mass of brass (8.3909 g/cm3). It is recommended that the 

conventional mass versus 8.0 g/cm3 be reported unless otherwise requested. The 

density of X, ρx, must be entered in g/cm3

. (See SOP No. 2) 

 3.10.1 Conventional mass 

3.10.2 Apparent mass versus brass (8.3909 g/cm3 at 20 °C) 

⎟

⎠

⎞ ⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

8.3909

0.0012 1-

0.0012

M 1 -

AM vs.8.4= x

x

x

ρ

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

1-

M 1- +d + M 1- - M 1-

M =

x

A

t

A

t

t

A x t

s

A s

x

x

x

s

s

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

0.999850

0.0012

M 1 -

CM vs.8.0= x

x

x

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

ρ
March 16, 2005

SOP 5 Page 9 of 15

4 Assignment of Uncertainty 

 The limits of expanded uncertainty, U, include estimates of the standard uncertainty of the 

mass standards used, us, plus the uncertainty of measurement, um, at the 95 percent level of 

confidence. See SOP 29, “Standard Operating Procedures for the Assignment of 

Uncertainty”, for the complete standard operating procedure for calculating the uncertainty. 

When the 3-1 weighing design is used in conjunction with the Mass Code for data reduction, 

see SOP 28, “Recommended Standard Operating Procedure for Using Advanced Weighing 

Designs”, for detailed instructions on calculating the uncertainty components which are 

required by the Mass Code program. 

 4.1 The standard uncertainty for the standard, us, is obtained from the calibration report. 

The combined standard uncertainty, uc, is used and not the expanded uncertainty, U, 

therefore the reported uncertainty for the standard will need to be divided by the 

coverage factor k. Since only one standard is used as the restraint for the 3-1 

weighing design, the uncertainty of the check standard is not included in assigning an 

uncertainty to the unknown mass. 

 4.2 Standard deviation of the measurement process from control chart performance (See 

SOP No. 9.) 

 The value for sp is obtained from the control chart data for check standards using 3-1 

weighing designs. Statistical control must be verified by the measurement of the 

check standard in the 3-1 design. 

 4.3 Other standard uncertainties usually included at this calibration level include 

uncertainties associated with calculation of air density and standard uncertainties 

associated with the density of the standards used. 

5 Report 

 5.1 Report results as described in SOP No. 1, Preparation of Test/Calibration Reports. 

 
March 16, 2005

SOP 5 Page 10 of 15

 Appendix 

 3-1 Weighing Design When Tare Weights Are Used 

 (Densities used to compute air buoyancy correction) 

 (Air buoyancy correction on the tare weights) 

Laboratory data and conditions:

Date Temperature 

Balance Pressure 

Load Relative Humidity 

Pooled within process s.d., sw= Calculated Air Density 

Check standard s.d., sp = 

Mass standard(s) data:

ID Mass = N + C 

(g) 

Density 

(g/cm3) 

Unc(k=1) 

(mg) 

ID Mass = N + C 

(g) 

Density 

(g/cm3) 

Unc(k=1) 

(mg) 

Nx tx 

Ms ts 

Msc tsc 

sw 

N = Nominal, C = Correction, M = True Mass

Laboratory observations:

Balance Observations 

S - X = a1 S - Sc = a2 X - Sc = a3

S + ts S + ts X + tx 

X + tx Sc + tsc Sc + tsc 

X + tx + sw Sc + tsc + sw Sc + tsc + sw 

S + ts + sw S + ts + sw X + tx + sw 

a1 = a2 = a3 = 

Note: dotted line represents decimal point. 

