Selected Laboratory and Measurement Practices, 

and Procedures, to Support Basic Mass 

Calibrations

(2012 Ed)

Georgia L. Harris

Office of Weights and Measures 

Physical Measurement Laboratory 

NISTIR 6969
 

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NISTIR 6969

Selected Laboratory and Measurement 

Practices, and Procedures, to Support Basic 

Mass Calibrations

(2012 Ed)

Georgia L. Harris

Office of Weights and Measures 

Physical Measurement Laboratory 

February 2012

U.S. Department of Commerce 

John E. Bryson, Secretary 

National Institute of Standards and Technology 

Patrick D. Gallagher, Under Secretary of Commerce for Standards and Technology and Director
 

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February 2012 

NISTIR 6969 i 

Foreword 

This NIST IR of Selected Publications was been compiled as an interim update for a number of Good 

Laboratory Practices, Good Measurement Practices, Standard Operating Procedures, Statistical 

Techniques and Reference Tables. This interim publication was used as a reference document for the 

CD-ROM Mass Metrology Course (SP 1001) prepared by the NIST Weights and Measures Division. 

Many of these procedures are updates to procedures that were originally published in NBS Handbook 

145, Handbook for the Quality Assurance of Metrological Measurements, in 1986, by Henry V. 

Oppermann and John K. Taylor. The 2003 updates incorporated many of the requirements noted for 

procedures in ISO Guide 25, ANSI/NCSL Z 540-1-1994, and ISO/IEC 17025 laboratory quality 

systems. The major changes incorporated 1) uncertainty analyses that comply with current 

international methods and 2) measurement assurance techniques using check standards. No 

substantive changes were made to core measurement processes or equations, with the exception of 

SOP 2, Standard Operating Procedure for Applying Air Buoyancy Corrections. Procedures were 

updated in 2003 in conjunction with Jose A. Torres-Ferrer. 

The CIPM 1981/91 equation for calculating air density was added to SOP 2 in 2003 and updated in 

2012 based on the 2007 CIPM equation. 

 

The following Practices and Procedures are new in this interim publication: 

Good Measurement Practices for: 

• Assignment and Adjustment of Calibration Intervals for Laboratory Standards (11) 

• Standard Operating Procedure Selection (12) 

• Ensuring Traceability (13) 

• Selection and Use of Sensitivity Weights in Weighing Procedures (14) 

Standard Operating Procedures for: 

• Assignment of Uncertainty (29) 

• Process Measurement Assurance Program (30) 

The essential portions of SOP 29, Standard Operating Procedure for Assignment of Uncertainty were 

first published in 1994 for the NCSL Conference and Symposium and to assist State weights and 

measures laboratories comply with the ISO Guide to the Expression of Uncertainty in Measurements 

(1993). 

Key updates in 2012 were made to address updated definitions of metrological traceability. 

 
February 2012 

NISTIR 6969 ii 

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NISTIR 6969 page iii 

Table of Contents 

Foreword i

Good Laboratory Practices 

1 Quality Assurance of the Measurement Process GLP 1-1 

4 Periodic Recalibration Of State Standards GLP 4-1 

9 Rounding Expanded Uncertainties and Calibration Values GLP 9-1 

11 Painting Weights GLP 11-1

Good Measurement Practices 

1 Reading Turning Points On An Equal Arm Balance GMP 1-1 

10 Understanding Factors Affecting Weighing Operations GMP 10-1 

11 Assignment and Adjustment of Calibration Intervals for Laboratory 

Standards 

GMP 11-1 

12 Standard Operating Procedure Selection GMP 12-1 

13 Ensuring Traceability GMP 13-1 

14 Selection and Use of Sensitivity Weights in Weighing Procedures GMP 14-1 

Standard Operating Procedures 

1 Preparation of Test/Calibration Reports SOP 1-1 

2 Applying Air Buoyancy Corrections SOP 2-1 

4 Weighing by Double Substitution Using a Single Pan Mechanical 

Balance, a Full Electronic Balance, or a Balance with Digital 

Indications and Built In Weights 

SOP 4-1 

7 Weighing by Single Substitution Using a Single Pan Mechanical 

Balance, a Full Electronic Balance, or a Balance with Digital 

Indications and Built In Weights 

SOP 7-1 

8 Medium Accuracy Calibration of Mass Standards by Modified 

Substitution 

SOP 8-1 

9 Control Charts for Calibration of Mass Standards SOP 9-1 

29 Assignment of Uncertainty SOP 29-1 

30 Process Measurement Assurance Program SOP 30-1 

33 Calibration of Weight Carts SOP 33-1 

Reference Materials 

Section 8 Statistical Techniques (Updated Section 8 of NBS Handbook 145, 

1986) 

8-1 

Section 9 Reference Tables (Updated Section 9 of NBS Handbook 145, 1986) 9-1 
NISTIR 6969 page iv 

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Good Laboratory Practices 

 
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GLP 1 Page 1 of 2 

GLP 1 

Good Laboratory Practice 

for the 

Quality Assurance of the Measurement Process 

 

Quality Assurance of the Measurement Process means establishing, documenting, implementing, 

and maintaining a quality system appropriate to the laboratory’s scope of activities. Having such 

a system in place will allow the laboratory to know, within the limits of a measurement process, 

that a measurement is valid with respect to its traceability, accuracy, and precision. 

The validity of tests and calibrations should be monitored with quality control procedures. 

Statistical techniques are used to record and monitor charted measurement results to permit the 

detection of trends. The metrologists and laboratory management should plan and review the 

results from quality assurance monitoring. 

Other steps taken to ensure the quality of the measurement process may include, but are not 

limited to: 

 the regular use of Standard Reference Materials (SRMs) and /or internal quality 

control using secondary reference materials; 

 participation in interlaboratory comparisons (round robins); 

 test replications with same and/or different methods; 

 recalibration of retained items; 

 correlation of different characteristics of an item; and 

 proper calibration intervals. 

Each measurement parameter in the laboratory’s scope of activities should be reviewed and 

analyzed to determine the validity of the measurement process. 

The standards and the measurement process for each parameter must be in a state of statistical 

control. Statistical control means that the variability of the measurement process is known and 

stable; when a process is in statistical control, we can assume that the reported measurement 

uncertainties are valid. The National Institute of Standards and Technology provides technical 

guidance and support to the State weights and measures laboratories to develop measurement 

control programs that provide measurement assurance. The objective of these programs is to 

evaluate the entire measurement process including: 

 procedures; 

 standards; 

 equipment; 

 personnel; and 

 environment. 
February 2012 

GLP 1 Page 2 of 2 

While other quality assurance programs could meet these objectives, the control programs 

developed for measurement assurance greatly increase the comprehensiveness of the program. 

 

The State laboratories typically provide measurement services in the disciplines of mass, 

volume, and length. Some laboratories provide services in other measurement areas. 

Approximately 89 % of their workload is in mass standards calibration.1

 Mass calibration 

demands the greatest precision and is the first discipline for which a measurement assurance 

program was developed and implemented. Nevertheless, all measurement disciplines should 

have a measurement assurance system in place. 

The most recent improvement in assuring the quality of each measurement parameter in the State 

Laboratories is the incorporation of a Process Measurement Assurance Program (PMAP).2

A measurement assurance program consists of duplicating the measurement process by including 

a check/control standard as surrogate for the test item. Measurements made throughout the year 

will show all the conditions that are likely to affect the measurement process and their combined 

effects. Controlled duplication of the process provides for the realistic evaluation of the 

variability of the measurement process as one of the components in the estimation of the 

measurement uncertainty. 

 

Measurement results that are collected over several years may be statistically evaluated with 

current results being compared to results from previous years. Any observed problems or 

changes in the measurement results are investigated and if necessary, corrective action can be 

taken. Ongoing monitoring establishes a continuous, comprehensive, internal measurement 

assurance program in a State laboratory. 

 

Data from internal measurement assurance programs is compared to the results of interlaboratory comparisons (round robins) in which the laboratory participates as part of the 

Regional Measurement Assurance Program (RMAP) groups. (These groups consist of the 

metrologists of the State laboratories and their counterparts in private industry in a geographic 

region of the country. Six regional measurement assurance groups operate in the United States 

and its territories. ) 

 

The strength of the measurement assurance approach lies in its applicability to a wide variety of 

measurements with sufficient flexibility to permit each measurement control program to be 

tailored to the particular needs of a given measurement area. The sophistication of the control 

program depends on the criticality of the measurement. 

 

1

 State Laboratory Program Workload Survey, 2011, NIST/OWM and CSLI Legal Metrology Committee. 2

 See NISTIR 6176 and SOP 30. 
February 2012 

GLP 4 Page 1 of 2 

GLP 4 

 

Good Laboratory Practice 

for 

Periodic Recalibration of State Standards 

 

A number of States have provisions in their weights and measures laws that require the periodic 

submission of their State standards to NIST for calibration. Those provisions are based on an 

early version of the Model Law (1911), which was considered appropriate for the circumstances 

that prevailed prior to the establishment of the New State Standards Program by NIST. Periodic 

calibration is necessary on a defined regular, periodic basis, and also when measurement control 

results from internal control charts or external round robins indicate questionable data. 

 

Standards of mass, volume, and length, fabricated from modern materials, kept in the controlled 

environment of a State metrology laboratory under the custody of trained metrologists, are 

generally stable and not subject to excessive change. The cooperative NIST-State audit programs 

often identify changes in ample time for corrective action in the unlikely event that such a 

change should occur. These same programs provide the necessary evidence of the ability to 

provide metrological traceability of measurement results at a level of confidence sufficient for 

the need. 

 

The process of packing, shipping, and unpacking exposes the standards to unnecessary hazards 

that could result in damage, compromising their integrity. The return and re-calibration could 

take several months causing an unavailability of State services that would be disruptive to the 

performance of the mission of the State laboratories. 

 

In order to develop a policy for the guidance of and implementation by all 50 States regarding 

this subject, the following actions are recommended: 

 

1. All States should recognize the fact that periodic return of their State standards 

to NIST (or another suitable and accredited calibration laboratory) for recalibration is essential to comply with ISO/IEC 17025, GMP 11, and when: 1) 

data regarding traceability is unavailable; 2) charted measurement results 

indicate that the standards are out of control; 3) measurement results on round 

robins or performance tests are suspect; or 4) NIST advises the State of the 

need. 

2. GMP 11 and GMP 13 provide the basis for documenting metrological 

traceability and baseline calibration intervals. These publications are templates 

that must be modified to specific measurement parameters and applications in 

each laboratory. 

 

3. References to the periodic re-calibration of State standards in the law such as, 

 
February 2012 

GLP 4 Page 2 of 2 

 "He (the director) shall maintain the State standards in good order and 

shall submit them, at least once in ten years, to the National Institute of 

Standards and Technology for calibration," 

 

 should be followed if present and may be amended to reflect a more 

frequent interval, plus calibration based on identified needs. In lieu 

thereof, the wording of Sections 3 and 12 of the present Uniform Law 

from NIST Handbook 1301

 may be substituted: 

 

 SECTION 3. PHYSICAL STANDARDS. -- Weights and measures that 

are traceable to the U.S. prototype standards supplied by the Federal 

Government, or approved as being satisfactory by NIST, shall be the state 

reference and working standards of weights and measures, and shall be 

maintained in such calibration as prescribed by the NIST as demonstrated 

through laboratory accreditation or recognition. All field standards may be 

prescribed by the Director and shall be verified upon their initial receipt, 

and as often thereafter as deemed necessary by the Director. 

 SECTION 12. POWERS AND DUTIES OF THE DIRECTOR. -- The 

director shall: 

 

 12 (a) maintain traceability of the state standards as demonstrated through 

laboratory accreditation or recognition. 

The approach described above is recommended by NIST because each State that participates in 

the Weights and Measures Division Laboratory Recognition Program has the capability of 

evaluating its own State standards with the necessary documentation referencing the national 

standards. The Laboratory Recognition Program provides interaction between the State standards 

laboratories and NIST, ensuring satisfactory laboratory conditions suitable for primary standards 

in addition to the proper use of NIST procedures in standards calibration. Thus, each State 

program is evaluated and, if found in compliance, is Recognized as being capable of performing 

the measurements listed on the Certificate of Measurement Traceability. 

 

1

 NIST Handbook 130, Uniform Laws and Regulations in the Areas of Legal Metrology and Engine Fuel Quality 

(2012 Edition). 
February 2012 

GLP 9 Page 1 of 3 

GLP 9 

 

Good Laboratory Practice 

for 

Rounding Expanded Uncertainties and Calibration Values 

 

A calibration is not complete until the expanded uncertainty associated with the calibration is 

determined and reported. Each Standard Operating Procedure (SOP) includes information 

regarding the calculation of uncertainties. The expanded uncertainty is generally reported with 

approximately a 95 % confidence interval. The confidence interval is determined by multiplying 

a coverage factor (usually 2 but based on the degrees of freedom) times the root sum square of 

properly combined Type A and Type B evaluated components according to the ISO/IEC Guide 

to the Expression of Uncertainty in Measurement (GUM). 

The steps for reporting corrections and uncertainties are as follows: 

 

1. Round the uncertainty to two significant figures following the rules given here. 

2. Round the correction/error to the last figure affected by the uncertainty. 

3. Report the rounded correction value and uncertainty to the same level of significance. 

 

Zeros, which follow a decimal point, when there are only zeros ahead of the decimal point, are 

not considered significant figures. Moving from left to right, the first non-zero number is 

considered the first significant digit. 

 

Rounding Rules 

Use the following rules to round measurement data, consistent with its significance: 

 

1. When the digit next beyond the one to be retained is less than five, keep the 

retained figure unchanged. For example: 2.541 becomes 2.5 to two significant 

figures. 

 

2. When the digit next beyond the one to be retained is greater than five, increase the 

retained figure by one. For example: 2.453 becomes 2.5 to two significant 

figures. 

 

3. When the digit next beyond the one to be retained is exactly five, and the retained 

digit is even, leave it unchanged; conversely if the digit is odd, increase the 

retained figure by one (even/odd rounding). Thus, 3.450 becomes 3.4 but 3.550 

becomes 3.6 to two significant figures. 

Note: Even/odd rounding of numbers provides a more balanced distribution of 

results. Use of computer spreadsheets to reduce data typically follows the practice 

of rounding up. 
February 2012 

GLP 9 Page 2 of 3 

4. When two or more figures are to the right of the last figure to be retained, 

consider them as a group in rounding decisions. Thus, in 2.4(501), the group (501) 

is considered to be greater than 5 while for 2.5(499), (499) is considered to be less 

than 5. 

Several examples to illustrate the proper method of reporting corrections and uncertainties 

follow. 

 

Example 1 

 

Suppose the correction for a weight is computed to be 1.357 8 mg and the uncertainty is 0.577 

5 mg. First, round the uncertainty to two significant figures, that is, 0.58 mg. Then state the 

correction as 1.36 mg. Notice that the uncertainty and the correction express the same number of 

decimal places. Report the correction as 1.36 mg ± 0.58 mg. 

 

Example 2 

 

The volume of a given flask is computed to be 2000.714 431 mL and the uncertainty is 

0.084 024 mL. First, round the uncertainty to two significant figures, that is, 0.084 mL. (Do not 

count the first zero after the decimal point.) Round the calculated volume to the same number of 

decimal places as the uncertainty statement, that is, 2000.714 mL. Report the volume as 

2000.714 mL ± 0.084 mL. 

 

Example 3 

 

The correction for a weight is computed to be 4.3415 mg and the uncertainty is 2.0478 mg. First, 

round the uncertainty to two significant figures, that is, 2.0 mg. (Notice that two significant 

figures are shown. The zero is a significant figure since it follows a non-zero number.) Then, 

round the correction to the same number of decimal places as the uncertainty statement, that is, 

4.3 mg. Report the correction as 4.3 mg ± 2.0 mg. 

 

Example 4 

 

The correction for a weight is computed to be 285.41 mg and the uncertainty is 102.98 mg. 

Because this uncertainty is a large number, we first convert both values to the next larger 

commonly reported unit (i.e., 0.28541 g and 0.10298 g respectively). First, round the uncertainty 

to 0.10 g. (The first nonzero digit (1) is the first significant figure and the remaining digits are 

rounded to the nearest number following the first nonzero digit.) Then, round the correction to 

the point where the rounding occurred in the uncertainty statement. Round the correction to 

0.29 g. Report the correction as 0.29 g ± 0.10 g. 

 

Example 5 

 

The correction for a weight is computed to be 285.41 mg and the uncertainty is 33.4875 mg. 

First, round the uncertainty to two significant figures, that is 33 mg. Then, round the correction 
February 2012 

GLP 9 Page 3 of 3 

to the same number of decimal places as the uncertainty statement, that is, 285 mg. Report the 

correction as 285 mg ± 33 mg. 

Example 6 

 

The length of a calibrated interval is computed to be 9.999 455 8 ft and the uncertainty is 

0.003 561 7 in. First, make sure both values are reported in the same unit (i.e., convert the 

uncertainty to ft, 0.000 296 808 ft.) Then, round the value to two significant figures, that is, 

0.000 30 ft. Then, round the length of the interval to the same number of decimal places as the 

uncertainty value, that is, 9.999 46 ft. Report the length of the interval as 9.999 46 ft ± 

0.000 30 ft. 

 

Rather than stating the uncertainty value with each correction, it is also proper to place the 

correction values in a column headed by the words "Correction" or "Error," etc., and place the 

uncertainties (without plus or minus signs) in a column headed "Expanded Uncertainty". 
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February 2012 

GLP 11 Page 1 of 1 

GLP 11 

Good Laboratory Practice 

for 

Painting Cast Iron Weights 

 

Large cast iron or steel weights should be painted both for their protection and to preserve their 

mass integrity. Unprotected weights are subject to corrosion. Furthermore, when corrosion is 

present, the extent and any changes resulting from it may be difficult to estimate. 

 

Thin even coats of aluminum paint are recommended for this purpose. Spray applications are 

best if large weights or a number of small weights are to be painted. In preparation for painting, a 

weight should be cleaned and loose scale should be removed using a wire brush (DO NOT 

remove old paint and corrosion by means of sand blasting or pressure washing. Severe damage 

may result.). The painting should be done before the weights are calibrated or tolerance tested, 

unless arrangements are made to obtain "as found" values. If “as found” values are desired, two 

tests may be required. 

 

Painting should be done in an area reserved for this purpose, or at least in a place which is 

removed from laboratory measurement operations. Painting operations must be in compliance 

with applicable safety standards. The weights should be protected from dust or dirt 

contamination while the coating is drying. 
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GMP 1 Page 1 of 1 

GMP 1 

 

Good Measurement Practice 

for 

Reading Turning Points on an Equal-Arm Balance 

 

 

The damping of the beam oscillations of an equal-arm balance is generally very slow. 

Consequently, it is practical to read the turning points (the highest and lowest value of the beam 

oscillation) on the graduated scale and use the sum of the turning points as the observation value 

rather than wait for the beam to come to rest. This GMP requires that the graduated scale be 

numbered such that adding weights to the left arm increases the readings. 