Calculate “a” values: 

O - O

M 1-

 2

( O - O +O - O )

a = 3 2

sw

A sw

1 2 4 3

x

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

ρ

ρ
March 16, 2005

SOP 5 Page 11 of 15

Calculate short term within process standard deviation and conduct F-test: 

sw = .577(a1 - a2 + a3) 

<value F -table 9.5

( Pooled s )

s F -statistic= 2

w

w

2

F-test passes? Yes No 

Evaluate check standard (by plotting on a control chart or with a t-test): 

3

- a - 2 a - a d = 1 2 3

sc

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

s

A

t

A

t

t

A

s t

s

A s

s

c

sc

sc

s

c s

c

1-

M 1- +d + M 1- - M 1-

M =

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

Check standard passes? Yes No 

If both F-test and check standard pass the tests, calculate the mass of the unknown test item: 

3

- 2 a - a +a d = 1 2 3

x

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

x

A

t

A

t

t

A x t

s

A s

x

1 -

M 1 - +d + M 1 - - M 1 -

M = x

x

s

s ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

8.0

0.0012 1 -

0.0012

M 1 -

CM vs = x

x

x ref

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

ρ

ρ

 where ρref refers to the reference density. 
March 16, 2005

SOP 5 Page 12 of 15

 Example 

 3-1 Weighing Design When Tare Weights Are Used 

 (Densities used to compute air buoyancy correction) 

 (Air buoyancy correction on the tare weights) 

Laboratory data and conditions:

Date 8/18/96 Temperature 21.7

Balance AT 1005 Pressure 753.5

Load 1 kg Relative Humidity 45

Pooled within process s.d., sw= 0.023 mg Calculated Air Density 1.182 mg/cm3

Check standard s.d., sp = 0.10 mg 

Mass standard(s) data:

ID Mass = N + C Density 

(g/cm3) 

Unc(k=1) 

(mg) 

ID Mass = N + C 

(g) 

Density 

(g/cm3) 

Unc(k=1) 

(mg) 

Nx 1000 g 7.84 TBD tx NA 

Ms 999.99850 g 8 0.0327 mg ts NA 

Msc 1000.0023 g 8 0.0327 mg tsc NA 

sw 50.086 mg 8.41 0.0010 mg 

N = Nominal, C = Correction, M = True Mass

Laboratory observations:

Balance Observations 

S - X = a1 S - Sc = a2 X - Sc = a3

S + ts 1000 S + ts 1030 X + tx 1550 

X + tx 1530 Sc + tsc 1400 Sc + tsc 1410 

X + tx + sw 6530 Sc + tsc + sw 6410 Sc + tsc + sw 6400 

S + ts + sw 6010 S + ts + sw 6040 X + tx + sw 6560 

a1 = - 5.25829 a2 = - 3.69845 a3 = 1.50538 

Note: dotted line represents decimal point. 
March 16, 2005

SOP 5 Page 13 of 15

Calculate “a” values: 

O - O

M 1-

 2

( O - O +O - O )

a= 3 2

sw

A sw

1 2 4 3

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

ρ

ρ

Calculate short term within process standard deviation and conduct F-test: 

sw = .577(a1 - a2 + a3) = -0.03142 

<value F -table 9.5

( Pooled s )

s F -statistic= 2

w

w

2

=1.87 < 7.31value F -table 9.5 d.f. =40

0.023

-0.03142 F -statistic= 2

2

F-test passes? Yes No 

Evaluate check standard (by plotting on a control chart or with a t-test): 

 =3.71660 mg 3

- 5.25829 - 2(-3.69845)- 1.50538 = 3- a - 2 a - a d = 1 2 3

sc

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

s

A

t

A

t

t

A

s t

s

A s

s

c

sc

sc

s

c s

c

1-

M 1- +d + M 1- - M 1-

M =

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=1000.002217 g

8

0.001182 1-

+0.00371660

8

0.001182 999.9985 g 1-

M = sc

⎟

⎠

⎞ ⎜

⎝

⎛

Check standard passes? Yes No 
March 16, 2005

SOP 5 Page 14 of 15

If both F-test and check standard pass the tests, calculate the mass of the unknown test item: 

=5.24014 mg 3

- 2(-5.25829)-(-3.69845)+1.50538 = 3- 2 a - a +a d = 1 2 3

x

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

x

A

t

A

t

t

A x t

s

A s

x

1 -

M 1 - +d + M 1 - - M 1 -

M = x

x

s

s ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=1000.006757 g

7.84

0.001182 1-

+0.00524014 g 8

0.001182 999.9985 g 1-

M x=

⎟

⎠

⎞ ⎜

⎝

⎛

=1000.003695 g

8

0.0012 1-

7.84

0.0012 1000.006757 g 1-

= 0.0012 1-

0.0012

M 1-

CM =

ref

x

x

x

⎟

⎠

⎞ ⎜

⎝

⎛ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

ρ

ρ

where ρref refers to the reference density 8.0 g/cm3

, or conventional mass. 

Uncertainty: 

+0.10 mg +0.005 mg 3

0.098 mg U = 2* U = 2* u + s +u = 2* 2 2

2

o

2

p

2

s

2

c ⎟

⎠

⎞ ⎜

⎝

⎛

U = 0.210638 mg = 0.21 mg 

Cx = 3.70 mg ± 0.21 mg (Conventional Mass vs 8.0 g/cm3

). 
March 16, 2005

SOP 5 Page 15 of 15

Calculate 

dx 

Mx,CMx 

Uncertainty 

Prepare Report 

Pass F-test? 

Gather Data: 

Standards 

Balance 

Procedures 

Calculate 

Air Density 

a1, a2, a3 

sw

F-ratio 

Measurements 

T, P, RH 

12 Balance Observations 

Plot MSc 

Calculate t-value 

Calculate 

DSc 

MSc

Pass t-test? 

Pass 

Measurement 

Control?

YES 

YES

NO 

NO

March 16, 2005 

SOP 28 Page 1 of 11 

SOP No. 28 

 

Recommended Standard Operations Procedure 

for 

Using Advanced Weighing Designs 

1 Introduction 

1.1 Purpose 

Advanced weighing designs use a combination of double substitution 

comparisons of weights of equal nominal value or a series of weights in ascending 

or descending order; standard(s), unknown weights, and an additional standard 

called a check standard. The weights are intercompared using an equal-arm, 

single-pan mechanical, full electronic, or a combination balance utilizing built-in 

weights and a digital indication. The specific SOP for the double substitution 

procedure for each balance is to be followed. Weighing designs provide two 

methods of checking the validity of the measurement using an F-test to check the 

measurement process and a t-test to evaluate the stability of the standard and 

check standard. Hence, the procedure is especially useful for high accuracy 

calibrations in which it is critical to assure that the measurements are valid and 

well documented. This procedure is recommended for precision calibration of 

laboratory working standards that are subsequently used for lower level 

calibrations, for routine calibration of precision mass standards used for 

calibration of other mass standards, and for surveillance of mass reference and 

working standards. 

1.2 Prerequisites 

1.2.1 Verify that valid calibration certificates are available for the standards 

used as restraints in the test. 

1.2.2 Verify that the standards to be used have sufficiently small standard 

uncertainties for the intended level of calibration. Reference standards 

should only be used to calibrate the highest level of working standards in 

the laboratory and should not be used to routinely calibrate customer 

standards. 

1.2.3 Verify that the balance used is in good operating condition with 

sufficiently small process standard deviation as verified by F-test values, 

pooled short term standard deviations, and by a valid control chart for 

check standards, or preliminary experiments to ascertain its performance 

quality when new balances are put into service. See NISTIR 5672

1

 for a 

discussion on the performance levels expected for use of these procedures 

 

1Fraley, K. L., Harris, Georgia G. L., NIST IR 5672, Advanced Mass Calibration and Measurement Assurance 

Program for State Calibration Laboratories, March 2005. 
March 16, 2005 

SOP 28 Page 2 of 11 

as part of a laboratory measurement assurance program to ensure 

traceability of laboratory standards. 

1.2.4 Verify that the operator is experienced in precision weighing techniques, 

and has had specific training in SOP 2, SOP 4, SOP 5, SOP 29, and is 

familiar with the concepts in GMP 10. Further, the operator must have 

been trained in the creation of data files and the operation of the NIST 

Mass Code when it is used for data reduction as recommended. Example 

data sets and sample observation sheets are available in the Advanced 

Mass Seminar offered by the NIST Weights and Measures Division. 