 

Suppose the graduated scale has 20 graduations that are numbered from 0 to 20. The loads on 

the balance arms should be adjusted so that the sum of the turning points is approximately twice 

the midscale reading. In this example, the sum of the turning points should be within one 

division of twenty. Turning points should be estimated to at least 0.1 division in this example, 

which is typical of the Russell balance. This means that the final rest point is approximately 10, 

the midscale reading. Motion to the beam may be induced or dampened so that the turning 

points can be read easily. Care should be taken so that the beam does not hit limiting stops 

during its normal oscillation while turning points are being read. The amount of the beam 

oscillation is not critical although a span of from three to ten divisions is adequate. 

 

Once motion has been induced for the beam oscillation, wait for at least one complete oscillation 

cycle for the beam motion to stabilize. After this time, the turning points can be read. The 

readings may begin with either the high or low turning point. The turning points for at least two 

consecutive oscillation cycles should be recorded. The turning points should reveal a consistent 

pattern of slow damping; that is, the turning points should gradually converge to the eventual rest 

point. For example, if the last high turning point was greater than the previous high turning 

point (assuming that the readings normally drop on successive readings), this would indicate that 

something has interfered with beam oscillation, hence the last reading was not valid with respect 

to the previous readings. Under these circumstances, turning points should continue to be read 

until a consistent damping pattern has been obtained. In some cases, the balance oscillation will 

dampen so slowly that the same readings may be obtained for several oscillations before a 

decrease is observed. These readings are valid and may be used to compute the sum of the 

turning points. 

 

When at least four satisfactory turning points have been obtained (two high and two low turning 

points), all but the last three readings should be discarded. This will leave two high and one low 

turning point or vice versa. The two readings for the high or low turning points, as the case may 

be, should be averaged and added to the single turning point to obtain the sum of the turning 

points. The sum should be carried out to two decimal places if the second decimal place digit is 

nonzero. As an example, the following readings were obtained: 15.5, 4.3, 15.4, 4.4. The sum of 

the turning points is computed as 15.4 19.75

2

4.3 4.4

   

 



  . 
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February 2012 

GMP 10 Page 1 of 6 

GMP 10 

Good Measurement Practice 

for 

Understanding Factors Affecting Weighing Operations

Good laboratory techniques and practices, as well as good standards and balances, are required 

to obtain good mass measurements. Errors can arise from a variety of sources, such as balance 

drift, air currents, lack of thermal equilibrium, magnetic or electrostatic fields, and to 

manipulative skills using the balance. The magnitude and significance of these errors depends 

upon the laboratory environment, the balance, and the accuracy required for the measurement. 

Different methods of weighing are used to eliminate or reduce the effects of sources of error. 

Particular sources of error and ways to eliminate the errors are discussed with the weighing 

procedures. The accuracy required for a measurement and the criticality of the measurement 

often influence the choice of the weighing method used in the measurement. Regardless of the 

procedure used, several practices should be followed to promote good measurements. 

Environment and Preparation 

First, the balance should be installed in a laboratory having good temperature and humidity 

control. The requirements for temperature stability increase as more precision is needed in the 

measurement. For high precision measurement, the temperature should not vary by more than 

± 1 C in a 24 h period throughout the year and should vary by less than 0.5 C during the course 

of any one measurement series (typically less than 1 h). General laboratory measurements can 

probably withstand temperature variations of ± 2 C per day. Varying temperatures result in 

balance drift and in unequal temperatures for the air, balance, and objects to be weighed. The 

relative humidity should be maintained between 40 % and 60 % which is the recommended 

limits provided by most balance manufacturers and by OIML R 111. If the relative humidity falls 

significantly below 40 %, electrostatic discharge may be generated both in and near the balance. 

This can result in erratic readings and make precision weighing impossible. If precision mass 

standards are cleaned, they should stabilize in the laboratory environment for at least seven to 

ten days. 

Thermal and environmental equilibrium are critical for mass calibrations. This is why each mass 

SOP will provide instruction that the environment must be stable and the weights set in or near 

the balance for 24 hours prior to a calibration. The test objects, balances, reference standards, 

and air should all be in thermal equilibrium. When possible, the objects to be weighed and the 

reference standards should be placed either in the balance chamber or adjacent to the balance so 

all can come to the same thermal equilibrium. If there is a lack of thermal equilibrium, 

convection currents will be generated when an object is on the balance pan and an erroneous 

reading may result. These types of errors are likely to go unnoticed when check standards are 

handled the same way, so care must be taken to reduce the likelihood of their occurrence. Tests 

have shown that these types of errors depend upon the temperature differences among the objects 

and the balance, and on the geometry of the objects being weighed. On 20 g standards of greatly 

differing geometry, studies have shown that errors as large as 0.2 mg can occur for a 4 C 

temperature difference among the standards and the balance. Errors as large as 3 mg have also 
February 2012 

GMP 10 Page 2 of 6 

been observed at 1 kg loads when standards were stored in a cabinet, and unknown test items 

near the balance where a temperature gradient of 2 C was present (when the uncertainty was 

less than 0.1 mg). When weights are delivered to the laboratory, care must be taken to ensure 

adeuquate environmental equilibration, especially when outside conditions deviate from those in 

the laboratory by more than 10 °C. The presence of frost, ice, or condensation on the weights is a 

common occurrence and may not be readily visible. 

The balance must be installed in an area free from air currents. Balances should be installed 

away from heating/cooling vents. It is not adequate to merely close vents when weighing 

because disrupting the temperature control system may result in temperature changes near the 

balances. 

Handling Weights 

The masses of standard weights or objects weighed can be affected significantly by the manner 

in which they are handled. Human contact can leave grease or oily films that affect the mass at 

the time of measurement and can even cause permanent damage due to corrosion. 

Small weights should never be touched by hand, but handled using forceps, clean gloves, or 

swatches of cloth. In the latter case, the cloth must be lint free. Instructions for cleaning weights 

and for removing adhering foreign material are described in GMP 5. 

Large weights of lower tolerance classes (NIST Class F) may be handled by bare hands. Large 

weights are a source of special problems. Fork lifts, portable cranes, hoists, or even overhead 

cranes may have to be used to move the weights within the laboratory and on or off the balances. 

Laboratory personnel must become expert in using such equipment, as necessary, to avoid 

damage to the laboratory facilities, to the balances used, and even to the weights themselves. 

Special hoist/crane hydraulics or multi-speed systems are available to gently set large weights in 

place on large comparators to avoid damage. The problem of temperature equilibrium for large 

weights is qualitatively the same as for small weights and needs consideration on an individual 

basis. 

Large weights must be clean at the time of use, but cleaning may be a problem. Minimally, they 

should be inspected to ensure that foreign material is not present. Cleaning is addressed further 

in GMP 5. 

Operation 

Analytical balances are high precision instruments and should be operated carefully. Excessive 

shock can damage a balance. Avoid shock loading the balance. Gently place all weights in the 

center of the weighing pan/platform. The dials on mechanical balances should be turned slowly 

and gently. Careful balance operation will improve the repeatability of measurements. 

Mechanical analytical balances are provided with partial and full release positions. The partial 

release position is used when determining if an unknown load will provide an on-scale reading. 

The balance beam has limited movement in this position. The partial release position provides 
February 2012 

GMP 10 Page 3 of 6 

some protection to the balance when the dialed-in weights are not close to the actual mass placed 

on the pan. It is good practice to arrest the pan each time a dial is being changed to protect the 

balance from shock loading. It is acceptable to change the dial representing the smallest built-in 

weights when in the partial release position because the small weight changes should not result 

in shock loading of the balance. 

When releasing the pan to either the full or partial release position, the action should be done 

slowly and carefully. The objective is to minimize disturbances to the balance as much as 

possible. 

Similarly, all loads should be placed on the balance pan carefully and centered on the pan. 

When a mechanical balance is released, the beam goes through a series of oscillations. The 

frequency of these oscillations diminishes as time passes until they are almost imperceptible to 

the naked eye. At this point, optimal stabilization is achieved. This stabilization of the balance 

typically lasts for a very short period of time, after which the balance reading will usually drift. 

A similar situation occurs when a mass is placed on the pan of an electronic balance. Therefore 

readings should be taken at the precise moment of achieving balance stability. This interval 

between the releasing of a pan on a mechanical balance, or the placing of a mass on an electronic 

balance, and the reading of the observation, varies from balance to balance. Stabilization time 

differs for all balances, even among those of the same type and model. Knowledge of the 

instrument is critical to correctly establish this time interval. Although manufacturers will 

usually state this value, it is necessary for the metrologist to verify its reliability. Many electronic 

balances have a stability signal incorporated into the design, but this also must be verified. All 

measurements in a calibration should be performed at the same time interval, and within the 

shortest time possible. 

All balances should be exercised before taking readings. A load equal to the load to be measured 

should be placed on the balance, a reading taken, and the pan arrested, if appropriate, or the 

weight removed from electronic balances. This operation should be repeated several times before 

readings are taken for measurement purposes. Once the balance has been "warmed-up", better 

repeatability will be achieved. Balances can be very accurate even when used without being 

exercised first, but improved results can be obtained by going through a "warm-up" procedure. If 

the larger variation present in initial weighings on a balance that has not been exercised is not 

significant to the measurement, the warm-up procedure may be minimized. 

To determine the repeatability of measurements when a balance has not been exercised versus its 

performance after being exercised, and to determine how much warm-up time is required, begin 

measurements starting with the unexercised balance and record the readings. Repeat a series of 

measurements until you have obtained several measurements after the balance has been 

exercised. This test can be repeated over several days using the same objects and the results 

compared. The readings obtained when using an unexercised balance are likely to show a 

slightly larger variation than those obtained after the balance has been exercised. Balance drift is 

likely to be larger initially and then reach a steady state when the balance has been "warmed-up". 
February 2012 

GMP 10 Page 4 of 6 

Comparison Weighing - Mechanical Balance 

For mass calibrations, the unknown object must be compared to a known mass standard. 

Comparison weighing eliminates the errors of the built-in weights, reduces disturbances during 

the measurement because dial settings are not changed during the measurement, and can cancel 

the effect of drift by selecting the appropriate weighing design. Comparing the unknown, X, to a 

standard, S, eliminates the built-in weights from the measurement: thus, the built-in weights act 

only as counterweights; they do not affect the difference measured between X and S. 

Consequently, the dial settings must not be changed during a comparison measurement; 

otherwise the built-in weights would be part of the measurement. 

When comparison measurements are made on a single pan mechanical balance, all readings are 

taken from the optical scale. The unknown and the standard must have nearly the same mass so 

that the difference between them can be measured on the optical scale. If the masses of the 

unknown and the standard are significantly different, small mass standards are used as tare 

weights with either the unknown or the reference standard or both to obtain an observed mass 

difference that is significantly less than one-fourth the range of the optical scale. 

As part of a comparison measurement, the mass value of a scale division is determined by 

performing a sensitivity determination. The small mass standard used as part of the weighing 

design to calibrate the optical scale is called the sensitivity weight and should have a mass 

between 1/5 and 1/2 the range of the optical scale, inclusive. Additionally, the mass of the 

sensitivity weight should have a mass that is at least twice the observed mass difference between 

the standard and the unknown. Since the maximum size of the sensitivity is limited to one-half 

the range of the optical scale, it may be necessary to carry tare weights with the standard and the 

unknown to ensure that the observed difference between them is less than one-half the mass of 

the sensitivity weight. The value of the sensitivity weight should include the correction plus its 

nominal value. For high precision calibrations, the inclusion of the air buoyancy correction for 

the sensitivity weight is critical. See GMP 14 for detailed guidance on the selection of sensitivity 

weights. 

The first readings for the standard and the unknown in a comparison on a single pan balance 

should fall in the first quarter of the optical scale, but well ahead of zero, so the balance drift will 

not result in negative values for any readings. Although negative numbers may be used in 

calculations, they are avoided to simplify calculations and reduce calculation errors. Because the 

sensitivity weight may have a mass as large as one-half the range of the optical scale and the 

measured difference between the standard and the unknown may be as large as one-fourth the 

range of the optical scale, it is necessary to obtain the first two readings in the first quarter of the 

optical scale so all readings will remain on-scale for the measurement. In this way it is not 

necessary to change the dial settings to measure the difference between the standard and the 

unknown. 
February 2012 

GMP 10 Page 5 of 6 

Comparison Weighing - Electronic Balance 

Measurements made on a full electronic balance are simplified because there are no built-in 

weights to consider. Although many electronic balances are equipped with a built-in calibration 

weight, the weight is not involved in the comparison weighing. 

The principles for comparison weighing on a full electronic balance are the same as when using 

a single pan mechanical balance. The balance indications are used to measure the mass 

difference between the standard and the unknown, and a sensitivity weight is used to establish 

the mass value for a digital division on the balance. Since there are no built-in weights in the full 

electronic balance, the entire range of the digital indications can be considered for "optical scale" 

of the balance. 

For comparison weighing the standard and the unknown should be "nearly the same mass." 

Since a full electronic balance has a much larger range for indicating mass values, the masses do 

not have to be as close together as when a mechanical balance is being used. When using an 

electronic balance, the difference in mass between the standard and unknown should be less than 

0.05 % of the balance capacity. Tare weights that are known standards should be used if the 

masses are not sufficiently close together. The sensitivity weight used to determine the mass per 

digital division should have a mass that is at least 10 to 20 times the mass difference between the 

standard and the unknown but not exceeding 1 % of the balance capacity. For high precision 

weighing, air buoyancy corrections must be made for all objects used in the intercomparison. 

Magnetic and Electrostatic Fields 

Care must be taken when weighing magnets or objects having electrostatic charges. A magnetic 

field will likely affect results of a mass measurement. A magnet is attracted to ferromagnetic 

materials in the balance and balance chamber. The magnetic field may also affect the magnetic 

field generated in an electronic balance that utilizes the principle of magnetic force restoration as 

its method of measurement. 

Weights made of ferromagnetic material can become magnetized during manufacture or during 

use if they are placed in a strong magnetic field. Magnetized weights can result in measurement 

errors that may go undetected. If a measurement problem is suspected, the weights should be 

checked for magnetism and may have to be rejected if excessively magnetized. 

If magnets or magnetized material must be weighed, the material should be placed in a 

magnetically shielded container to prevent the magnetic field from generating measurement 

errors. If balance design and conditions permit, an alternative is to position the magnetized 

material a relatively large distance from the balance pan using a non-ferromagnetic object on the 

pan to serve as a platform for the magnetic. Since the strength of the magnetic field drops off at a 

rate of the cube of the distance from the magnetic, it may be possible to effectively isolate the 

magnet from other ferromagnetic material in the balance. 

Electrostatic fields can also cause errors in measurements. If there is a static charge in a 

mechanical balance, the balance may give erratic readings and lack repeatability. If the object 
February 2012 

GMP 10 Page 6 of 6 

being weighed has a significant electrostatic charge, it may result in measurement errors and 

may leave a static charge on the balance. Electrostatic charges are of particular concern when 

plastic containers are placed on the balance. 

Care should be taken to remove electrostatic charges from objects being weighed by grounding 

the objects, if necessary, before placing them on the balance. To prevent the build-up of static 

electricity in a balance, the relative humidity in the laboratory should be maintained between 

40 % and 60 %. The water vapor in the air will serve to drain off electrostatic charges from the 

balance. 

Balances utilizing the magnetic force restoration principle for weighing should be checked to 

verify that the magnetic field generated by the magnetic cell in the balance does not exist around 

the balance pan. If the shielding of the magnetic cell is inadequate, measurement errors may 

occur when weighing ferromagnetic objects or when the balance is placed on a surface 

comprised of ferromagnetic material. 
February 2012 

GMP 11 Page 1 of 6 

GMP 11 

Good Measurement Practice 

for 

Assignment and Adjustment of Calibration Intervals for Laboratory Standards 

1. Introduction 

1.1. Purpose 

Measurement processes are dynamic systems and often deteriorate with time or use. 

The design of a calibration system would be incomplete without some established 

means of determining how often to calibrate instruments and standards. A 

calibration performed only once establishes a one-time reference of uncertainty. 

Recalibration detects uncertainty growth and serves to reset values while keeping a 

bound on the limits of errors. A properly selected interval assures that an item will 

receive recalibration at the proper time. Proper calibration intervals allow 

specified confidence intervals to be selected and they support evidence of 

metrological traceability. The following practice establishes calibration intervals 

for standards and instrumentation used in measurement processes. 

1.2. Prerequisites 

1.2.1. Calibration history for laboratory standards. 

1.2.2. Expected tolerance limits and uncertainties as applicable. 

1.3. Safety 

1.3.1. No outstanding safety concerns 

2. Methodology 

2.1. Summary 

Recommended calibration intervals are based on various examples of traceability as 

described in GMP 13. As data is collected and evaluated, the laboratory technical 

manager may adjust the calibration intervals to ensure that measurement results are 

not invalidated by the intervals selected. 

2.2. Apparatus 

None. 
February 2012 

GMP 11 Page 2 of 6 

2.3. Procedure 

2.3.1. Identification of Parameters 

Critical Parameters 

Components that contribute more than 25 % of a measurement’s uncertainty 

are identified as critical parameters. To ensure an accurate evaluation of 

performance, calibration intervals are determined to meet a 99 % reliability 

target. Critical parameters are checked and defined below: 

Mass Critical Parameters (The example provided below corresponds to 

Example A in Mass Traceability in GMP 13.) 

Balance Performance 

Balance performance in Echelon I weighing processes is evaluated in every 

measurement series. An F-test ratio evaluates the observed standard 

deviation of the process against the accepted standard deviation of the 

process. The balance performance component is also tested with a check 

standard in each weighing series. The check standard value is evaluated with 

a t-test by compared the observed value to the accepted reference value. All 

other weighing processes have incorporated measurement control procedures 

and control charts that are evaluated as data is collected. 

Mass Standards 

Mass standards are dynamic with use. Wear, contamination and other factors 

can cause drift from accepted values. Thus, the following intervals have 

been set: 

 

 Table 1. Calibration intervals for mass standards. 

Item Initial Cal Interval (months) Source 

R 1. kg and R 1.. kg 48 NIST 

Sc 1. kg and Sc 1.. kg (alternating 2 years) 48 NIST 

R 30 kg to R 2 kg 12 Lab 

R 500 g to R1 mg 6 Lab 

Sc 500 g to to Sc 1 mg 6 Lab 

W 25 kg to W 1 mg 12 Lab 

R Pound Standards 24 Lab 

W Pound Standards 12 Lab 

R = Reference; Sc = check/control; W = working standards 
February 2012 

GMP 11 Page 3 of 6 

Length Critical Parameters 

Length Comparison Performance 

The measurement performance of each length calibration is evaluated with a 

check standard (when performing tape-to-tape comparison). 

Initial Length Intervals 

Length standards are dynamic with use. Wear, contamination and other 

factors can cause drift from accepted values. The following intervals have 

been set due to these factors: 

 Table 2. Calibration intervals for length standards. 

Item Initial Cal Interval (months) Source 

100 ft Tape #1 60 NIST 

100 ft Tape #2 60 NIST 

25 ft or 7 m Tape 60 NIST 

18 in Steel Rule 120 NIST 

Length Bench 24 (if used or moved) Lab 

Volume Critical Parameters (Example shown corresponds to 

Example A for Volume calibration in GMP 13.) 

Volume Comparison Performance 

The measurement performance of a volume transfer calibration is evaluated 

in each use with a repeatability check. Use of check standards is the 

preferred method for evaluating the measurement process over time. 