1.2.5 Verify that the laboratory facilities meet the following minimum 

conditions to meet the expected uncertainty possible with this procedure: 

Table 1. Environmental conditions 

Echelon Temperature Relative Humidity (percent) 

I 20 °C to 23 °C, allowable variation of ± 1 °C 

maximum change of 0.5 °C/h 40 to 60 ± 5 

 

2 Methodology 

2.1 Scope, Precision, Accuracy

This method can be performed on any type of balance using the appropriate 

double substitution SOP for the particular balance. Because considerable effort is 

involved, this weighing design is most useful for calibrations of the highest 

accuracy. The weighing design utilizes a combination of double substitutions to 

calibrate a single unknown weight, or a group of related weights in a decade. 

This method introduces redundancy into the measurement process and permits 

two checks on the validity of the measurement; one on accuracy and stability of 

the standard and the other on process repeatability. A least-squares best fit 

analysis is done on the measurements to assign a value to the unknown weights. 

The standard deviation of the process depends upon the resolution of the balance 

and the care exercised to make the required weighings. The accuracy will depend 

upon the accuracy and uncertainty of the calibration of the restraint weights and 

the precision of the comparison. 

2.2 Summary 

 

A restraint weight, S, in some cases two restraint weights, S1 and S2, an unknown 

weight, X, or group of unknown weights, and a check standard, Sc are compared 

in a specific order typically using the double substitution procedure although 

other procedures may be appropriate. The balance and the weights must be 

prepared according to the appropriate double substitution SOP for the particular 

balance being used. Once the balance and weights have been prepared, all 

readings must be taken from the reading scale of the balance without adjusting the 
March 16, 2005 

SOP 28 Page 3 of 11 

balance or weights in any way. Within a double substitution all weighings are 

made at regularly spaced time intervals to minimize effects due to instrument 

drift. Because of the amount of effort required to perform weighing designs, the 

procedure includes an air buoyancy correction using the average air density as 

determined immediately before and after the weighings, drift-free equation for 

calculating the observed differences, correction for the cubical coefficient of 

expansion when measurements are not made at 20 °C, an average sensitivity for 

the balance over the range of measurements made, and the international formula 

for air density.

2

2.3 Apparatus/Equipment Required

2.3.1 Precision analytical balance or mass comparator with sufficient capacity 

and resolution for the calibrations planned. 

2.3.2 Reference standard weights (usually starting at 1 kg or 100 g), calibrated 

check standards for each decade (e.g., 1 kg, 100 g, 10 g, 1 g, 100 mg, 10 

mg, 1 mg for the seven series between 1 kg and 1 mg), working standard 

weights and sensitivity weights with valid calibrations traceable to NIST. 

2.3.3 Small standard working standards with valid calibrations traceable to 

NIST to be used as tare weights. Note: The calculations performed by the 

mass code do not take into consideration the value of any tare weights 

used in the weighing design. Additional calculations will be required 

when tare weights are used. 

2.3.4 Uncalibrated weights to be used to adjust the balance to the desired 

reading range or adequate optical or electronic range for the intended load 

and range. 

2.3.5 Forceps to handle the weights or gloves to be worn if the weights are 

moved by hand. 

2.3.6 Stop watch or other timing device to observe the time of each 

measurement or the operator is experienced with determining a stable 

indication. If an electronic balance is used that has a means for indicating a 

stable reading, the operator may continue to time readings to ensure 

consistent timing that can minimize errors due to linear drift. 

2.3.7 Thermometer accurate to 0.10 °C with recent calibration certificate 

traceable to NIST to determine air temperature.

3

 

 

2Formula for the Density of Moist Air, (CIPM-81/91). This equation is published in SOP 2. The difference 

between Option A and Option B in SOP 2 is less than the uncertainty associated with air density equations. 

3The thermometer, barometer, and hygrometer are used to determine the air density at the time of the measurement. 

The air density is used to make an air buoyancy correction. The accuracies specified are recommended for high 

precision calibration. Less accurate equipment can be used with only a small degradation in the overall accuracy of 

the measurement (See SOP 2). 
March 16, 2005 

SOP 28 Page 4 of 11 

2.3.8 Barometer accurate to 0.5 mm of mercury (66.5 Pa) with recent calibration 

certificate traceable to NIST to determine air pressure. 

2.3.9 Hygrometer accurate to 10 percent with recent calibration certificate 

traceable to NIST to determine relative humidity. 