Traceability for volume standards may be established through gravimetric 

calibrations using traceable mass standards. 

Initial Volume Intervals 

Volume standards are dynamic with use. Wear, contamination and other 

factors can cause drift from accepted values. Calibration intervals are as 

follows: 
February 2012 

GMP 11 Page 4 of 6 

Table 3. Calibration intervals for volume standards. 

Item Initial Cal Interval (months) Source 

R 100 gal standard 60 NIST 

R 25 gal standard * 60 Lab 

R 5 gal standard * 12 Lab 

GlasswareAutopipetes 5 L to 100 mL 120 Lab 

*Gravimetric calibration for volumes 5 gallon or smaller, and all “slicker plate” 

standards. Laboratory must be qualified for performing gravimetric calibrations. 

Volume transfer is acceptable above 5 gallon. 

*May be a “slicker plate” type. None are hand-held, “dump” style, test measures. 

Temperature Critical Parameters 

Temperature Comparison Performance 

The measurement performance of each temperature comparison calibration is 

evaluated with a check standard and can be verified periodically using triple 

point cells, melting point cells, and ice baths (using documented procedures). 

Initial Intervals 

Temperature standards are dynamic with use. Shock, contamination and 

other factors can cause drift from accepted values. Recalibration intervals are 

as follows: 

 Table 4. Calibration intervals for temperature standards. 

Item Initial Cal Interval 

(months) Source 

25.5 ohm SPRT 36 NIST 

100 ohm PRT′s 12 Lab 

Standard Thermistor 12 Lab 

Check Standards 6 Lab 

Liquid-in-glass standards 6 Lab 

Secondary Parameters 

Components that contribute less than 25 % but more than 1 % of a 

measurement’s uncertainty are identified as secondary parameters. Secondary 

parameters are assigned calibration intervals designed to meet a 95 % 

reliability target. Secondary parameters are defined below: 
February 2012 

GMP 11 Page 5 of 6 

Mass and Gravimetric Volume Secondary Parameters 

Environmental Measurement Equipment 

 Table 5. Calibration intervals for environmental equipment. 

Item Initial Calibration Interval 

(months) 

Source 

Barometer 12 NIST 

Hygrometer 24 NIST 

Thermometer 12 Lab 

Length Secondary Parameters - No secondary parameters. 

Volume Secondary Parameters 

Water Temperature Measuring Device 

 Table 6. Calibration intervals for volume secondary parameters. 

Item Initial Cal Interval (months) Source 

Thermometer 12 Lab 

Temperature Secondary Parameters - No secondary parameters. 

3. Calculations 

3.1. Initial Intervals 

3.1.1. Assignment of Initial Intervals 

Assignment of initial intervals is based on these recommendations. 

Subsequent intervals may be adjusted based on analysis of Check standard 

data that is recorded on control charts, demonstrated control through 

surveillance assessments, and ongoing stability as demonstrated through 

interlaboratory comparison. The initial interval is equivalent to the amount 

of time that passes before a metrologist makes the first observation of a result 

lying outside the warning limits of the chart when the investigation yields no 

apparent correctable cause. Warning limits are established at  2 standard 

deviations of the measurement process around the accepted value of the 

check standard. 

3.1.2. Initial Intervals – Additional Standards 
February 2012 

GMP 11 Page 6 of 6 

A calibration interval must be documented for all standards used in the 

laboratory to comply with the definition and interpretation of metrological 

traceability. Statements such as “as needed” are not acceptable alone without 

additional qualifications. A statement such as “calibrations are conducted on 

an annual basis and may be extended based on demonstrated control through 

use of control charts, surveillance testing, or interlaboratory comparisons, but 

will not exceed five years” would be acceptable. 

3.1.3. Absence of Control Charts 

If no control charts are available, the laboratory’s Technical Manager will 

assign the initial interval based on engineering evidence, manufacturers 

specifications, NIST recommendations, and experience. 

3.2. Adjustment of Intervals 

3.2.1. Intervals will be adjusted when determined to be necessary by the 

laboratory’s Technical Manager. 

3.2.2. The intervals will be adjusted by taking the following factors into 

consideration as appropriate: 

 calibration history; 

 measurement assurance data; 

 data for the population of similar equipment; 

 NIST recommendations; 

 statistical analysis methods; and 

 manufacturers recommendations. 

4. Assignment of Uncertainty 

The uncertainty associated with the establishment of calibration intervals is not included as a 

part of the uncertainties associated with the respective measurement SOPs. See SOP 29 for 

calculating uncertainties for standards and the instruments. 
February 2012 

GMP 12 Page 1 of 4 

GMP 12

Good Measurement Practice 

 on 

 Standard Operating Procedure Selection

Good laboratory practices, use of proper standards and equipment, and selection of standard 

operating procedures are essential for providing calibration results with accurate and traceable 

values with appropriate and suitable uncertainties. The following matrix recommends SOPs based on 

the parameter, type of test items, and level of uncertainty needed. 

Parameter Test Item Recommended SOP 

Mass

Railroad test cars SOP 27, Railroad Test Cars using a Master 

Track Scale 

Weight carts SOP 33, Calibration of Weight Carts 

(References SOP 4 and SOP 7) 

HB 105-1, Class F 

ASTM, OIML 

Class F 

Class 5, 6, 7 

Class M1, M2, M3 

e.g., 10 kg to 250 kg 

(500 lb) cast iron 

1 mg to 5 kg 

(1 lb to 10 lb) 

stainless steel 

SOP 8, Modified Substitution - may be used if 

expanded uncertainty is less than 1/3 of the 

tolerance 

SOP 7, Single Substitution - to be used, as a 

minimum, if conditions given for SOP 8 cannot 

be met 

NOTE: Balances and standards must be 

selected properly for these conditions to be met. 

ASTM, OIML Class 3, 4 (P) 

Class F1, F2 

e.g., 1 kg kit, 100 g kit 

SOP 7, Single Substitution - may be used if 

expanded uncertainty is less than 1/3 of the 

tolerance. If uncertainty is greater than 1/3 of 

the tolerance, then use SOP 4. 

SOP 4, Double Substitution - to be used for 

buoyancy corrections and if expanded 

uncertainty is less than 1/3 of the tolerance 

Modified SOP 4/5, to be used to incorporate 

measurement control into SOP 4 

NOTE: Balances and standards must be 

selected properly for these conditions to be met. 

ASTM, OIML Class 1, 2 (S, S-1) 

Class E2 

for use in balance 

calibration 

SOP 5, 3-1 Weighing Design (preferred) 

Modified SOP 4/5, to be used to incorporate 

measurement control with SOP 4 
February 2012 

GMP 12 Page 2 of 4 

Parameter Test Item Recommended SOP 

ASTM, OIML Class 0, 1 (S) 

Class E1 

for use as laboratory 

standards 

Weighing Designs per TN 952, TN 844, 

NISTIR 5672, SOP 28, with appropriate check 

standards 

Volume

HB 105-2 Glass flasks SOP 16, Volume Transfer (acceptable, SOP 14 

preferred) 

HB 105-3 20 L test measures 

(5 gal or 10 gal) 

SOP 18, Volume Transfer (single delivery from 

slicker plate type standard; if glass standards 

SOP 19). When temperature instability is 

observed during the calibration process, use 

SOP 19. 

HB 105-3 Large graduated neck 

type provers - used for 

meter verification 

SOP 19, Volume Transfer for Graduated Neck 

Type Provers 

HB 105-4 LPG provers SOP 21, Volume Transfer for LPG Provers 

HB 105-7 Dynamic Small 

Volume Provers 

SOP 26, Gravimetric Calibration of Dynamic 

Small Volume Provers 

Laboratory standards 

Glassware: burets, 

pipetes, flasks 

SOP 13, Gravimetric Calibration with a Single 

Pan Mechanical Balance OR 

SOP 14, Gravimetric Calibration with an 

Electronic Balance 

Laboratory standards 

Laboratory slicker 

plate standards 

SOP 14, Gravimetric Calibration with an 

Electronic Balance OR 

SOP 15, Gravimetric Calibration with an Equal 

Arm Balance 

Micropipetes SOP 13, Gravimetric Calibration with a Single 

Pan Mechanical Balance OR 

SOP 14, Gravimetric Calibration with an 

Electronic Balance 
February 2012 

GMP 12 Page 3 of 4 

Parameter Test Item Recommended SOP 

HB 105-3 Large graduated neck 

type provers - used as 

laboratory standards 

SOP 19, Volume Transfer for Graduated Neck 

Type Provers OR 

SOP 14, Gravimetric Calibration with an 

Electronic Balance OR 

SOP 15, Gravimetric Calibration with an Equal 

Arm Balance 

Length

Tapes SOP 11, Tape to Tape OR 

SOP 12, Bench Method (lower uncertainties) 

Rigid Rules SOP 10, Rigid Rule 

Pi Tapes SOP 23, Pi Tape Calibration 

Liquid-in-Glass Thermometers 

HB 105-6 Field standards for 

weights and measures 

SOP 25 

Timing Devices

HB 105-5 Field standards for 

weights and measures 

SOP 24 

Traffic Speed Gun Tuning Forks

For highway official 

use 

SOP 22 

Parameter Process Recommended SOP 

Measurement Assurance

All Process 

Measurement 

Assurance 

SOP 30 Use of process measurement assurance 

programs 

Mass SOP 5, 28 Use of check standards in procedure 

TN 952, TN 844, NISTIR 5672 

Mass SOP 4, 6, 7, 8 SOP 9 

 Redundancy built into procedures 
February 2012 

GMP 12 Page 4 of 4 

Parameter Process Recommended SOP 

Measurement Assurance

Length SOP 10, 11, 12, 23 

Volume SOP 13, 14, 15, 16, 18, 

19, 21, 26 

SOP 17, laboratory check standards OR 

SOP 20, range charts and standard deviation 

charts 

Temperature SOP 25 Use of check standards in procedure 

Uncertainty

All parameters All SOPs SOP 29, Calculation of Uncertainty 
February 2012 

GMP 13 Page 1 of 16 

GMP 13 

Good Measurement Practice 

for 

Ensuring Metrological Traceability 

1. Introduction 

1.1 Purpose 

The purpose of this Good Measurement Practice is to enable compliance with 

essential elements of Metrological Traceability. Traceability ensures that the 

measurements are accurate representations of the specific quantity subject to 

measurement, within the uncertainty of the measurement. 

To ensure metrological traceability, suitably calibrated standards that are 

appropriately maintained and cared for, proper standard operating procedures, 

continuous measurement control, surveillance, and suitable documentation must all 

be present. 

Test numbers issued by NIST should not be used nor required as proof of the 

adequacy or traceability of a test or measurement. Having a NIST number does not 

provide evidence that the measurement value provided by another organization has 

the property of metrological traceability. 

1.1.1 Metrological Traceability is defined1

 as the “property of a measurement result 

whereby the result can be related to a reference through a documented unbroken 

chain of calibrations, each contributing to the measurement uncertainty.” 

NOTE 1: For this definition, a ‘reference’ can be a definition of a measurement unit 

through its practical realization, or a measurement procedure including the 

measurement unit for a non-ordinal quantity, or a measurement standard. 

NOTE 2: Metrological traceability requires an established calibration hierarchy. 

NOTE 3: Specification of the reference must include the time at which this reference 

was used in establishing the calibration hierarchy, along with any other relevant 

metrological information about the reference, such as when the first calibration in the 

calibration hierarchy was performed. 

NOTE 4: For measurements with more than one input quantity in the measurement 

model, each of the input quantity values should itself be metrologically traceable and 

 

1 These definitions are provided in the “International vocabulary of metrology — Basic and general concepts and 

associated terms (VIM)” JCGM 100:2008. 
February 2012 

GMP 13 Page 2 of 16 

the calibration hierarchy involved may form a branched structure or a network. The 

effort involved in establishing metrological traceability for each input quantity value 

should be commensurate with its relative contribution to the measurement result. 

NOTE 5: Metrological traceability of a measurement result does not ensure that the 

measurement uncertainty is adequate for a given purpose or that there is an absence 

of mistakes. 

NOTE 6: A comparison between two measurement standards may be viewed as a 

calibration if the comparison is used to check and, if necessary, correct the quantity 

value and measurement uncertainty attributed to one of the measurement standards. 

NOTE 7: The ILAC considers the elements for confirming metrological traceability 

to be an unbroken metrological traceability chain to an international measurement 

standard or a national measurement standard, a documented measurement 

uncertainty, a documented measurement procedure, accredited technical competence, 

metrological traceability to the SI, and calibration intervals (see ILAC P-10:2002). 

NOTE 8: The abbreviated term “traceability” is sometimes used to mean 

‘metrological traceability’ as well as other concepts, such as ‘sample traceability’ or 

‘document traceability’ or ‘instrument traceability’ or ‘material traceability’, where 

the history (“trace”) of an item is meant. Therefore, the full term of “metrological 

traceability” is preferred if there is any risk of confusion. 

1.1.2 A Metrological Traceability Chain is defined as “traceability chain sequence of 

measurement standards and calibrations that is used to relate a measurement result to 

a reference”. 

NOTE 1: A metrological traceability chain is defined through a calibration 

hierarchy. 

NOTE 2: A metrological traceability chain is used to establish metrological 

traceability of a measurement result. 

NOTE 3: A comparison between two measurement standards may be viewed as a 

calibration if the comparison is used to check and, if necessary, correct the quantity 

value and measurement unit . 

1.1.3 Metrological Traceability to a Measurement Unit is defined as: metrological 

traceability to a unit metrological traceability where the reference is the definition of 

a measurement unit through its practical realization 

NOTE: The expression “traceability to the SI” means ‘metrological traceability to a 

measurement unit of the International System of Units’. (See section 1.3.3.) 
February 2012 

GMP 13 Page 3 of 16 

1.2 Prerequisites 

Metrological traceability can be characterized by the following seven essential 

elements: 

1.2.1 Unbroken chain of comparisons. A documented system of comparisons with 

each step having the essential elements of metrological traceability going 

back to a standard acceptable to the parties, usually a national or international 

standard; 

1.2.2 Documented Measurement Uncertainty. The measurement uncertainty for 

each step in the traceability chain must be calculated according to defined 

methods and must be stated so that an overall uncertainty for the whole chain 

may be calculated (see SOP 29); 

1.2.3 Documented Measurement Procedure. Each step in the chain must be 

performed according to documented and generally acknowledged procedures 

(see GMP 12) and the results must be documented (i.e., in a calibration 

report, see SOP 1); 

1.2.4 Accredited Technical Competence. The laboratories or bodies performing one 

or more steps in the chain must supply evidence of technical competence 

(e.g., by maintaining appropriate training records, participating in 

interlaboratory comparisons, and by demonstrating that they are accredited 

by a recognized accreditation body); 

 

1.2.5 Realization of SI Units. The primary national, international or intrinsic 

standards must be primary standards for the realization of the International 

System of Units (SI); 

1.2.6 Documented Calibration Intervals. Calibrations of standards (and equipment 

where appropriate) must be repeated at established (may be defined through 

measurement assurance) and appropriate intervals to preserve metrological 

traceability of the standard over time and use (see GLP 4, GMP 11); and 

1.2.7 Measurement assurance. A proper measurement assurance program must be 

established to ensure the validity of the measurement process and the 

accuracy of standard used at the time of the measurement (see SOPs 9, 17, 20, 

30). 

1.3 Responsibility 

1.3.1 Provider. Providing support for the claim of traceability of the result of a 

measurement or value of a standard is the responsibility of the calibration 

provider. Calibration reports and certificates must contain a statement 
February 2012 

GMP 13 Page 4 of 16 

regarding metrological traceability. See Appedices B and C for a form that 

may be used to assess evidence supporting metrological traceability. 

1.3.2 User. Assessing the validity of a claim of traceability is the responsibility of 

the user of that result or value. Verifying claims of traceability often includes 

obtaining a calibration directly from a national metrology institute or another 

laboratory that has achieved recognition or accreditation through a 

recognized accrediting body.See Appendix D for a form that may be used to 

assess calibration reports for weights and measures applications. 

1.3.3 Use of, or reference to, official test numbers of a national metrology institute. 

Having an authentic test number does not provide assurance or evidence that 

the measurement value provided by another organization is traceable. Not 

only must there be an unbroken chain of comparisons, but each measurement 

must be accompanied by a statement of uncertainty associated with the value. 

Test report numbers should not be used nor required as proof of the adequacy 

or traceability of a test or measurement. National and international standards 

dealing with test and measurement quality requirements, such as ISO 10012, 

ISO/IEC 17025 and the ISO 9000 series do not require the use or reporting of 

specific test numbers to establish traceability. 

 

1.4 Safety 

No outstanding safety concerns 

2. Methodology 

2.1 Summary 

Traceability is maintained through comparison to appropriate standards with suitable 

procedures and measurement uncertainties. Procedures are outlined in SOPs and 

GMPs. Examples of possible hierarchies of the standards leading to the traceability 

of a calibration are provided in this document. 

2.2 Procedure 

The charts in this GMP show examples of the traceability hierarchy for mass, length, 

volume, and temperature measurement disciplines. Each laboratory must define the 

exact traceability system in their operating quality manual. (A worksheet is included 

as an Appendix to help in the definition of the traceability system.) 

3. Calculations 

 There are no calculations in this GMP. 
February 2012 

GMP 13 Page 5 of 16 

4. Assignment of Uncertainty 

The uncertainty associated with reported calibration values is included within the uncertainty 

analysis for each SOP and in SOP 29, Calculating and Reporting Uncertainties. 
February 2012 

GMP 13 Page 6 of 16 

Mass - Example A

NIST

W 1-kg.

W 1-kg..

W 100 g.

W 100 g..

C 1-kg.

C 1-kg..

C 100 g.

C 100 g..

P 1- kg.

P 1-kg..

P 100 g.

P 100 g..

Client

Echelon I, II

Calibration

Workload

Laboratory Field

Level Working

Standards

Client

Echelon III

Calibration

Workload

Remaining

Laboratory

Standards

Metric/Avdp
February 2012 

GMP 13 Page 7 of 16 

Mass - Example B 

NIST

Monitoring kg., kg.. Primary Metric &

Avdp Standards

Client

Echelon I, II

Calibration

Workload

Laboratory Field

Level Working

Standards

Client

Echelon III

Calibration

Workload

Laboratory

Standards

Metric/Avdp
February 2012 

GMP 13 Page 8 of 16 

 Mass - Example C 

NIST

Accredited

Laboratory

Laboratory Field

Level Working

Standards

Client

Echelon III

Calibration

Workload

Laboratory

Primary

Standards

Metric/Avdp
February 2012 

GMP 13 Page 9 of 16 

 Volume - Example A

 

Volume Standards

5 gal , 25 gal,

100 gal

NIST

Volume Group or

Mass Group

Client Volume

Transfer

Calibration

Workload

(5 gal or larger)

Client

Gravimetric

Volume Calibration

(5 gal or smaller)

Laboratory

Mass Standards

Metric/Avdp

Mass Standards
February 2012 

GMP 13 Page 10 of 16 

Volume - Example B 

Volume Standards

5 gal , 25 gal,

100 gal

NIST

Volume Group

Client Volume

Transfer

Calibration

Workload

(5 gal or larger)

Volume Standards

Glassware

Standards

Client Volume

Transfer

Calibration

Workload

(Glassware)
February 2012 

GMP 13 Page 11 of 16 

Length Example

18" Rigid Rule

Primary Standard

NIST

Client

Calibration

Workload

Length Bench

Working Standard

25' Tape

Primary Standard

100 ' Tape

Primary Standard

Client

Calibration

Workload
February 2012 

GMP 13 Page 12 of 16 

 Temperature Example

NIST

SPRT

Primary Standard

Liquid-in-Glass

Primary Standard

Client

Calibration

Workload

Laboratory Fixed

Points

Laboratory

Working Standards
February 2012 

GMP 13 Page 13 of 16 

Appendix A - Traceability worksheet 

Parameter: Mass, Volume, Length, Temperature, Other 

R Metric (g, L, m, C) Cal. date Cal. Lab Interval 

Range 

 

 Customary (lb, gal, ft, F) 

 Range 

 

W Metric (g, L, m, C) Cal. date Cal. Lab Interval 

Range 

 

 Customary (lb, gal, ft, F) 

 Range 

 

Sc

Metric (g, L, m, C) Cal. date Cal. Lab Interval 

Range 

 

 Customary (lb, gal, ft, F) 

 Range 

 

R = Reference Standards; W = Working Standards; Sc = Check Standards. 
February 2012 

GMP 13 Page 14 of 16 

Appendix B: Traceability Evaluation Form 

Assess whether all of the essential elements of traceability are available for the entire Scope by 

completing the table below. Specific Analysis: Include specific comments on what was observed and 

what evidence is available. 

Manual 

Reference Description Specific Example or Evidence Complies? Yes/No 

Unbroken Chain of 

Evidence. 

Standards (Reference, 

Working, and Check or 

Control) demonstrate an 

unbroken chain to SI 

references. 

 

Documented 

Uncertainty. Needed for 

each step in the 

hierarchy. 

 

Documented 

Procedures. 

Suitable, applicable, 

documented and 

validated procedures. 

 

Accredited Technical 

Competency. 

Demonstration of 

competency is 

documented. 

 

Realization to SI Units. 

Calibration intervals. 

No calibration intervals 

are “past due.” 

 

Measurement Assurance. 
February 2012 

GMP 13 Page 15 of 16 

Appendix C: Sample Technical Audit for Traceability Evidence 

Select 3 calibration reports for the past year, and identify the following specific information to support 

traceability for each calibration. On each chart, next to each block for your primary and working 

standards, please list evidence that is available in your laboratory: 

Item Evaluated Evidence 

Calibration selected (Measurement 

Type, Lab Test Number) 

Test Number(s) for all standards used 

Calibration Date(s) for the standards 

Calibration Interval 

Uncertainty analysis components 

included(s) 

Uncertainty file name where data is 

stored 

Documented SOP used 

Evidence of competency 

Measurement assurance type 

Measurement Assurance file name 

where data is stored 

Report contains a Traceability 

Statement 

You should be able to identify any gaps and corrective action needed in your laboratory in the area of 

traceability, uncertainty analysis, and measurement assurance. Note it below. 
February 2012 

GMP 13 Page 16 of 16 

Appendix D: Evaluating Supplier Calibration Reports for Metrological Traceability for 

Weights and Measures Applications 

Step Notes Accept 

(Yes/No) 

Legal Does your law allow acceptance of reports for the applicable use 

(laboratory standards, field standards, service company 

standards)? 

Does your law specify traceability to NIST? To the SI? 

Is there a required calibration interval? 

Specifications Does your law require assessment against published 

specifications like the NIST Handbook 105-series documentary 

standards? IF SO, have the customer standards been evaluated 

for compliance to specifications (are the standards suitable for 

use?) Note: This is NOT common for non-governmental 

laboratories, even when accredited, and cannot be done if there 

are no applicable specifications. Accreditation has nothing to do 

with evaluating compliance to specifications.

Documented 

Calibration Report 

Is there a valid calibration report for all of the standards? Does 

it (do they) comply with laboratory requirements? (Checklist 

available.) 

Specifically, traceability components: reference to SI and 

standards used, traceability statement, uncertainty statement, 

measurement results, measurement uncertainty, reference to 

documented procedures. 

Is the report from a 

National Metrology 

Laboratory (NMI) 

like NIST? 

Can you accept a calibration report for a standard that was tested 

by Measurement Canada that is traceable to NRC for example? 

Has that NMI signed the CIPM MRA and is this 

range/scope/uncertainty of measurements published in the 

Appendix C listing maintained by BIPM? 

(List available.) 

Is the report from a 

NIST OWM 

Recognized 

laboratory? 

OWM is NOT a recognized Accreditation Body and has not 

signed the ILAC MRA. 

Is the measurement parameter on their Scope? 

Is the reported uncertainty acceptable for your needs? 

(Posted on our website, includes those Accredited by NVLAP.) 

Is the report from 

an Accredited 

laboratory? 

Is the Accreditation Body a signatory of the ILAC MRA? 

(List available.) 

Is the measurement parameter on their Scope now? 

Is the currently reported uncertainty small enough to meet 

requirements for the laboratory or for Fundamental 

Considerations (less than 1/3 the applicable tolerance)?? 

(Directories available.) 
February 2012 

GMP 14 Page 1 of 4 

GMP 14 

 Good Measurement Practice 

for the 

Selection and Use of Sensitivity Weights in Weighing Procedures

 

1. Introduction 

Mass calibration procedures are based on comparing the unknown mass, X, to a 

standard mass, S, utilizing the balance as a comparator. This comparison relies on the 

accuracy of balance indications. Most balance indications are not accurate enough for 

precision mass calibrations, and they tend to drift with time. Drift can often be assumed 

to be linear over a short period. Concerns over balance inaccuracy and drift result in two 

possible causes of errors in mass determination procedures. Inaccuracy of the balance 

indications can be corrected by incorporating a sensitivity weight in the procedure that 

calibrates the range of use of the optical scale (mechanical balances) or of the digital 

indications (electronic balances). Errors due to drift can be minimized by using the 

correct comparison method, selecting a suitable sensitivity weight, and by consistent 

timing within the procedure. The proper selection of procedures (GMP 12), the 

adherence to those procedures, and equal time intervals between weighing operations 

will allow the measured difference between X and S to be corrected for inaccuracy of 

the balance indications and for balance drift. 

1.1. Purpose 

The following practice will guide you through the process of selecting and 

using a correct sensitivity weight for mass determination procedures. 

 

1.2. Prerequisites 

1.2.1. Verify that valid calibration certificates are available for the masses to 

be used as standards, sensitivity weights, and tare weights. 

1.2.2. Verify that weight-handling equipment is available and in good 

operational condition. 

1.2.3. Verify that the operator is familiar with the design and the operation 

of the balances and familiar with weighing procedures. 

1.3. Safety 

1.3.1. Handling of large or small weights can represent a hazard to either the 

weights or personnel if the weights are dropped. 
February 2012 

GMP 14 Page 2 of 4 

2. Methodology 

2.1. Summary 

A sensitivity weight is selected to calibrate the balance over the range to be 

used in the measurement procedure. Minimizing the difference in mass values 

between X and S is critical when choosing an appropriate sensitivity weight. 

Therefore, tare weights may be necessary whenever the difference in mass 

values is significant. Minimizing the difference between X and S works to our 

benefit since the range of the measurements is minimized and reduces potential 

errors that can be introduced by nonlinearity of the balance. 

2.2. Apparatus 

2.2.1. Sensitivity weights with accurate and recent calibration values and 

which have demonstrated metrological traceability to the international 

system of units (SI), which may be to the SI through a National 

Metrology Institute such as NIST. 

2.2.2. Tare weights with accurate recent calibration values and which have 

demonstrated metrological traceability to the international system of 

units (SI), which may be to the SI through a National Metrology 

Institute such as NIST. 

2.2.3. Clean forceps to handle the weights, or gloves to be worn if the 

weights are to be moved by hand. 

2.3. Procedure for selection 

2.3.1. Conduct preliminary measurements to determine the approximate 

mass value for the difference between the standard and the unknown 

(X – S). 

2.3.2. Define the range of use for the balance to be used: 

2.3.2.1. Equal arm – number of scale divisions 

2.3.2.2. Mechanical – optical scale 

2.3.2.3. Combination – digital indications 

2.3.2.4. Fully electronic – capacity 

2.3.2.5. Comparators – digital indications 
February 2012 

GMP 14 Page 3 of 4 

2.3.3. Determine the need for tare weights if the difference between X and S

exceeds the values shown in Table 1. 

Table 1. Maximum allowed difference between X and S.

Balance (X – S) 

Equal arm balance each other within 

one division on the scale 

Mechanical 1/10 optical scale 

Combination 1/10 digital range 

Fully electronic 0.05 % capacity 

Comparator 1/10 digital range 

2.3.4. Select tare weights, if necessary, making sure that the difference 

between X and S, with the appropriate tare weights, do not exceed the 

values shown in Table 1. 

2.3.5. Select a sensitivity weight according to Table 2. 

Table 2. Selection of Sensitivity Weight. 

Balance Procedure Sensitivity Weight 

Equal Arm SOP 3, 5, 6, 7, 8, 28 change turning points by 20 % 

Mechanical 

SOP 4, 5, 7, 28 

SOP 8 

≥ 4 times (X – S);  1⁄2 optical 

scale 

 1⁄4 optical scale 

Combination 

Electro-mechanical 

SOP 4, 5, 7, 8, 28 ≥ 4 times (X – S);  1⁄2 digital 

Range 

Fully Electronic SOP 4, 5, 7, 28 

SOP 8 

 1 % capacity 

2 times the applicable tolerance 

Comparator* SOP 4, 5, 7, 8, 28 ≥ 4 times (X – S); < 1⁄2 digital 

range 

*A sensitivity weight is not required if the electronic mass comparator that is used has been 

tested (with supporting data available) to determine that the balance has sufficient accuracy, 

resolution, repeatability, and stability so that no advantage is gained using a sensitivity 

weight. For example, any possible errors must be less than the last digit retained in the 

expanded uncertainty. When a mass comparator is used without a sensitivity weight, the 

sensitivity must be periodically verified and documented (e.g., prior to each use). 

Monitoring will be required to verify metrological traceability. 

2.4. Use of sensitivity weight 

The sensitivity weight is used to ensure that the mass differences determined 

with the optical scale, or electronic range, have valid accuracy and traceability. 

The sensitivity weight calibrates the range of use of the balance used for 

making the mass determinations. Using a sensitivity weight provides us with a 

sensitivity value in terms of mass units per division. 
February 2012 

GMP 14 Page 4 of 4 

deflection

M divisions

mass units sensitivity sw  

Where Msw represents the mass of the sensitivity weight. 

3. Calculations 

No special calculations are associated with this practice other than the one noted in 

section 2.4. See each mass SOP for calculation of sensitivity as it is included in the 

procedure. 

4. Uncertainty 

No uncertainty calculations are associated with this practice. See the appropriate SOP 

for the calculations of uncertainty. The uncertainty of the sensitivity weight does not 

need to be included in calculations of uncertainty since the uncertainty value is 

distributed across the range of use. However, it does no harm to incorporate it in 

uncertainty calculations when spreadsheets are set up to handle all of the data. 

Sensitivity errors that may be incorporated in SOP 8 where sensitivity is assessed but 

not included in the calculations need to be evaluated and included as an uncorrected 

systematic error in the uncertainty, according to instructions in SOP 8. 
Standard Operating Procedures 

 
This page intentionally made blank.
February 2012 

SOP 1 Page 1 of 8 

SOP 1 

Recommended Standard Operating Procedures 

for 

Preparation of Test/Calibration Reports 

1. Introduction 

1.1. Test/calibration reports are the visible outputs of the testing laboratory. They should be 

prepared with utmost care to ensure that they accurately convey all information 

pertaining to the testing so that reports may be used with maximum benefit by all 

concerned. Carefully prepared test reports will contain or refer to all information 

necessary to justify the test results. 

1.2. The test report may consist of filling in the blanks in a form in the case of a routine 

measurement. A more detailed report, including narrative information, may be required 

for special calibrations or tests. 

1.3. Regardless of the final form, the test report must contain the basic information 

described in the following sections (see also Section 5.10 of ISO/IEC 17025 and the 

attached Checklist for review of template reports and those provided by calibration 

suppliers). 

2. Content 

2.1. Title (e.g., “Test Report” or “Report of Calibration”). 

2.2. Name and address of the laboratory, or location at which tests were performed. 

2.3. Unique identification of the test report or calibration certificate, and on each page an 

identification in order to ensure that the page is recognized as part of the test report or 

calibration certificate, and a clear identification of the end of the report or certificate. 

2.4. Name and address of the client. 

2.5. Method used – Describe how test was made by reference to SOP(s). In the absence of 

SOP’s, brief but informative descriptions of the methodology should be included. 

Information describing deviations from previously agreed upon procedure must also be 

included. 

2.6. Description of, the condition of, and unambiguous identification of the item calibrated 

(e.g., description and/or serial numbers). A laboratory number should be assigned and 

attached to each test item at the time of its acceptance for testing. The use of the 

laboratory number will facilitate the internal control of test items during the testing 

process. 
February 2012 

SOP 1 Page 2 of 8 

2.7. Date of receipt of calibration item where this is critical to the validity and application of 

the results, and the date of performance of calibration. 

2.8. Reference to sampling plan or procedures, where relevant. 

2.9. Calibration results with the units of measurement in tabular or other convenient form. 

(When an instrument or standard has been repaired or adjusted the calibration results 

before and after repair or adjustment are reported.) 

2.10. Name, title, and signature of person authoring the report or certificate. Other signatures 

may be required, at the discretion of the laboratory director. Each signer accepts 

his/her share of responsibility for the technical accuracy of the report contents. 

2.11. Where relevant, a statement to the effect that the results relate only to the items tested 

or calibrated. 

2.12. Conditions (e.g., environmental) under which the calibrations were made that have an influence 

on the measurement results.

2.13. A statement of the estimated measurement uncertainty, components that were 

considered and included, a rationale for their inclusion, and the coverage factor and 

estimated confidence interval. 

2.14. Include evidence to support metrological traceability and a traceability statement. E.g., 

identification of the standards used and their traceability to national or international 

standards and to the International System of Units (SI), through a national metrology 

institute as appropriate. 

2.15. Where relevant, a statement of compliance/non-compliance with requirements and/or 

specifications. Compliance refers to all criteria of a referenced standard and not just 

portions (e.g., compliance to all specifications and tolerances versus tolerance only). 

2.16. Where appropriate and needed, opinions and interpretations. When included, opinions 

and interpretations are clearly stated. 

2.17. Additional information which may be required by specific methods, clients or groups of 

clients. 

2.18. Hard copies of test reports should also include the page number and total number of 

pages. 

2.19. A statement specifying that the test report or calibration certificate shall not be 

reproduced except in full, without written approval of the laboratory. 

2.20. Calibration intervals only when required for legal applications or where requested by 

the customer. 
February 2012 

SOP 1 Page 3 of 8 

3. Records 

3.1. File all test reports in a systematic manner for ease of retrieval, as necessary. 

3.2. Retain copies of all test reports for a minimum period of five years, until superseded by 

a subsequent report, or as stated in the laboratory quality manual, or until deemed by 

the laboratory director as having no future value. 
February 2012 

SOP 1 Page 4 of 8 

APPENDIX A - Recommended Format for Report of Test 

Report of Test 

Issued by 

Name of Testing Laboratory 

Laboratory Report No. ______________ 

 

Test Item(s)/Lab No(s): ______________________________________________________________________ 

Submitted by: ____________________________________________________________________________ 

 

Date: __________________________________________________________________________________ 

 

Purpose of Test: ____________________________________________________________________________ 

 

Condition: __________________________________________________________________________________ 

 

________________________________________________________________________________________ 

Test Results*: 

 

As Found After Adjustment Uncertainty 

 

 

 _______________________________________________________________________________________ 

Reference Information: 

 

Test Method: ______________________________________________________________________________ 

 

Traceability Statement: ________________________________________________________________________ 

Uncertainty Statement: ________________________________________________________________________ 

Conditions of Test: ___________________________________________________________________________ 

 

Data 

Reference: ______________________________________________________________________________ 

 

Test Results Approved by (name, title, date): ___________________________________________ 

 

 

*Report, as appropriate 

The results stated on this report relate only to the items specifically identified. 

This test report or calibration certificate shall not be reproduced except in full, without written approval of the laboratory.
February 2012 

SOP 1 Page 5 of 8 

APPENDIX B – Recommended Format for Calibration Certificate 

A B C COMPANY 

123 Utopia Street 

Anywhere, USA 

 

Report of Calibration 

 

Report Number: 

Name of Device: 

 

Model: Serial No: 

 

Submitted by: 

 

 

Calibration (date) . 

The ambient conditions were C, % relative humidity and mm Hg Barometric pressure. 

The item tested was/was not in tolerance at time of calibration. Adjustments are noted and any out of tolerance data are 

attached. 

 

Data: 

 

Nominal Value As Found Correction 

or Error 

As Left Correction 

or Error 

Expanded 

Uncertainty 

 

 

The primary standards to which the above data are traceable are identified in this report. The calibration of these 

standards is traceable to the International System of Units (SI) through the National Institute of Standards and 

Technology. 

 

Identity of Standard Traceability Reference Calibration Date Next Calibration Due 

 

Test Method: ______________________________________________________________________________ 

 

Uncertainty Statement: ______________________________________________________________________ 

 

Test Results Approved by (name, title, date): ___________________________________________ 

 

 

The results stated on this report relate only to the items specifically identified. 

This test report or calibration certificate shall not be reproduced except in full, without written approval of the laboratory.
February 2012 

SOP 1 Page 6 of 8 

APPENDIX C - Example 

BUREAU OF STANDARDS 

PO Box 12345 

City, State 12345-1234 

COMPLIANT CALIBRATION LABORATORY 

123 Some Ave. 

City, State 12312-1231 

REPORT OF CALIBRATION 

FOR 

1 kg to 10 mg weight kit 

(Twenty-one metric weights) 

Maker: DENTROM LAKE Lab Test No.: TI-01-056 

Serial No.: 27269 NMI Test No.: 822/1234 

SUBMITTED BY 

YOUR CUSTOMER, INC. 

Customer’s Address 

City, State 

Nominal 

(g) 

Conventional Mass 

(g) 

Conventional Mass 

Correction 

(mg) 

Expanded 

Uncertainty 

(mg) 

1 000 1 000.000 82 0.82 0.92 

 500 500.000 71 0.71 0.53 

 300 299.999 87 - 0.13 0.27 

 200 200.000 67 0.67 0.18 

 100 100.000 411 0.411 0.091 

 50 50.000 318 0.318 0.051 

 30 30.000 117 0.117 0.028 

 20 19.999 987 - 0.013 0.023 

 10 10.000 011 0.011 0.018 

 5 5.000 022 0.022 0.015 

 3 3.000 112 0.112 0.013 

 2 1.999 965 - 0.035 0.012 

 1 1.000 117 0.117 0.010 

 0.500 0.500 013 2 0.013 2 0.005 1 

 0.300 0.300 022 3 0.022 3 0.004 8 

 0.200 0.200 001 7 0.001 7 0.004 3 

 0.100 0.100 001 3 0.001 3 0.004 2 

 0.050 0.050 001 8 0.001 8 0.004 0 

 0.030 0.030 001 1 0.001 1 0.003 7 

 0.020 0.020 000 9 0.000 9 0.003 3 

 0.010 0.009 999 7 - 0.000 3 0.003 1 

The data in the above table of this report only applies to those items specifically listed on this report. 