2.3.10 Computer with sufficient processing capability and memory. 

 

2.4 Procedure 

2.4.1 Place the test weight and standards in the balance chamber or near the 

balance overnight to permit the weights and the balance to attain thermal 

equilibrium, or use a thermal soaking plate next to the balance with 

weights covered. Thermal equilibration time is particularly important 

with weights larger than 1 gram. An alternative heat source such as a heat 

lamp may further improve temperature stability in front of the balance. 

Conduct preliminary measurements to determine the size of the sensitivity 

weight and any tare weights that are required, adjust the balance to the 

appropriate reading range of the balance indications, and to exercise the 

balance. Refer to the appropriate double substitution SOP for details. 

2.4.2 Weighing Designs 

The table below shows the most common comparisons to be made as 

referenced in NBS Technical Note 952, Designs for the Calibration of 

Standards of Mass, J. M. Cameron, M. C. Croarkin, and R. C. Raybold, 

1977. Each series is characterized by the number of observations, n, the 

degrees of freedom, d.f. associated with the standard deviation, the 

number of weights in each design, k (not shown in this table), the number 

of restraints (standards), and check standards, along with appropriate 

positions within the design*. 

*Positions for check standards must be carefully considered as subsequent 

equations may be dependent on the position of use. 
March 16, 2005 

SOP 28 Page 5 of 11 

Table 2. Common weighing designs 

Design 

ID 

Description n

Observations

d.f.

Degrees of 

freedom 

Restraint 

Position 

Check Std 

Position* 

A.1.1 3-1 Weighing Design

4 3 1 1 3 

A.1.2 4-1 Weighing Design 6 3 1, 2 3 or 4 

A.1.4 5-1 Weighing Design 10 6 1, 2 3,4, or 5 

A.2.1 6-1 Weighing Design 8 3 1, 2 3,4,5, or 6 

C.1** 5, 3, 2, 1, 1 Design (descending) 8 4 1, 2, 3 5 or 4 

C.1 5, 3, 2, 1, 1 Design (ascending) 8 4 5 4 

C.2 5, 3, 2, 1, 1, 1 Design (descending) 11 6 1, 2, 3 4,5, or 6 

C.2 5, 3, 2, 1, 1, 1 Design (ascending) 11 6 6 4 or 5 

C.9** 5, 2, 2, 1, 1 Design (descending) 8 4 1, 2, 3, 4 5 

C.9 5, 2, 2, 1, 1 Design (ascending) 8 4 5 4 

C.10 5, 2, 2, 1, 1, 1 Design (descending) 8 3 1, 2, 3, 4 5 or 6 

C.10 5, 2, 2, 1, 1, 1 Design (ascending) 8 3 6 4 or 5 

**If these designs are NOT the last in a series, there is no position for a check standard. 

The “restraint” is another name for the standard used in the comparison. 

Matrices are shown in Technical Note 952. Determine the best design 

prior to beginning the series. The series shown allow calibration of any 

commonly found set of mass standards in either the 5, 2, 2, 1 combination 

or the 5, 3, 2, 1 combination. 

2.4.3 Measurement Procedure 

Record the pertinent information for all weights being intercompared on a 

suitable data sheet unless an automated data collection system is being 

used to collect the data and create a data file. Record or collect the 

laboratory ambient temperature, barometric pressure, and relative 

humidity immediately before and immediately after each series of 

intercomparisons. 

3 Calculations 

Calculations are completed by the NIST Mass Code as described in NBS Technical Note 

1127, National Bureau of Standards Mass Calibration Computer Software, R. N. Varner, 

and R. C. Raybold, July 1980, with updates to conform to the international formula for 

calculating air density and the ISO Guide to the Expression of Uncertainties, 1993, and 

 

4Design a.1.1. with inverted order (y3, y2, and y1), with restraint in position 1 (B) is detailed in SOP 5. 
March 16, 2005 

SOP 28 Page 6 of 11 

minor error corrections to the original code. The code is the same as that used by the 

NIST Mass Group for routine calibrations. The code performs two statistical tests (t-test 

and F-test) to verify both the value of the restraints and check standards, and to verify 

that the measurement process was in control during the comparisons. 