Uncertainty statement: 

The combined standard uncertainty includes the standard uncertainty reported for the standard, the standard uncertainty for the measurement 

process, the standard uncertainty for any uncorrected errors associated with buoyancy corrections, and a component of uncertainty to account for 

any observed deviations from NIST values that are less than surveillance limits. The combined standard uncertainty is multiplied by a coverage 

Test No.: TI-01-056 

Page 1 of 2
February 2012 

SOP 1 Page 7 of 8 

factor of 2 to provide an expanded uncertainty, which defines an interval having a level of confidence of approximately 95 percent. The expanded 

uncertainty presented in this report is consistent with the ISO/IEC Guide to the Expression of Uncertainty in Measurement. The expanded 

uncertainty is not to be confused with a tolerance limit for the user during application. 

Traceability statement: 

The Standards of the Compliant Calibration Laboratory are traceable to the International System of Units (SI) through the National Institute of 

Standards and Technology, and are part of a comprehensive measurement assurance program for ensuring continued accuracy and measurement 

traceability within the level of uncertainty reported by this laboratory. The laboratory test number identified above is the unique report number to 

be used in referencing measurement traceability for artifacts identified in this report only. 

Supplemental Information 

Description of artifacts submitted for testing: 

Twenty one metric weights from 1 kg to 10 mg, marked ASTM Class 4. Weights from 1 kg to 1 g: two-piece weights, with assumed density of 

8.0 g/cm3

. Weights from 500 mg to 50 mg: sheet weights, with assumed density of 16.6 g/cm3. Weights from 30 mg to 10 mg: sheet weights, 

with assumed density of 2.7 g/cm3

. 

Conditions of artifacts submitted for testing: 

Artifacts showed evidence of improper handling. Fingerprints and dents were visible on the surface of the weights. 

Treatment of artifacts prior to testing: 

Artifacts were cleaned with cheesecloth and ethyl alcohol. Thermal equilibrium time/conditions: ten days next to balances in mass lab. 

Equipment & Standards: 

Balance Range Standards Used Calibration due 

AT1005 1 kg to 200 g Set H 2/31/2002 

AT106 100 g to 10 g Set H 2/31/2002 

UMT5/6 5 g to 10 mg Set H 2/31/2002 

Assumed Density of Reference Standards: 

1 kg to 1 g: 7.94 g/cm3

 500 mg to 10 mg: 8.41 g/cm3

Procedure used:

Double Substitution (NIST HB 145, SOP 4) 

Environmental conditions at time of test: 

Temperature: 20.1 C to 20.2 C Barometric Pressure: 752.7 mm Hg Relative Humidity: 43.35 % to 43.40 % 

Date artifacts were received: February 15, 2001 Date of report preparation: March 3, 2002 

Date of test: February 25, 2001 

Due date per customer’s request: February 25, 2002 

Josh Balani II 

Test performed by: Josh Balani II 

 Metrology Expert 

Member: ARMAP 

NCSLI 

NCWM 

ASQ 

This document does not represent or imply endorsement by the Compliant Calibration Laboratory, NMI, or any agency of the State and/or 

national governments. This document may not be reproduced, except in full, without the written permission of the Compliant Calibration 

Laboratory. 

Test No.: TI-01-056 

Page 2 of 2 
February 2012 

SOP 1 Page 8 of 8 

Appendix D - Calibration Certificate and Test Report Review Checklist 

ISO/IEC 17025:2005 (NIST HB 143:2007), Section 5.10 - Reporting the Results 

Criteria Guide Compliance 

5.10.1 Reported accurately, clearly, unambiguously and objectively; in accordance with method instructions; 

include all info requested by customer, necessary for interpretation of the results, & required by the method 

Test Reports and Calibration Certificates Include at least the following information, unless valid reasons for not

5.10.2 a) Title 

5.10.2 b) Laboratory name & address; location where tests/calibrations carried out, if different from lab 

5.10.2 c) Unique certificate/test report ID (e.g., SN #); ID on each page; clearly ID document end 

Note: Hard copies should include page number and total number of pages 

5.10.2 d) Customer name & address 

5.10.2 e) ID method used 

5.10.2 f) Description of, the condition of, & unambiguous identification of the item(s) calibrated/tested 

5.10.2 g) Item(s) receipt date where critical to validity/application of results; date(s) of cal/test performance 

5.10.2 h) Reference to sampling plan & procedures used where relevant to validity or application of results 

5.10.2 i) Calibration/test results with units of measurement, where appropriate 

5.10.2 j) Name(s), function(s) & signature(s) or equivalent ID of person(s) authorizing the certificate/report 

5.10.2 k) Where relevant, a statement to the effect that the results relate only to the items tested or calibrated 

Note: Recommend statement shall not be reproduced except in full, without written lab approval 

Test Reports In addition to 5.10.2, test reports shall include the following

5.10.3.1a) Deviations, additions, exclusions from method; info on specific test conditions (e.g., environmental 

conditions) 

5.10.3.1b) Statement of compliance/non-compliance with requirements and/or specifications, where relevant 

5.10.3.1c) Statement on estimated uncertainty, where applicable; uncertainty info needed when: relevant to results 

validity/application, customer’s instructions requires, or uncertainty affects specification limit compliance 

5.10.3.1d) Opinions and interpretations, where appropriate and needed 

5.10.3.1e) Additional information which may be required by specific methods, customers or groups of customers 

Test Reports Containing the results of sampling

5.10.3.2 a-f) 

Sampling date, ID substance, material or product, location, diagrams, sketches, photographs; reference plan 

& procedures; environmental conditions; method/procedure, deviations, additions, or exclusions from 

specification 

Calibration Certificates In addition to 5.10.2, where necessary

5.10.4.1 a) Calibration conditions (e.g., environmental) that have an influence on the measurement results 

5.10.4.1 b) Uncertainty and/or statement of compliance with an identified metrological specification or its clauses 

5.10.4.1 c) Evidence that the measurements are traceable 

5.10.4.2 Relate only to quantities/ results of functional tests. IF compliance statement is made: ID clauses met/not 

met & account for uncertainty; Record & maintain results for reference if results/uncertainties omitted 

5.10.4.3 Report before and after adjustment or repair results, if available, when adjusted/repair are made 

5.10.4.4 No calibration interval on certificate/label except if agreed w/customer (superseded by legal regulations) 

Opinions and Interpretations 

5.10.5 Document basis of opinions and interpretations when included; clearly mark them 

Format of Reports and Certificates 

5.10.8 The format shall be designed to accommodate each type of test or calibration carried out and to minimize 

the possibility of misunderstanding or misuse 

Amendments to Test Reports and Calibration Certificates 

5.10.9 Amendments after issue made as further document/data transfer with statement, meet all requirements; New 

report/certificate is uniquely identified & reference to the original that it replaces 
February 2012 

SOP 2 Page 1 of 20 

SOP 2 

 

Recommended Standard Operating Procedure 

for 

Applying Air Buoyancy Corrections 

1. Introduction 

1.1. Purpose 

If uncorrected, the effect of air buoyancy on the objects being compared is frequently 

the largest source of error in mass measurement. This SOP provides the equations to 

be used to correct for the buoyant effect of air. The significance of the air buoyancy 

correction depends upon the accuracy required for the measurement, the magnitude 

of the air buoyancy correction relative to other sources of error in the overall 

measurement process, and the precision of the mass measurement. An air buoyancy 

correction should be made in all high accuracy mass determinations. The gravimetric 

volume procedure uses a high accuracy mass determination with the corresponding 

buoyancy corrections. The Appendix to this SOP provides a brief theoretical 

discussion of this subject. 

1.2. Prerequisites 

1.2.1.Verify that (true) mass values or corrections are available for the standards 

used in the measurement process and that they have demonstrated metrological 

traceability to the international system of units (SI), which may be to the SI 

through a National Metrology Institute such as NIST. 

1.2.2.Verify that the thermometer, barometer, and hygrometer used have been 

calibrated, and that they have demonstrated metrological traceability to the 

international system of units (SI), which may be to the SI through a National 

Metrology Institute such as NIST, and are in good operating condition as 

verified by periodic tests or cross-checks with other standards. 

 

2. Methodology 

2.1. Scope, Precision, Accuracy 

This procedure is applicable to all weighings using a comparison of mass standards. 

The precision will depend upon the accuracy of the thermometer, barometer, and 

hygrometer used to determine the air density. When the calculations for the air 

density and air buoyancy corrections are made, a sufficient number of decimal places 

must be carried so the error due to the rounding of numbers is negligible relative to 

the error in the measurement process. Typically, carrying six to eight decimal places 

is sufficient, but depends upon the precision of the measurement. 
February 2012 

SOP 2 Page 2 of 20 

2.2. Apparatus/Equipment Required1

2.2.1.Barometer accurate to ± 66.5 Pa (0.5 mm Hg) to determine air pressure. 

2.2.2.Thermometer accurate to ± 0.10 C to determine air temperature. 

2.2.3.Hygrometer accurate to ± 10 % to determine relative humidity. 

2.3. Estimating the Magnitude of the Air Buoyancy Correction 

2.3.1. Estimate the magnitude of the air buoyancy correct, MABC, using the following 

formula: 

MABC ( ) (V V )  a  n x  s

2.3.2. The equation may also be represented as follows: 















    1 1 MABC m 

x s

o a n  

(   )

 Table 1. Variables for MABC equation. 

Variable Description of Variable 

a air density at the time of the measurement in 

mg/cm3

n density of "normal" air; i.e., 1.2 mg/cm3

mo nominal mass (in grams) 

Vx volume of the unknown weight, X in cm3

Vs volume of the reference standard, S in cm3

s density of reference standard, S in g/cm3

x density of unknown weight, X in g/cm3

The relative magnitude of the correction can be compared to the expanded 

measurement uncertainty to determine the importance of the air buoyancy correction 

and the uncertainty components for a particular measurement. 

 

1

 The barometer, thermometer, 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.
February 2012 

SOP 2 Page 3 of 20 

2.4. Procedure 

2.4.1.Record the temperature, pressure, and relative humidity at the start and at the 

end of the measurement process as near the location of the measurement as 

necessary and practical. If these parameters change significantly during the 

measurement process, it may be necessary to wait for more stable operating 

conditions or to use average values to compute the air density. Use of the 

average environmental values may influence the uncertainty of the 

measurement result and must be evaluated for significance. 

2.4.2.Determine the air density using the equation given in Section 8 of the 

Appendix to this SOP. 

3. Calculations 

3.1. Calculate the mass, Mx, of the unknown weight, X, using the following equation, 

where d represents the “difference” obtained with buoyancy corrections applied to 

the sensitivity weight. 















 

 













 



1 

1 

 

s

x

a

s

a

x

M d

M









If tare weights were carried with X and/or S, use the following equation: 















 















   















   













 



1 

M 1 d M 1 M 1 

M 

x

a

t

a

t

t

a

t

s

a

s

x

x

x

s

s
















February 2012 

SOP 2 Page 4 of 20 

Table 2. Variables not previously defined. 

Variable Description of Variable 

d measured difference between X and the reference standard, S, using one of 

the weighing designs given in other SOPs 

Ms [true] mass of the reference standard 

s M t [true] mass of the tare weights carried with S 

x M t [true] mass of the tare weights carried with X 

s density of the reference standard, S 

x density of the unknown standard, X 

s t density of the tare weights carried with S 

x t density of the tare weights carried with X 

3.2. If reporting the conventional mass2

, CMx, compute it using the following. 





 



 















 



8 0

0 0012 1 

0 0012 M 1 

CM x

x

x

.

.

.



3.3. If reporting the apparent mass, AMx, versus brass, compute it using the following. 





 



 















 



8 3909

0 0012 1 

0 0012 M 1 

AM x

x

x vs brass

.

.

.



3.4. The conventional and apparent mass values are related by the following: 





 



 





 



 



8 0

0 0012 1 

8 3909

0 0012 AM 1 

CM 

x vs brass

x

.

.

.

.

 

2

 Conventional Mass: “The conventional value of the result of weighing a body in air is equal to the mass of a 

standard, of conventionally chosen density, at a conventionally chosen temperature, which balances this body at this 

reference temperature in air of conventionally chosen density.” The conventions are: reference density 8.0 g/cm3

; 

reference temperature 20 C; normal air density 0.0012 g/cm3. Conventional mass was formerly called “Apparent 

Mass versus 8.0 g/cm3

” in the United States and is not recommended. See OIML D28 (2004). 
February 2012 

SOP 2 Page 5 of 20 

4. Assignment of Uncertainty 

The uncertainty in determining the air buoyancy correction is usually negligible relative to 

the precision of the measurement process itself. Consequently, the uncertainty for the 

measurement is based upon the uncertainty for the measurement process used and the 

uncertainty associated with the environmental measurement standards. The uncertainty in 

the CIPM3

 2007 air density equation is 0.000 026 4 mg/cm3 (or 0.002 2 % of normal air 

density.) 

 

Table 3. Tolerances for measurements related to air density estimation. 

 Uncertainty of air density values in % of air density 

Variable ± 0.1 % of air 

density 

± 1.0 % of air density Recommended 

(Section 2.2) 

Air pressure (Pa) ± 101 ± 1010 ± 66.5 

Air pressure (mm Hg) ± 0.76 ± 7.6 ± 0.5 

Air temperature ( C) ± 0.29 ± 2.9 ± 0.1 

Relative Humidity (%) ± 11.3 ............ ± 10 

 

3

 CIPM References: A Picard, R.S. Davis, M. Gläser, K. Fujii, Metrologia 45:149-155 (2008).

 
February 2012 

Appendix SOP 2 Page 6 of 20 

Appendix A 

 

Based on “The Basic Theory of Air Buoyancy Corrections” 

 

by Richard S. Davis4 

 

1. Introduction 

In performing measurements of mass, the balance or scale used acts as a force transducer. 

The force produced by an object to be weighed in air has two components: one 

proportional to the mass of the object, and the other proportional to its volume. The latter 

component, or buoyant force, may under some circumstances be large enough to require 

correction. The following shows under what circumstances buoyancy corrections are 

required as well as how they are made. 

2. Scope 

The method for applying buoyancy corrections presented below applies to mass 

measurements made in air. The density of air is computed from auxiliary measurements of 

temperature, pressure and relative humidity after which the buoyancy corrections are 

calculated directly from the Principle of Archimedes. The following weighing situations 

are considered. 

2.1. Two-Pan Balance 

 

2.2. Single-Pan Balance 

2.2.1.With Built-In Weights 

2.2.2.With Electronic Control 

3. Summary of Method 

In general, buoyancy corrections are applied to mass measurements by calculating the 

difference in volume between the unknown weight and the standard, multiplying this 

volume difference by the density of air at the balance or scale, and adding the product to 

the mass of the standard. The density of air is computed from an equation of state using 

measured values for the temperature, pressure and relative humidity of the air. 

4. Significance and Use 

Buoyancy corrections generally must be applied when determining the mass of an 

unknown object to high accuracy. The corrections may become important even at modest 

 

4

Richard S. Davis, formerly of the National Institute of Standards and Technology, Mass Group, International 

Bureau of Weights and Measures (BIPM), retired. 
February 2012 

Appendix SOP 2 Page 7 of 20 

accuracies if the unknown object whose mass is to be determined has a density that differs 

widely from that of the standards (weighing of water, for example). Many mass standards 

are calibrated in terms of a so-called "apparent mass" [conventional mass] scale (See 

Chapter 7.3, Handbook 1451

). Use of this scale does not indiscriminately eliminate the 

need for buoyancy corrections as is sometimes assumed. 

5. Terminology 

5.1. Weighing by Substitution 

 

Substitution weighing is the procedure by which one determines the assembly of 

standard weights that will produce nearly the same reading on a one-pan balance as 

does the unknown object. The balance thus serves as a comparator. A two-pan 

balance may be used in this mode if one of the pans contains a counterpoise and the 

standards and unknown are substituted on the second pan. (See SOP No. 3.) 

5.2. Mass 

The term "mass" is always used in the strict Newtonian sense as a property intrinsic 

to matter. Mass is the proportionality constant between a force on a material object 

and its resulting acceleration. This property is sometimes referred to as "true mass", 

"vacuum mass", or "mass in vacuum" to distinguish it from conventional [apparent] 

mass. 

5.3. Conventional [Apparent] Mass5

The mass of material of a specified density that would exactly balance the unknown 

object if the weighing were carried out at a temperature of 20 C in air of density 

0.0012 g/cm3

. The mass, MN, of an object, N, is related to its apparent mass MN,A by 

the equation: 















 















 

 0.0012 1 

0.0012 1 

 ,

B

N

N

N A

M

M





 

5

Pontius, P. E., Mass and Mass Values, NBS Monograph 122, 1974, pp 12-23, 26-33 described the concept of 

apparent mass. The term conventional mass is described by OIML D28, the Conventional Value of Mass in Air. 
February 2012 

Appendix SOP 2 Page 8 of 20 

Table A-1. Variables for conventional (apparent) mass equation.

Variable Description 

N density of the object N at 20 °C in g/cm3

B

density of the conventional (apparent) mass scale at 

20 °C in g/cm3

 

There are at present two apparent mass scales in wide use. The older is based on 

B = 8.4000 g/cm3 at 0 C with a coefficient of volumetric expansion of 0.000054 /C 

and the more recent (Conventional Mass) specifies B = 8.0000 g/cm3

 at 20 C. The 

quantity MN,A is a function of the particular conventional or apparent mass scale, 

which has been used in its calculation. OIML D28 only recognizes Conventional 

Mass. 

5.4. Sensitivity 

The response of a balance under load to an additional small weight: 

R

M

R

M V sensitivity sw

a

sw

sw a sw

















 

 

 

1 

 





Table A-2. Variables for sensitivity equation. 

 Variable Description 

sensitivity balance sensitivity (mass per division) 

Msw mass of the small, additional weight 

a density of the air 

sw density of the small, additional weight 

Vsw volume of the small, additional weight 

R change in balance reading due to the addition of the small weight, 

balance deflection 

6. Apparatus 

In order to ascertain the density of air at the balance, the following measuring instruments 

are necessary: thermometer, barometer, and hygrometer. Ideally, these instruments should 

be placed in or next to the balance case (as near the measurement location as is practical). 

It may only be practical for the thermometer or temperature sensor to actually be placed 

inside the balance chamber. A calculator or computer will be extremely useful for this 

procedure. 
February 2012 

Appendix SOP 2 Page 9 of 20 

7. Procedure 

Weigh the unknown object as directed by the balance manufacturer or in accordance with 

accepted procedure. Record the temperature, pressure and relative humidity of the air in 

the balance at the time of weighing. Do not correct the barometric pressure to sea level. 

8. Calculation 

8.1 Air density, Option A (Option B is preferred) 

The density of air, in g/cm3

, can be approximated for lesser accuracy from the 

following formula:6

   

 

3 x 10

273.15 

0.464 60 0.003 796 0 















  tP U es a (1) 

 Table A-3. Variables for air density equation. 