3.1 Calculating Effective Densities and Coefficients of Expansion for Summations

5

: 

Some designs use a summation mass and sometimes the individual masses of this 

summation will be constructed from different materials that have different 

densities and coefficients of expansion. The following equations will be used to 

calculate the effective density and effective coefficient of expansions for the 

summation that will be needed as input for the data file. The subscripts 5, 3, and 

2 refer to the individual masses that comprise the summation. This approach may 

also be needed with a 5, 2, 2, 1 combination. 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ + =

 

 

2

2

3

3

5

5

5 3 2

ρ ρ ρ

M M M

M M M Effective Density

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

 

 

 

2

2

3

3

5

5

2

2

2

3

3

3

5

5

5

ρ ρ ρ

α

ρ

α

ρ

α

ρ

M M M

M M M

Effective Cubical Coefficient of Expansion

 Table 3. Variables for equations above 

Variable Description 

M Mass (g) 

ρ Density (g/cm3

) 

α Cubical Coefficient of Expansion ( /°C) 

4 Assignment of Uncertainty 

The NIST Mass Code generates uncertainties as a part of the data reduction. Proper input 

in the data file is critical for obtaining valid results and is dependent upon a well 

characterized measurement process. See NIST IR 5672 for a discussion on the input for 

standard uncertainties in the data file. 

4.1 Calculating the standard uncertainty, us, of the starting restraint in the first series: 

Usually the starting restraint will be one or several 1 kg (or 100 g) mass standards 

that have NIST calibrations and density determinations. The uncertainty of the 

 

5Jaeger, K B., and R. S. Davis, NIST Special Publication 700-1, A Primer for Mass Metrology, November 1984. 
March 16, 2005 

SOP 28 Page 7 of 11 

standard as stated on a calibration report is divided by two or three, dependent on 

the confidence interval stated in the calibration report. 

One starting restraint scheme (a single starting standard), where Us is the 

uncertainty from NIST which must be divided by the proper coverage factor, k. 

factor

s

s k

U

u =

Multiple starting restraint scheme with standards calibrated at the same time 

against the same starting standards, i.e., dependent calibration (more than one 

starting standard): 

 , . k k k 

 

 

, k k 

factor3

3

factor2

2

factor1

1

factor2

2

factor1

1

etc

U U U

u

or

U U

u

s s s

s

s s

s

= + +

= +

Multiple starting restraint scheme with standards NOT calibrated at the same time 

as the starting standards, i.e., independent calibration (more than one starting 

standard): 

, . k k k 

 

 

, k k 

2

factor3

3

2

factor2

2

2

factor1

1

2

factor2

2

2

factor1

1

etc

U U U

u

or

U U

u

s s s

s

s s

s

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =
March 16, 2005 

SOP 28 Page 8 of 11 

4.2 Calculating the within-process standard deviation, sw, for a particular series: 

For each particular weighing design, the observed within process standard 

deviation, sw, along with its degrees of freedom, d.f., is pooled using the 

technique described in NIST Handbook 145 section 8.4. 

() ( ) ( )

k

k k

w df df df

df s df s df s

s ... 

 ... 

1 2

2 2

2 2

2

1 1

+ + +

+ + + =

4.3 Calculating the between-time standard deviation for each particular series (sb): 

Establish a standard deviation (st) for each check standard over time. If a plot of 

the check standard shows no apparent drift, the between-time standard deviation 

may be calculated. The following formulae are used to calculate the between-time 

standard deviation for the particular series. If sb

2

 is less than zero, then sb equals 

zero. 

4.3.1 For the 3-1 design with a single restraint, and a check standard that is either 

another single weight or a summation, the between time standard deviation is 

calculated using the following formula. The check standard may be in any 

position. 