Variable Description 

a density of air, g/cm3

P barometric pressure, mm Hg 

U % relative humidity, entered as a whole number 

t temperature, °C 

es 1.314 6 x 109

 x e[-5 315.56/(t + 273.15)]

 

Note: es can be written as follows in a spreadsheet and in some calculators: 

1.3146E9*@EXP(-5315.56/(t+273.15)) 

Small errors (of order 0.01 %) in this equation occur for locations well above sea level or 

under conditions in which the concentration of carbon dioxide differs greatly from the 

global average. See the references for a more general formulation of the equation. 

 

6

Jones, F.E., "The Air Density Equation and the Transfer of the Mass Unit," Journal of Research, National Institute 

of Standards and Technology, Vol. 83, 1978, p. 419. 
February 2012 

Appendix SOP 2 Page 10 of 20 

8.2. Air density, Option B – Preferred 

The density of air should be calculated with the following formula.7

Table A-4. Variables for CIPM air density equation. 

Variable Description 

Ma molar mass of the air within laboratory 28.965 46 x 10-3 kg/mol 

p ambient barometric pressure in Pascal 

T ambient temperature in Kelvin 

R universal gas constant: 8.314 472 J mol-1 K-1

h relative humidity in % 

f 1.000 62 + (3.14 x 10-8) p + (5.6 x 10-7)t

2

t ambient temperature in degrees Celsius 

psv 1 Pascal x exp (AT2 + BT + C + D/T) 

A 1.237 884 7 x 10-5 K-2

B -1.912 131 6 x 10-2 K-1

C 33.937 110 47 

D -6.343 164 5 x 103 K 

a0 1.581 23 x 10-6 K Pa-1

a1 -2.933 1 x 10-8 Pa-1

a2 1.104 3 x 10-10 K-1 Pa-1

b0 5.707 x 10-6 K Pa-1

b1 -2.051 x 10-8 Pa-1

c0 1.989 8 x 10-4 K Pa-1

c1 -2.376 x 10-6 Pa-1

d 1.83 x 10-11 K2 Pa-2

e -0.765 x 10-8 K2 Pa-2

 

7

 CIPM References: Giacomo, P. Metrologia 18: 33-40 (1982), Davis, R.S., Metrologia 29: 67-70 (1992), A Picard, 

R.S. Davis, M. Gläser, K. Fujii, Metrologia 45:149-155 (2008). 

 

 1        

 / 100 

M

M 1 1-

2

2

2

2

0 1 0 1

2

0 1 2

a

v

v v v

sv

v

v

a

a

d ex

T

p a a t a t b b t x c c t x

T

p Z

p

x h f

x

ZRT

pM

         

































  




February 2012 

Appendix SOP 2 Page 11 of 20 

Calculate the density of air at the balance during the weighing. Then determine the 

mass of the unknown, Mx, as follows: 

8.3 If a two-pan balance is used: 

 

 (3c)

 

M V sensitivity M

M 

or

 (3b)

 

M sensitivity M

M 

or

M M V V sensitivity M ( a)

x

a

s a s opt

x

x

a

opt

s

a

s

x

x s a x s opt















 

   















 

  













 



    

















1

1

1

3

Table A-5. Variables not previously defined. 

Variable Description 

Mx mass of the unknown object 

Ms mass of the standard weights 

s density of the standard weights, 

S

S

V

M

Mopt off-balance indication read on the optical scale 

Vx volume of the unknown object 

Vs volume of the standard weights 

x density of the unknown object, 

X

X

V

M

Volumes and densities are, in general, a function of temperature. The thermal 

coefficients of volumetric expansion of the unknown object and the standard may be 

significant in very accurate work.8

 The coefficient of volumetric expansion is 

 

8

In general, Vt = V20 [1 + ß(t - 20 C)] where t is the temperature of the weight, V20 is the volume at 20 C, and ß is 

the cubical coefficient of expansion. 
February 2012 

Appendix SOP 2 Page 12 of 20 

usually estimated as three times the linear coefficient of expansion of the weight 

material. 

The error in Mx incurred by ignoring the buoyancy correction is a (Vx - Vs). To 

estimate quickly whether such an error is of consequence in a particular 

measurement, (assume a = 1.2 x 10-3g/cm3

). 

If the mass and volumes of the standards have been adjusted to a conventional mass 

scale, then 















 

  













 



 

CM sensitivity M

M 

x

a

opt

B

a

s

x









1

1

 (4) 

Table A-6. Variables not previously defined. 

Variable Description 

CMs conventional mass of the standard 

The symbol  signifies an approximation 

8.4 If a single-pan balance with built-in weights is used, it is probable that the built-in 

weights have been adjusted on an apparent mass or conventional mass scale9

. 

Determine which apparent mass scale has been used and calculate the mass of the 

unknown from the equation 















 

 













 



 

M sensitivity M

M 

x

a

opt

B

a

D

x









1

1

 (5) 

Table A-7. Variables not previously defined. 

Variable Description 

MD mass indicated by dial or digital readings 

Mopt mass indicated on the optical scale when present 

 

9

Schoonover, R. M. and Jones, F. E., "Air Buoyancy in High-Accuracy Weighing on Analytical Balances," Anal. 

Chem., 53, 1981, p. 900. 
February 2012 

Appendix SOP 2 Page 13 of 20 

If the balance has been used only as a comparator, that is, to compare the mass of the 

unknown object with that of some external standard, then: 

 

  x s a x s opt M  M   V  V  sensitivity M'

Table A-8. Variable not previously defined. 

Variable Description 

M’opt

difference in optical scale reading between observations of the 

standard and the unknown 

For some balances, operation requires that the user restore the balance to null by 

means of a manually controlled dial. The portion of the mass reading controlled by 

this dial should be treated, for purposes of buoyancy corrections, as an optical scale. 

8.5 If a single-pan balance with full-range electronic control is used, the following 

should be noted. As part of its calibration, the electronic gain has been adjusted by 

means of a weight of known mass. For example, if the range of electronic control is 

100 g, the electronics have been adjusted so that a 100-g standard weight produces 

an indication of precisely 100 g. This procedure effectively builds an apparent mass 

calibration into the balance. The reference density of the apparent mass scale is the 

density of the standard mass used for the calibration and the reference air density is 

the air density at the time of calibration. 

The mass of an unknown object weighed on the balance is then 















 















 



 

M 

M 

x

a

c

a

R

x









1

1

 (6) 

Table A-9. Variables not previously defined. 

Variable Description 

MR readout displayed on the balance 

a density of air at the time of balance calibration 

c

density of the standard used to calibrate the balance (or B if the 

apparent [conventional] mass of the standard was used instead of the 

true mass 

 

If the balance includes both an electronic control system and built-in weights, the 

buoyancy considerations for the built-in weights are as described in section 8.2 and 

the considerations for the electronically determined mass are those given directly 

above. 
February 2012 

Appendix SOP 2 Page 14 of 20 

8.6 Top-loading balances may be considered a form of single-pan balance and the 

appropriate procedure for buoyancy correction followed. 

9. Precision 

The contribution of the random error of the evaluation of air density to the precision of 

mass measurement may be estimated as follows: 

For mechanical balances, or electronic balances used in weighing by substitution, the 

contribution is: 

 V V   a x  s

Table A-10. Variables for above equation. 

Variable Description 

 a random error of evaluation of a

Vs

volume of standards, if weighing by substitution 

Vs = MD / DB, if using the built-in weights on a single pan balance. 

Vx volume of object weighed 

The quantity, a will have contributions from the measurements of temperature, pressure 

and relative humidity which are required for the calculation of a. Equation (1) may be 

used to estimate the effects of imprecision in measurements of P, t, and U. It is unrealistic 

to expect a /a ever to be less than 0.05 % even using the best techniques available. 

 

10. Accuracy 

Inattention to problems of buoyancy most often results in systematic errors. For a 

substitution weighing, for example, the buoyancy correction is of the order of a(Vx - Vs). 

While this quantity may be significant to a measurement, day-to-day variation in a

(usually no more than 3 %) may not be noticeable and hence need not be measured. For the 

most accurate work, not only must a be accurately determined, but the volumes of the 

unknown and standard may have to be measured to better than 0.05 % - the minimum 

systematic uncertainty attainable in the calculation of a. 

 

If the standards have been calibrated in terms of conventional mass, complete neglect of 

buoyancy corrections will produce an error in the measured result Mx of order: 

  



 



































   













    CM x 

x

a

B

B x

s B

s a 





 

 



1 1 1.2 10 3 (7) 

This error is often unacceptably large. 
February 2012 

Appendix SOP 2 Page 15 of 20 

Use of equation (4), on the other hand, introduces only an error of approximately 

  













    CM x 

s B

s a  



1 1 1.2 10 3 (8) 

It is a requirement for manufacture that the actual density of standard weights be near 

enough to the assumed density of the apparent mass scale to which they are adjusted that 

the magnitude of (8) will always be small under normal conditions in laboratories near sea 

level. 

 

The fact that there are two apparent mass scales widely used - one based on density 

8.0 g/cm3

 and an older one based on 8.4 g/cm3 - means that some caution is required on the 

part of the user. Conventional mass is generally preferred and reported for all calibrations 

where mass standards will be used to calibrate weighing instruments. For the most accurate 

work, the apparent mass scale should be abandoned in favor of substitution weighing with 

standards of known mass, density, and volume. 

 

The user must decide the accuracy required of the particular mass measurement and choose 

a buoyancy correction technique commensurate with that accuracy. 

 

The same considerations, which apply to the accuracy of buoyancy corrections in weighing 

by substitution, are easily extended to the other types of weighing indicated above. 

 

There are many factors, which affect the accuracy of a mass measurement. The above has 

dealt only with those arising from problems of buoyancy. 

 
February 2012 

Appendix SOP 2 Page 16 of 20 

Appendix B 

Examples 

 

Example 1: 

 

The weight set of Table B-1 is used with an equal-arm balance to find the mass of a piece of 

single-crystal silicon. The following weights were used to balance the silicon: 10 g, 3 g. The 

balance pointer showed the silicon side to be light by 3.5 divisions. The 10 mg weight of Table 1 

was used to find the sensitivity of the balance. When the weight was added, the pointer moved 

10.3 divisions. At the time of the weighing, the following measurements were taken: 

P = 748.1 mm Hg 

t = 22.3 C 

 

U = 37 % relative humidity 

 

What is the mass of the silicon? 

Answer: 

 

From Eq. 1, or Table B-2, calculate 

  1.314 6 10 20.194 mm Hg 22.3 273.15

5 315.56

9   



















e x x e s

     

 

3 10

273.15 22.3

0.464 60 748.1 0.003 796 0 37 20.2 















 a 

a = 1.171 94 x 10-3 g/cm3 

The density of silicon at 20 C is 2.329 1 g/cm3

 and its coefficient of linear expansion is 

0.000 002 6 /C. 

 

Make use of Eq. 3c and Table B-1. 

  10.000126 4  3.000 045 9  13.000172 g Ms

  3 Vs  1.267  0.380  1.647 cm 
February 2012 

Appendix SOP 2 Page 17 of 20 

Calculate the sensitivity: 

Sensitivity =   

10.3

0.010 003 0.003 70 1.172 103

g  x g/division 

  0.000 971g/division 0.971

10.3

0.009 999    mg/division 

       















 

    

 

2.3291

1.172 10 1 

13.000172 1.647 1.172 10 0.971 10 3.5 3

3 3

x

x x M x

Mx = 13.001 385 g 

Note that the thermal expansion is insignificant in this example. 

Example 2: 

 

Let us again consider the weighing performed in Example 1. This time, all we know about our 

weight set is that it has been adjusted to the 8.4 apparent mass scale at the best available 

commercial tolerance. 

Using Eq. (4), 











































 





 



  



 



















   



 



















 

 

 

2.3291

1.172 10 1 

10.3

3.5

8.390 9

1.172 10 0.010 1 8.390 91.172 10 13.00 1 

 3

3 3

x

x x

M x

Mx = 13.001 329 g 

For routine weighing, it is sometimes satisfactory to assume that the temperature is 20 C and the 

density of air is 1.2 x 10-3 g/cm3

. Had this been done, the computed value for the silicon would 

be 

 











































 





 



  



 



















   



 



















 

 

 

2.3291

1.2 10 1 

10.3

3.5

8.390 9

1.2 10 0.010 1 8.390 91.2 10 13.00 1 

 3

3 3

x

x x

M x
February 2012 

Appendix SOP 2 Page 18 of 20 

Mx = 13.001 442 g 

which is within 100 μg of the answer found in Example 1. 

 

Example 3: 

Another piece of silicon is measured on a single-pan microbalance. The balance weights were 

adjusted by the manufacturer to the conventional mass scale. The sensitivity of the balance has 

been determined to be exactly 1.000. This particular laboratory is well above sea level. At the 

time of the weighing, the following measurements were recorded: 

 

P = 612.3 mm Hg 

 

 t = 23.4 C 

 

U = 23 % relative humidity 

 

The balance reading was 15.00 g on the built-in weights and 0.000 358 g on the optical screen. 

What is the mass of the silicon? 

Answer: 

 

First, calculate es and a: 

 

es = 21.59 mm Hg 

 

a = 0.956 x 10-3 g/cm3 

Then, use Eq. 5: 

 















 

















 

 



2.3291

0.956 10 1 

8.0

0.956 10 15.00 0.000 358 1 

 3

3

x

x

M x

Mx = 15.004 724 g 

 

Example 4: 

The built-in weights in Example 3 are actually stainless steel of density 7.78 g/cm3

 at 20 °C. 

What is the approximate error caused by using the apparent mass scale? 
February 2012 

Appendix SOP 2 Page 19 of 20 

Answer: 

 

Using (7), the error is approximately 

  11 g 8 0 g cm

1 7 78 g cm1 15 g 0 2 x 10 g cm 3 3

3 3   













  

. / . / . /

 

This discrepancy, though larger than the precision of the best analytical balances, is actually well 

within the tolerance of ASTM Class 1 weights. 

Table B-1. Example of calibration report data. 

Mass 

(g) 

Uncertainty 

(g) 

Vol at 20 °C 

(cm3

) 

Cubical 

Coefficient of 

Expansion (/°C) 

 100.000 941 0.000 025 12.674 39 0.000 045

 50.000 463 0.000 016 6.337 19 0.000 045

 30.000 293 0.000 014 3.802 32 0.000 045 

 20.000 158 0.000 011 2.534 87 0.000 045

 10.000 130 0.000 013 1.267 44 0.000 045

 5.000 042 0.000 0069 0.633 72 0.000 045

 3.000 046 0.000 0046 0.380 23 0.000 045

 2.000 006 3 0.000 003 3 0.253 49 0.000 045

 1.000 014 4 0.000 003 0 0.126 74 0.000 045

 0.499 953 8 0.000 001 6 0.030 12 0.000 020

 0.299 961 4 0.000 001 2 0.018 07 0.000 020

 0.199 949 84 0.000 000 87 0.012 05 0.000 020

 0.099 963 78 0.000 000 91 0.006 02 0.000 020

 0.049 986 59 0.000 000 72 0.003 01 0.000 020

 0.029 991 00 0.000 000 77 0.001 81 0.000 020

 0.020 005 70 0.000 000 66 0.007 41 0.000 069

 0.010 002 77 0.000 000 86 0.003 70 0.000 069

 0.004 997 06 0.000 000 70 0.001 85 0.000 069

 0.003 002 99 0.000 000 76 0.001 11 0.000 069

 0.002 001 97 0.000 000 66 0.000 74 0.000 069

 0.001 000 83 0.000 000 86 0.000 37 0.000 069

 
February 2012 

Appendix SOP 2 Page 20 of 20 

Table B-2. es approximation in terms of temperature. 

Temperature (°C) es (mm Hg) 

 18 15.48 

 18.5 15.97 

 19 16.48 

 19.5 17 

 20 17.54 

 20.5 18.09 

 21 18.65 

 21.5 19.23 

 22 19.83 

 22.5 20.44 

 23 21.07 

 23.5 21.72 

 24 22.38 

 24.5 23.07 

 25 23.77 

 25.5 24.49 

 26 25.23 

 26.5 25.99 

 27 26.77 

 27.5 27.57 

 28 28.39 

 28.5 29.23 

 29 30.09 

 29.5 30.98 

 

 

1

 Handbook 145 is out of print as of 2012. Section 7 was updated and published as a part of NISTIR 6969 in 2003, 

updated in 2012. 
February 2012 

SOP 4 Page 1 of 20 

SOP No. 4 

 

Recommended Standard Operating Procedure 

for 

Weighing by Double Substitution Using a Single-Pan Mechanical 

Balance, a Full Electronic Balance, or a Balance with Digital 

Indications and Built-In Weights 

1.0 Introduction 

1.1. Purpose 

The double substitution procedure is one in which a standard and an unknown 

weight of equal nominal value are compared twice to determine the average 

difference between the two weights. Errors in any built-in weights or in the 

balance indications are minimized by using the balance only as a comparator and 

by calibrating the balance indications over the range of use for the measurement 

with a sensitivity weight. Accordingly, the procedure is especially useful for high 

accuracy calibrations. The procedure does not incorporate measurement control 

steps to ensure the validity of the standards and the measurement process; 

therefore, additional precautions must be taken. 

1.2. Prerequisites 

 

1.2.1. Verify that valid calibration certificates are available for the working 

standards used in the test. 

1.2.2. Verify that the working standards to be used have sufficiently small 

standard uncertainties for the intended level of calibration. Primary 

laboratory reference standards should not be used at this level. 

1.2.3. Verify that the balance is in good operating condition with sufficiently 

small process standard deviation as verified by a valid control chart or 

preliminary experiments to ascertain the performance quality when a new 

balance is put into service. 

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

and has had specific training in SOP 2, SOP 4, SOP 29, GMP 4, and GMP 

10. 

1.2.5. Verify that the laboratory facilities comply with the following minimum 

conditions to meet the expected uncertainty possible with this procedure 

and to comply with the balance manufacturer’s operating conditions 

specified for the balance. 
February 2012 

SOP 4 Page 2 of 20 

Table 1. Environmental conditions. 

Echelon1 Temperature Relative Humidity 

(percent) 

II 20 C to 23 C, a set point ± 2 C, maximum change 1.0 C/h 40 to 60 ± 10 / 4 h 

III 18 C to 27 C, maximum change 2.0 C/h 40 to 60 ± 20 / 4 h 

2. Methodology 

2.1. Scope, Precision, Accuracy 

This method is applicable to all weighings utilizing a mass comparator, a 

single-pan mechanical balance, a full electronic balance, or a balance that 

combines digital indications with the use of built-in weights (combination 

balance). The precision will depend upon the sensitivity of the balance and the 

care exercised to make the required weighings. The accuracy achievable with this 

procedure depends on the accuracy of the calibration of the working standards and 

the precision of the intercomparison. 

2.2. Summary 

The balance is adjusted if necessary, to obtain balance indications for all 

measurements that will be within the range of the optical scale or digital 

indications of the balance without changing the dial settings for the built-in 

weights, if present. The standard and the test weight are each weighed. A small, 

calibrated weight, called a sensitivity weight, is added to the test weight and these 

are weighed. The standard and the same sensitivity weight are then weighed. The 

latter two weighings provide both second weighings of the standard and the test 

weight as well as a determination of the sensitivity of the balance under the load 

conditions at the time of the intercomparison. All weighings are made at regularly 

spaced time intervals to average out any effects due to instrument drift. 

 The double substitution procedure is the same for all of the balances mentioned 

above, but the adjustment of the balance to prepare for the intercomparison and 

the selection of the sensitivity weight differs slightly depending upon the balance 

used. When steps specific to a particular balance are required, they are given in 

subsections of the procedure identified by a, b, and c along with the balance type. 

2.3. Apparatus/Equipment Required 

 

2.3.1. Precision balance with sufficient capacity and sensitivity for the 

calibrations planned. 

 

1 Echelon II corresponds to weights of Classes OIML F1 and F2. Echelon III corresponds to weights of Classes 

OIML M1, M2, and M3. This procedure does not provide adequate redundancy for OIML Classes E1 or E2 and SOP 

5 or SOP 28 should be used. 
February 2012 

SOP 4 Page 3 of 20 

 

2.3.2. Calibrated working standard and sensitivity weights with recent 

calibration values and which have demonstrated metrological traceability 

to the international system of units (SI), which may be to the SI through a 

National Metrology Institute such as NIST. 2.3.3. Calibrated small standard weights with recent calibration certificates 

which have demonstrated metrological traceability to the international 

system of units (SI), which may be to the SI through a National Metrology 

Institute such as NIST to be used as tare weights. 2.3.4. Uncalibrated weights to be used to adjust the balance to the desired 

reading 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 (calibration not required; this is used to ensure consistent 

timing of the measurement). 2.3.7. Calibrated barometer accurate to ± 66.5 Pa (0.5 mm Hg) with recent 

calibration values which have demonstrated metrological traceability to 

the international system of units (SI), which may be to the SI through a 

National Metrology Institute such as NIST to determine air pressure. 

2.3.8. Calibrated thermometer accurate to ± 0.10 C with recent calibration 

values which have demonstrated metrological traceability to the 

international system of units (SI), which may be to the SI through a 

National Metrology Institute such as NIST to determine air temperature. 

2.3.9. Calibrated hygrometer accurate to ± 10 percent with recent calibration 

values which have demonstrated metrological traceability to the 

international system of units (SI), which may be to the SI through a 

National Metrology Institute such as NIST to determine relative 

humidity.2

2.4. Symbols 

Table 2. Symbols used in this procedure.

Symbol Description 

S standard reference weight 

X weight to be calibrated 

 

2The barometer, thermometer, 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. 
February 2012 

SOP 4 Page 4 of 20 

t small calibrated tare weight, A subscript s or x is used to indicate the larger 

weight with which it is associated 

sw small calibrated weight used to evaluate the sensitivity of the balance 

M the mass (true mass) of a specific weight. Subscripts s, x, t, sw are used to 

identify the weight (equals Nominal plus Correction) 

N the nominal value of a specific weight. Subscripts s, x, are used to identify the 

weight. 

C the correction for a specific weight. Subscripts s, x, are used to identify the 

weight. 

CM the conventional mass of a specific weight. Subscripts s, x, t, sw are used to 

identify the weight. 

a density of air at time of calibration 

n density of normal air (1.2 kg/m3

) 

 density of masses; subscripts s, x, ts, tx, sw are used to identify the weight. 

2.5. Procedure 

2.5.1. Preliminary Procedure 

2.5.1.1. Weights are visually inspected for cleanliness and damage. 

2.5.1.2.If cleaning weights, it is important to clean weights before any 

measurements are made because the cleaning process may change 

the mass of the weight. Cleaning should not remove any significant 

amounts of weight material. Weights should be handled and stored 

in such a way that they stay clean. Before calibration, dust and any 

foreign particles shall be removed. Care must be taken not to 

change the surface properties of the weight (i.e. by scratching the 

weight). If a weight contains significant amounts of dirt that cannot 

be removed by the methods cited above, the weight or some part of 

it can be washed with clean alcohol, distilled water or other 

solvents. Weights with internal cavities should normally not be 

immersed in the solvent to avoid the possibility that the fluid will 

penetrate the opening. If there is a need to monitor the stability of a 

weight in use, the mass of the weight should, if possible, be 

determined before cleaning. 

2.5.1.3.If weights are cleaned with solvents they must be stabilized for the 

times given in the following table (better class weights need to 

stabilize for 7 to 10 days): 
February 2012 

SOP 4 Page 5 of 20 

Table 3. Cleaning Stabilization. 

Weight class F1 F2 to M3

After cleaning with alcohol 1–2 days 1 hour

After cleaning with distilled water 1 day 1 hour

2.5.1.4.Prior to performing any calibration tests, the weights need to be 

acclimated to the ambient conditions of the laboratory. In 

particular, weights of classes F1 (or better) should be close to the 

temperature in the weighing area. The minimum times (in hours) 

required for temperature stabilization (depending on weight size, 

weight class and on the difference between the initial temperature 

of the weights and the room temperature in the laboratory) are 

shown in the table below (with appropriate documented evidence). 

As a practical guideline, a waiting time of 24 hours is 

recommended. If weights are extremely hot or frozen additional 

equilibration may be needed to address problems with 

condensation and frozen surfaces. Weights must be completely dry 

prior to calibration. 

Table 4. Equilibration Times. 

ΔT* Nominal Mass OIML Class F1 

(time in h)

OIML Class F2 to M3 

(time in h)

± 20 °C 

1 000, 2 000, 5 000 kg 79 5 

100, 200, 500 kg 33 4 

10, 20, 50 kg 12 3 

1, 2, 5 kg 6 2 

100, 200, 500 g 3 1 

10, 20, 50 g 1 1 

< 10 g 1 0.5 

± 5 °C 

1 000, 2 000, 5 000 kg 4 1 

100, 200, 500 kg 4 1 

10, 20, 50 kg 4 1 

1, 2, 5 kg 3 1 
February 2012 

SOP 4 Page 6 of 20 

ΔT* Nominal Mass OIML Class F1 

(time in h)

OIML Class F2 to M3 

(time in h)

100, 200, 500 g 2 1 

< 100 g 1 1 

± 2 °C <100 g to 5 000 kg 1 0.5 

± 0.5 °C <100 g to 5 000 kg 0.5 0.5 

*ΔT = Initial difference between weight temperature and laboratory temperature. 

2.5.1.5. Conduct preliminary measurements to obtain an approximate 

value for the difference between the standard and the unknown, 

to determine where the readings occur on the balance, to 

determine if tare weights are required, to determine the 

sensitivity weight that must be used, and to determine the time 

interval required for the balance indication to stabilize. 

 

Tare weights are rarely needed for high precision mass standards 

that are within applicable tolerances. When unequal nominal 

weights are compared, tare weights are often required. When 

tare weights are required, carry tare weights, ts and tx, with the 

standard and the unknown, S and X, respectively. The tare 

weights must be calibrated standards with valid uncertainties 

that are evaluated in the process of determining calibration 

uncertainties. The standard and its tare weight, S + ts, should be 

"nearly the same mass" as the unknown with its tare weight, X + 

tx. "Nearly the same mass" depends upon the balance used (See 

GMP 14, Table 1). Select ts and tx such that the difference in 

mass between S + ts and X + tx is: 

a. Single-pan mechanical balance - less than 10

1 the range 

of the optical scale. 

b. Full electronic balance - less than 0.05 percent of the 

balance capacity. 

c. Combination balance - less than 

10

1 the range of the 

digital indications. 

d. Mass comparator – less than 10

1 digital range 

A sensitivity weight must be used on equal-arm balances, and is 

normally used on single-pan mechanical and electronic 
February 2012 

SOP 4 Page 7 of 20 

balances, to ensure that the measured differences determined 

through the use of the optical scale or electronic range have 

valid accuracy and traceability (See GMP 14, Table 2). (e.g., 

The optical scale is calibrated each time the procedure is 

performed through the use of a sensitivity weight). The 

uncertainty of the sensitivity weight does not generally need to 

be included in calculations of uncertainty since the uncertainty 

value is distributed across its range of use. 

If a sensitivity weight will be used, select one that is: a. Single-pan balance - between 1⁄4 and 1⁄2 the range of the 

optical scale, and at least 4 times the mass difference 

between X and S. 

b. Full electronic balance - at least 4 times the mass difference 

between X and S, but not exceeding 1 percent of the 

balance capacity. 

c. Combination balance - between 1⁄4 and 1⁄2 the range of the 

digital indications, and at least 4 times the mass difference 

between X and S. 

d. Mass comparator – at least 4 times the mass difference 

between X and S, but not exceeding 1⁄2 of the digital range. 

A sensitivity weight is not required if the electronic mass 

comparator that is used has been tested (with supporting 

data available) to determine that the balance has sufficient 

accuracy, resolution, repeatability, and stability so that no 

advantage is gained through the use of a sensitivity weight. 

For example, any possible errors must be less than what 

contributes to the uncertainty. When a mass comparator is 

used without a sensitivity weight, the sensitivity must be 

periodically verified and documented. 

2.5.1.6. Determine which optional sequence will be used, A or B. 

Optional sequence A uses the standard on the balance for the 

first and fourth observations and the unknown on the balance for 

the second and third observations; this is often called the 

“SXXS” sequence. Optional sequence B starts with the unknown 

on the balance first and last with the standard on the balance for 

the second and third observations; this is often called the 

“XSSX” sequence. The primary advantage of sequence B is less 

handling of the mass standards. The advantage of sequence A is 

in the case where the unknown is a summation of weights that 

require careful arrangement on the balance pan only once. 2.5.1.7. Adjust the single pan balance or the combination balance so the 

first two readings of the double substitution fall in the first 

quarter of the optical scale or digital indications. The zero 

adjustment and tare adjustment may be used. Small weights may 

be placed on the balance pan to reach the desired reading range. 

These weights remain on the pan throughout the double 
February 2012 

SOP 4 Page 8 of 20 

substitution. Once the balance has been adjusted to the desired 

position, neither the balance dials, the zero and tare adjustments, 

nor the small weights placed on the balance pan, are to be 

changed during the measurement. 2.5.1.8. If the balance is equipped with a pan arrestment mechanism, 

arrest the pan between each observation. 

2.5.2. Measurement Procedure, Optional Sequence A (SXXS) 

 

 Table 5. Optional Sequence A. 

Measurement No. Weights on Pan Observation 

1 S + ts O1

2 X + tx O2

3 X + tx + sw O3

4 S + ts + sw O4

 

All observations should be recorded on suitable data sheets, such as those in the 

appendix. Record the laboratory ambient temperature, barometric pressure, and 

relative humidity. 

2.5.2.1. Observation 1. Place the standard weight(s), S, along with ts on 

the balance pan. If equipped with a pan arrestment mechanism, 

release the balance pan. When the pan is released, start the 

stop-watch and record observation O1 once the balance 

indication has stabilized. 

 

2.5.2.2. Observation 2. Remove weight(s) S and ts and replace with test 

weight X and its tare weight, tx. Release the pan, time the 

interval, and record observation O2. 

 

2.5.2.3. Observation 3. Add the sensitivity weight, sw, to the weights of 

observation 2. Release the pan, time the interval, and record 

observation O3. 

2.5.2.4. Observation 4. Remove weights X and tx and replace with S and 

ts. The sensitivity weight, sw, remains on the balance pan. 

Release the pan, time the interval, and record observation O4. 

2.5.2.5. Compare the two differences (O2 – O1) and (O3 – O4); they 

should not differ from one another by more than 2 standard 

deviations of the balance for this process and load. If this 

difference is exceeded, reject the data and redo the 
February 2012 

SOP 4 Page 9 of 20 

measurements. Investigate possible causes of excess variability 

if measurements do not agree within these limits. 

 

2.5.2.6. If repeated double substitutions are performed, the values 

between successive trials should not differ from one another by 

more than ± 2 standard deviations of the balance for this process 

and load. If this difference is exceeded, reject the data and take a 

new series of measurements that agree. 

2.5.3. Measurement Procedure, Optional Sequence B (XSSX) 

Table 4. Optional Sequence B. 

Measurement No. Weights on Pan Observation 

1 X + tx O1

2 S + ts O2

3 S + ts + sw O3

4 X + tx + sw O4

Measurements for Option B are made as described in Option A except that X, S, tx, and ts are interchanged. 

3. Calculations 

3.1. If no air buoyancy correction is performed, calculate the conventional mass 

correction, Cx, for the test weight as follows, according to the optional sequence 

used. Incorporate an uncorrected systematic uncertainty in the uncertainty 

calculations if using this equation. In each case, the conventional mass corrections 

for the standard weight(s), Cs, the conventional mass of the tare weights, s CMt

and x CMt , and the conventional mass of the sensitivity weight, CMsw, are 

included. The symbols Ns and Nx refer to the nominal values of S and X, 

respectively. If no tare weights, and equal nominal values are used, those terms 

may all be deleted from the equations. 

. 3.1.1. Optional Sequence A (SXXS) 

3.1.2. Optional Sequence B (XSSX) 

    s x

3 2

2 1 3 4 sw

x s t t N - N

O - O

CM

2

O - O O - O C C CM s - CM x  



 









 



    

    s x

3 2

1 2 4 3 sw

x s t t N - N

O - O

CM

2

O - O O - O C C CM s - CM x  



 









 



    
February 2012 

SOP 4 Page 10 of 20 

3.2. Mass Calculation with Air Buoyancy Correction 

3.2.1. Calculate the air density, a, as described in the Appendix to SOP No. 2. 

3.2.2. Calculate the mass Mx of the test weight, and its mass correction Cx using 

the mass of the standard weight(s), the tare weights and the sensitivity 

weights according to the optional sequence used. 

3.2.2.1. Optional Sequence A (SXXS) 

3.2.2.2. Optional Sequence B (XSSX) 

3.2.3. Calculate the mass correction Cx, as follows: 

where Nx is the nominal value for X. 

 

3.2.4. Calculate the conventional mass3 of X, CMx. It is recommended that the 

conventional mass be reported. 

 

3 Conventional Mass: “The conventional value of the result of weighing a body in air is equal to the mass of a 

standard, of conventionally chosen density, at a conventionally chosen temperature, which balances this body at this 

reference temperature in air of conventionally chosen density.” The conventions are: reference density 8.0 g/cm3

; 

reference temperature 20 C; normal air density 0.0012 g/cm3. Conventional mass was formerly called “Apparent 

Mass versus 8.0 g/cm3

” in the United States. See OIML D28 (2004).

Cx = Mx - Nx

  

































































 



 

 































 

















ρ

ρ - 

 O - O

ρ

ρ M - 

 O - O O - O ρ

ρ - M - ρρ M - ρ

ρ M - 

M 

x

a

sw

a

sw

t

a

t

t

a

t

s

a

s

x

x

x

s

s

1

1

2

1 1 1

3 2

2 1 3 4

  

































































 



 

 































 

















ρ

ρ - 

 O - O

ρ

ρ M - 

 O - O O - O ρ

ρ - M - ρρ M - ρ

ρ M - 

M 

x

a

sw

a

sw

t

a

t

t

a

t

s

a

s

x

x

x

s

s

1

1

2

1 1 1

3 2

1 2 4 3
February 2012 

SOP 4 Page 11 of 20 

3.2.4.1. Conventional mass 

3.2.5. If requested, the apparent mass versus the reference density of brass may 

be calculated. This value should only be provided when requested by the 

customer for use when calibrating mechanical balances that have been 

adjusted to this reference density. 

3.2.5.1. Apparent mass versus brass 

4. Measurement Assurance 

4.1. Duplicate the process with a suitable check standard (See GLP 1, SOP 9, SOP 30, 

and Sec. 7.4). 

4.2. Plot the check standard value and verify that it is within established limits; a t-test 

may be incorporated to check observed value against accepted value. 

4.3. The mean of the check standard is used to evaluate bias and drift over time. 

4.4. Check standard observations are used to calculate the standard deviation of the 

measurement process, sp. 

5. Assignment of Uncertainty 

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

the mass standards used, us, estimates of the standard deviation of the measurement 

process, sp, and estimates of the effect of other components associated with this 

procedure, uo. These estimates should be combined using the root-sum-squared method 

(RSS), and the expanded uncertainty, U, reported with a coverage factor of two (k=2), to 

give us an approximate 95 percent level of confidence. See SOP 29 for the complete 

standard operating procedure for calculating the uncertainty. 

5.1. The expanded uncertainty for the standard, U, 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 usually 

need to be divided by the coverage factor k. 





 























8.3909

1 - 

M 1 - 

AM n

x

n

x

x vs brass 









 























8.0

1 - 

M 1 - 

CM n

x

n

x

x 




February 2012 

SOP 4 Page 12 of 20 

5.2. The value for sp is obtained from the control chart data for check standards using 

double substitution measurements. (See SOP No. 9.) 

5.3. Include an uncorrected systematic standard uncertainty if no buoyancy correction 

was performed. Calculate the magnitude of the air buoyancy correction per SOP 2 

and use a rectangular distribution. 

5.4. Other standard uncertainties usually included at this calibration level include 

uncertainties associated with calculation of air density, standard uncertainties 

associated with the density of the standards used, and any noted bias that has been 

determined through analysis of control charts and round robin data. See SOP 2 

and 29 for additional details. 

5.5. Evaluate compliance to applicable tolerances as needed by the customer. The 

expanded uncertainty, U, must be  1/3 of the applicable tolerances published in 

ASTM E617-97 and OIML R111 standards. Additionally, the mass value plus the 

expanded uncertainty must be less than the applicable tolerance to confidently 

state that mass standards are in or out of tolerance. 

6. Report 

 Report results as described in SOP No. 1, Preparation of Calibration/Test Reports. 
February 2012 

SOP 4 Page 13 of 20 

Appendix 

Double Substitution Data Sheet 

(Optional Sequence A, SXXS) 

Laboratory data and conditions: 

Operator Before After 

Date Temperature 

Balance Pressure 

Load Relative Humidity 

Standard deviation of the process, 

from control chart, sp

Degrees of Freedom

Mass standard(s) data:

ID Nominal Mass Correction* Expanded Unc: 

From cal. report 

Unc: 

k factor 

Density 

g/cm3 

S 

ts 

X

tx 

Sc

tSc

sw 

*Mass Correction = True Mass if using buoyancy correction. Mass Correction = Conventional Mass if NOT using buoyancy 

correction. Density is used only with buoyancy corrections. 

Observations: 

Observation No. Weights Balance Observations, Units _____ 

Time: 

1 (O1) S + ts

2 (O2) X + tx 

3 (O3) X + tx + sw 

4 (O4) S + ts+ sw 

Time: 

Measurement Assurance (Duplication of the Process): 

Observation No. Weights Balance Observations, Units _____ 

Time: 

1 (O1) S + ts

2 (O2) Sc + tSc 

3 (O3) Sc + tSc + sw 

4 (O4) S + ts+ sw 

Time: 

Check Standard in Control? _____Yes_____No 
February 2012 

SOP 4 Page 14 of 20 

Appendix 

Double Substitution Data Sheet 

(Optional Sequence B, XSSX) 

Laboratory data and conditions: 

Operator Before After 

Date Temperature 

Balance Pressure 

Load Relative Humidity 

Standard deviation of the process, 

from control chart, sp

Degrees of Freedom

Mass standard(s) data:

ID Nominal Mass Correction* Expanded Unc: 

From cal. report 

Unc: 

k factor 

Density 

g/cm3 

S 

ts 

X

tx 

Sc

tSc

sw 

*Mass Correction = True Mass if using buoyancy correction. Mass Correction = Conventional Mass if NOT using buoyancy 

correction. Density is used only with buoyancy corrections. 