2 2 2

2

1

2 2

1

2

2

0.8165

1.4142

1

1.4142

0.8165

1

b t w

b t w

s s s

K

K

s K s

K

s

= −

=

=

= −

4.3.2 Using a 4-1 design with two restraints, and the check standard is the 

difference between the two restraints, the next equation may be used to calculate 

the between-time standard deviation. If another weight in the series is used as the 

check standard, another equation is needed. 
March 16, 2005 

SOP 28 Page 9 of 11 

2 2 2

2

1

2 2

1

2

2

0.7071

1.4141

1

1.4141

0.7071

1

b t w

b t w

s s s

K

K

s K s

K

s

= −

=

=

= −

4.4.3 Using a 4-1 design with two restraints, and with a single check standard 

occupying any of the remaining positions, the next equation may be used to 

calculate the between-time standard deviation. 

2 2 2

2

1

2 2

1

2

2

0.6124

1.2247

1

1.2247

0.6124

1

b t w

b t w

s s s

K

K

s K s

K

s

= −

=

=

= −

4.3.4 Using a 5-1 design with two restraints, and the check standard is the 

difference between the two restraints, the next equation may be used to calculate 

the between-time standard deviation. If another weight in the series is used as the 

check standard, another equation is needed. 

2 2 2

2

1

2 2

1

2

2

0.6325

1.4142

1

1.4142

0.6325

1

b t w

b t w

s s s

K

K

s K s

K

s

= −

=

=

= −

4.3.5 Using a 5-1 design with two restraints, and with a single check standard 

occupying any of the remaining positions, the next equation may be used to 

calculate the between-time standard deviation. 

2 2 2

2

1

2 2

1

2

2

0.5477

1.2247

1

1.2247

0.5477

1

b t w

b t w

s s s

K

K

s K s

K

s

= −

=

=

= −
March 16, 2005 

SOP 28 Page 10 of 11 

4.3.6 In the second series (C.2), six weights are involved (500 g, 300 g, 200 g, 

100 g, Check 100 g, and a summation 100 g). Calculate the standard deviations 

of the mass values for the Check 100 g (st) and plot the results to evaluate the 

presence or lack of drift. If no drift is present, the following formula is used to 

calculate the between-time standard deviation for this series and all subsequent 

C.2 series. Subsequent series include the following check standards: 100 g, 10 g, 

1 g, 100 mg, 10 mg, 1 mg. If sb

2

 is less than zero, then sb equals zero. 

2 2 2

2

1

2 2

1

2

2

0.3551

1.0149

1

1.0149

0.3551

1

b t w

b t w

s s s

K

K

s K s

K

s

= −

=

=

= −

4.3.7 If a C.1 series is used, the following equation is used to calculate the 

between-time standard deviation when the check standard is in either of the last 

two positions: 

2 2 2

2

1

2 2

1

2

2

0.4253

1.0149

1

1.0149

0.4253

1

b t w

b t w

s s s

K

K

s K s

K

s

= −

=

=

= −

4.3.8 The between-time formulae shown here are those that are most common 

and are for descending series only. If another restraint or check standard is used, 

or if an ascending series is used, another formula will be needed. These formulae 

are statistically derived, based on the least squares analysis of the weighing 

design, and assume a normal, non-drifting distribution of measurement results. 

Equations for some other weighing designs may be calculated using the NIST 

Electronic Engineering Statistics Handbook. Section 2.3.3.2 “Solutions to 

Calibration Designs” gives an overview for deriving the solutions to weighing 

designs. It also provides the unifying equation for sb (it is called sdays in the 

electronic handbook). To clarify the difference in terminology and notation the 

unifying equation for sb is presented as: 

2

1

2

1

2

2

2

1

s K s

K

sdays = −

t

w

days b

s s

s s

s s

≡

≡

≡

2

1

2 2

1

2

2

1

b t w s K s

K

s = −
March 16, 2005 

SOP 28 Page 11 of 11 

Section 2.3.4.1 “Mass Weights” provides the solutions for 17 weighing designs 

used for decreasing weight sets, 6 weighing designs for increasing weight sets and 

1 design for pound weights. K1 is located in the portion of the solution titled 

“Factors for Repeatability Standard Deviations”, and K2 is located in the portion 

titled “Factors for Between-Day Standard Deviations”. 

5 Report 

Report results as printed in Tables I and II as generated by the Mass Code. Actual text of 

the mass code report must be modified for each laboratory in order to be ISO/IEC 17025 

compliant.