Observations: 

Observation No. Weights Balance Observations, Units 

Time: 

1 (O1) X + tx

2 (O2) S + ts 

3 (O3) S + ts+ sw 

4 (O4) X + tx + sw 

Time: 

Measurement Assurance (Duplication of the Process): 

Observation No. Weights Balance Observations, Units 

Time: 

1 (O1) Sc + tSc

2 (O2) S + ts 

3 (O3) S + ts+ sw 

4 (O4) Sc + tSc + sw 

Time: 

Check Standard in Control? _____Yes_____No 
February 2012 

SOP 4 Page 15 of 20 

Example: With Buoyancy Corrections 

Double Substitution Data Sheet 

(Optional Sequence A, SXXS) 

Laboratory data and conditions: 

Operator HO Before After 

Date 8/24/86 Temperature (°C) 22.3 22.2

Balance M5SA Pressure (mm Hg) 753.5 753.7

Load 10 g Relative Humidity (%) 45 47

Standard deviation of the process, 

from control chart, sp

0.002 9 mg Degrees of Freedom 132

Mass standard(s) data:

ID Nominal Mass Correction* Expanded Unc: 

From Cal. Rpt. (mg) 

Unc: 

k factor 

Density 

g/cm3 

Set 3 S 10 g -0.679 mg 0.014 mg 3 8.00 

 ts --- --- --- --- --- 

Set 432 X 10 g TBD TBD 2 7.84 

tx --- --- --- --- --- 

Set C Sc 10 g 0.321 mg 0.025 mg 2 8.0 

tSc --- --- --- --- --- 

Set 3 sw 5 mg -0.0227 mg 0.000 28 2 8.5 

*Mass Correction = True Mass if using buoyancy correction. Mass Correction = Conventional Mass if NOT using buoyancy 

correction. Density is used only with buoyancy corrections. 

Observations: 

Observation No. Weights Balance Observations, Units mg 

Time: 8:35 AM 

1 (O1) S + ts 1.268 

2 (O2) X + tx 1.821 

3 (O3) X + tx + sw 6.798 

4 (O4) S + ts+ sw 6.245 

Time: 8:47 AM 

Measurement Assurance (Duplication of the Process): 

Observation No. Weights Balance Observations, Units mg 

Time: 9:00 AM 

1 (O1) S + ts 1.270 

2 (O2) Sc + tSc 2.271 

3 (O3) Sc + tSc + sw 7.248 

4 (O4) S + ts+ sw 6.248 

Time: 9:10 AM 

Check Standard in Control? _____Yes_____No 
February 2012 

SOP 4 Page 16 of 20 

Calculate the air density (SOP 2): 

a = 1.179 5 mg/cm3 = 0.001 179 5 g/cm3

Use equation 3.2.2.1 for optional sequence A (SXXS) with buoyancy corrections4: 

  





 



























 



 









 



     

 







 7.84

0.0011795 1 - 

6.798 - 1.821

 8.5

0.0011795 0.004 9773 1 - 

 2

1.821 - 1.268 6.798 - 6.245 0 - 0 8.0

0.0011795 9.999 321 1 - 

M X

9.999 9041g 0.99984949

( 9.997 8461 0.000 552 957 ) Mx   

Calculate the mass (true mass) correction: 

Cx = Mx – Nx 

Cx = 9.999 904 1 g – 10 g = - 0.000 095 9 g = - 0.095 9 mg

Calculate the conventional mass value: 





 























8.0

1 - 

M 1 - 

CM n

x

n

x

x 





0.999 850

 7.84

0.0012 9.999 9041 1 - 

 





 





CM x 

 

4 Keep in mind that these equations may be truncated for the purpose of this example and minor differences 

may be seen in the ending decimal places due to the use of calculators or spreadsheets. 

  

































































 



 

 































 

















ρ

ρ - 

 O - O

ρ

ρ M - 

 O - O O - O ρ

ρ - M - ρρ M - ρ

ρ M - 

M 

x

a

sw

a

sw

t

a

t

t

a

t

s

a

s

x

x

x

s

s

1

1

2

1 1 1

3 2

2 1 3 4
February 2012 

SOP 4 Page 17 of 20 

  9.999 87351g 0.999 850

9.999 9041 0.999 846 9 CM x  

Cx = CM x – Nx 

Cx = 9.999 873 51 g – 10 g

Cx = - 0.000 126 49 g = - 0.126 49 mg

Calculate the uncertainty for the calibration: 

2

o

2

p

2

uc  us  s  u

The uncertainty for the standard, U, must be divided by the k factor to determine the us. 

2 2 2  ( 0.004 667 )  ( 0.002 9 )  ( 0.000 000 032 ) uc

U = 0.005 494 623 6 * 2 = 0.010 989 247 3 mg 

Uncertainty Statement 

The uncertainty reported is the root sum square of the standard uncertainty of the standard, the 

standard deviation of the process, and an uncorrected systematic error for lack of buoyancy 

corrections, multiplied by a coverage factor of 2 (k=2) for an approximate 95 percent confidence 

interval. Factors not considered in the evaluation: magnetism (weights are considered to meet 

magnetism specifications unless measurement aberrations are noted), balance eccentricity and 

linearity (these factors are considered as a part of the measurement process when obtaining the 

standard deviation of the process). 

Compliance Evaluation

We have to evaluate the correction with its expanded uncertainty to determine if the weight is in 

tolerance or not. The magnitude of the expanded uncertainty must be less than 1/3 of the 

tolerance to be able to perform that evaluation according to ASTM E617-97 and OIML R111. 

U = uc * 2 

uc = 0.005 494 623 6 mg 
February 2012 

SOP 4 Page 18 of 20 

Load = 10 g 

ASTM E617 OIML R111 

Class Tolerance (mg) Class Tolerance (mg) 

3 0.25 F1 0.20 

4 0.50 F2 0.60 

5 2 M1 2.0 

If we look at three times the uncertainty: 0.011 mg x 3 = 0.033 mg, we realize that the 

uncertainty complies with the 1/3 rule for the ASTM and OIML shown here. 

Next, we look at the correction with the uncertainty: - 0.126 mg  0.011 mg. 

We can see that the absolute value of the correction is within: 0.115  Cx  0.137, therefore, it 

only complies with OIML class F1.

Reporting 

The conventional mass correction and uncertainty would be reported as follows: 

CX = - 0.126 mg ± 0.011 mg 
February 2012 

SOP 4 Page 19 of 20 

Example: Without Buoyancy Corrections 

Double Substitution Data Sheet 

(Optional Sequence B, XSSX) 

Laboratory data and conditions: 

Operator HO Before After 

Date 8/24/86 Temperature (°C) 22.3 22.3

Balance CB 100 Pressure (mm Hg) 753.5 753.4

Load 30 g & 1 troy oz Relative Humidity (%) 45 45

Standard deviation of the 

process, from control chart, sp

0.018 mg Degrees of Freedom 207

Mass standard(s) data:

ID Nominal Mass Correction* Expanded Unc: 

From cal. report 

Unc: 

k factor 

Density 

g/cm3 

S 30 g 0.407 mg 0.022 mg 3 8 

ts 1.1 g 0.359 6 mg 0.006 3 mg 3 8.04 

X 1 t oz TBD TBD 2 7.84 

tx None --- --- --- --- 

Sc 30 g 0.907 mg 0.030 mg 2 8 

tSc None --- --- --- --- 

sw 50 mg -0.084 00 mg 0.000 65 mg 2 8.5 

*Mass Correction = True Mass if using buoyancy correction. Mass Correction = Conventional Mass if NOT using buoyancy 

correction. Density is used only with buoyancy corrections. 

Observations: 

Observation No. Weights Balance Observations, Units mg 

Time: 9:00 AM 

1 (O1) X + tx 20.93 

2 (O2) S + ts 17.21 

3 (O3) S + ts+ sw 67.08 

4 (O4) X + tx + sw 70.81 

Time: 9:05 AM 

Measurement Assurance (Duplication of the Process): 

Observation No. Weights Balance Observations, Units 

Time: 9:15 AM 

1 (O1) S + ts 20.95 

2 (O2) Sc + tSc 21.45 

3 (O3) Sc + tSc + sw 71.32 

4 (O4) S + ts+ sw 70.83 

Time: 9:20 AM 

Check Standard in Control? _____Yes_____No 
February 2012 

SOP 4 Page 20 of 20 

Use equation 3.1.2 for optional sequence B (XSSX) with NO buoyancy corrections5: 

    s x

sw

x s t t N - N

O - O

CM O - O O - O C C CM - CM s x  



 









 



    

3 2

1 2 4 3

2

Note: be careful to combine like units only! 

    30 g - 1 t oz

67.08 - 17.21

49.916 mg 2

20.93 -17.21 70.81- 67.08 0.407 mg 1.100 359 6 g - 0  



 









 



  Cx   

Cx  0.407 mg  1.100 359 6 g - 0  3.728 435 9 mg  30 g - 31.103 47 68 g

Cx = 1 104.495 036 mg – 1.103 476 8 g 

Cx = 1.018 236 mg

Calculate the uncertainty for the calibration: 

 

2

o

2

p

2

c s u  u  s  u

The uncertainty for the standard, U, must be divided by the k factor for the standard and the tare 

weight to determine each us. The additional uncertainty for not performing the air buoyancy 

correction can be determined using the magnitude of the air buoyancy correction from SOP 2. 

2 2 2 2  ( 0.007 33 )  ( 0.0021)  ( 0.018 )  ( 0.0016 ) uc

uc = 0.019 617 15 mg 

U = 0.019 617 15 * 2 = 0.039 234 mg 

The conventional mass correction and uncertainty would be reported as follows: 

Cx = 1.018 mg ± 0.039 mg 

OR 

Cx = 0.000 032 7 t oz  0.000 001 3 t oz 

 

5 Keep in mind that these equations may be truncated for the purpose of this example and minor differences 

may be seen in the ending decimal places due to the use of calculators or spreadsheets. 

U = uc * 2
February 2012 

SOP 7 Page 1 of 16 

SOP No. 7 

 

Recommended Standard Operating Procedure 

for 

Weighing by Single Substitution 

Using a Single-Pan Mechanical Balance, a Full Electronic Balance, or a 

Balance with Digital Indications and Built-In Weights 

 

1.0 Introduction 

1.1. Purpose 

In the single substitution procedure a standard and an unknown weight of equal 

nominal value are compared once to determine the difference in weights. Errors in 

any built-in weights or in the balance indications are minimized by using the 

balance only as a comparator and by calibrating the balance indications over the 

range of use for the measurement with a sensitivity weight. This procedure is 

suitable for calibration when moderate accuracy is required and as a single 

substitution, does not eliminate errors due to drift. The procedure does not 

incorporate measurement control steps to ensure the validity of the standards and 

the measurement process; therefore, additional precautions must be taken. 

1.2. Prerequisites 

1.2.1. Verify that valid calibration certificates are available for the working 

standards used in the test. 

1.2.2. Verify that the working standards to be used have sufficiently small 

standard uncertainties for the intended level of calibration. Primary 

laboratory reference standards should not be used at this level. 

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

sufficiently small process standard deviation as verified by a valid control 

chart or preliminary experiments to ascertain its performance quality when 

a new balance is put into service. 

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

and has had specific training in SOP 2, SOP 7, SOP 29, GMP 4, and GMP 

10. 

1.2.5. Verify that the laboratory facilities meet the following minimum 

conditions to meet the expected uncertainty possible with this procedure 

and to comply with the balance manufacturer’s operating conditions 

specified for the balance. 
February 2012 

SOP 7 Page 2 of 16 

Table 1. Environmental conditions. 

Echelon1 Temperature Relative Humidity 

(%) 

II 20 C to 23 C, a set point ± 2 C, maximum change 1.0 C/h 40 to 60  10 / 4 h 

III 18 C to 27 C, maximum change 2.0 C/h 40 to 60  20 / 4 h 

2. Methodology 

2.1. Scope, Precision, Accuracy 

This method is applicable to all weighings utilizing a single-pan mechanical balance, a 

full electronic balance, or a balance that combines digital indications with the use of 

built-in weights (combination balance). The precision depends upon the sensitivity of the 

balance and the care exercised in making the required weighings. The accuracy 

achievable with this procedure depends on the accuracy of the calibration of the working 

standards and the precision of the intercomparison. 

2.2. Summary 

The balance is adjusted, if necessary, to obtain balance indications for all 

measurements that will be within the range of the optical scale or digital 

indications of the balance without changing the dial settings for the built-in 

weights, if present. The standard and the test weight are each weighed. A small, 

calibrated weight, called a sensitivity weight, is added to the test weight and these 

are weighed 

 The single substitution procedure is the same for all of the balances mentioned 

above, but the adjustment of the balance to prepare for the intercomparison and 

the selection of the sensitivity weight differ slightly depending upon the balance 

used. When steps specific to a particular balance are required, they are given in 

subsections of the procedure identified by a, b, and c along with the balance type. 

2.3. Apparatus/Equipment Required 

 

2.3.1. Precision balance with sufficient capacity and sensitivity for the 

calibrations planned. 

 

2.3.2. Calibrated working standard and sensitivity weights with recent 

calibration values and which have demonstrated metrological traceability 

to the international system of units (SI), which may be to the SI through a 

National Metrology Institute such as NIST. 

 

2.3.3. Calibrated small standard weights with recent calibration values and 

which have demonstrated metrological traceability to the international 

 

1 Echelon II corresponds to weights of Classes OIML F1 and F2. Echelon III corresponds to weights of Classes 

OIML M1, M2, and M3. This procedure does not provide adequate redundancy for OIML Classes E1 or E2 and SOP 

5 or SOP 28 should be used. 
February 2012 

SOP 7 Page 3 of 16 

system of units (SI), which may be to the SI through a National Metrology 

Institute such as NIST to be used as tare weights. 

 

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

reading 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. 2.3.7. Calibrated barometer accurate to ± 66.5 Pa (0.5 mm Hg) with recent 

calibration values which have demonstrated metrological traceability to 

the international system of units (SI), which may be to the SI through a 

National Metrology Institute such as NIST to determine air pressure. 

2.3.8. Calibrated thermometer accurate to ± 0.10 C with recent calibration 

values which have demonstrated metrological traceability to the 

international system of units (SI), which may be to the SI through a 

National Metrology Institute such as NIST to determine air temperature. 

2.3.9. Calibrated hygrometer accurate to ± 10 % with recent calibration values 

which have demonstrated metrological traceability to the international 

system of units (SI), which may be to the SI through a National Metrology 

Institute such as NISTto determine relative humidity.2

2.4. Symbols 

Table 2. Symbols used in this procedure. 

Symbol Description 

S standard weight 

X weight calibrated 

t small calibrated tare weight, A subscript s or x is used to indicate the larger 

weight with which it is associated 

sw small calibrated weight used to evaluate the sensitivity of the balance 

M the mass (true mass) of a specific weight. Subscripts s, x, t, sw are used to 

identify the weight (equals Nominal plus Correction) 

N the nominal value of a specific weight. Subscripts s, x, are used to identify 

the weight. 

 

2 The barometer, thermometer, 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. 
February 2012 

SOP 7 Page 4 of 16 

C the correction for a specific weight. Subscripts s, x, are used to identify the 

weight. 

CM the conventional mass of a specific weight. Subscripts s, x, t, sw are used to 

identify the weight. 

a density of air at time of calibration 

n density of normal air (1.2 kg/m3

) 

 density of masses; subscripts s, x, ts, tx, sw are used to identify the weight 

2.5. Procedure 

2.5.1. Preliminary Procedure 

2.5.1.1.Weights are visually inspected for cleanliness and damage. 

2.5.1.2.If cleaning weights, it is important to clean weights before any 

measurements are made because the cleaning process may change 

the mass of the weight. Cleaning should not remove any significant 

amounts of weight material. Weights should be handled and stored 

in such a way that they stay clean. Before calibration, dust and any 

foreign particles shall be removed. Care must be taken not to 

change the surface properties of the weight (i.e. by scratching the 

weight). If a weight contains significant amounts of dirt that cannot 

be removed by the methods cited above, the weight or some part of 

it can be washed with clean alcohol, distilled water or other 

solvents. Weights with internal cavities should normally not be 

immersed in the solvent to avoid the possibility that the fluid will 

penetrate the opening. If there is a need to monitor the stability of a 

weight in use, the mass of the weight should, if possible, be 

determined before cleaning. 

2.5.1.3.If weights are cleaned with solvents they must be stabilized for the 

times given in the following table (better class weights need to 

stabilize for 7 to 10 days): 

Weight class F1 F2 to M3

After cleaning with alcohol 1 to2 days 1 hour

After cleaning with distilled water 1 day 1 hour

2.5.1.4.Prior to performing any calibration tests, the weights need to be 

acclimated to the ambient conditions of the laboratory. In 

particular, weights of classes F1 (or better) should be close to the 

temperature in the weighing area and equilibrate for a minimum of 

24 hours. The minimum times (in hours) required for temperature 

stabilization (depending on weight size, weight class and on the 

difference between the initial temperature of the weights and the 

room temperature in the laboratory) are shown in the table below 
February 2012 

SOP 7 Page 5 of 16 

(with appropriate documented evidence). As a practical guideline, 

a waiting time of 24 hours is recommended. If weights are 

extremely hot or frozen additional equilibration may be needed to 

address problems with condensation and frozen surfaces. Weights 

must be completely dry prior to calibration. 

ΔT* Nominal Mass OIML Class F1 

(time in h) 

OIML Class F2 to M3 

(time in h) 

± 20 °C 

1 000, 2 000, 5 000 kg 79 5 

100, 200, 500 kg 33 4 

10, 20, 50 kg 12 3 

1, 2, 5 kg 6 2 

100, 200, 500 g 3 1 

10, 20, 50 g 1 1 

< 10 g 1 0.5 

± 5 °C 

1 000, 2 000, 5 000 kg 4 1 

100, 200, 500 kg 4 1 

10, 20, 50 kg 4 1 

1, 2, 5 kg 3 1 

100, 200, 500 g 2 1 

< 100 g 1 1 

± 2 °C <100 g to 5 000 kg 1 0.5 

± 0.5 °C <100 g to 5 000 kg 0.5 0.5 

*ΔT = Initial difference between weight temperature and laboratory temperature. 

2.5.1.5. Conduct preliminary measurements to obtain an approximate 

value for the difference between the standard and the unknown, 

to determine where the readings occur on the balance, to 

determine if tare weights are required, to determine the 

sensitivity weight that must be used, and to determine the time 

interval required for the balance indication to stabilize. 

 
February 2012 

SOP 7 Page 6 of 16 

Tare weights are rarely needed for high precision mass standards 

that are within applicable tolerances. When unequal nominal 

weights are compared, tare weights are often required. When 

tare weights are required, carry tare weights, ts and tx, with the 

standard and the unknown, S and X, respectively. The tare 

weights must be calibrated standards with valid uncertainties 

that are evaluated in the process of determining calibration 

uncertainties. The standard and its tare weight, S + ts, should be 

"nearly the same mass" as the unknown with its tare weight, X + 

tx. "Nearly the same mass" depends upon the balance used (See 

GMP 14, Table 1). Select ts and tx such that the difference in 

mass between S + ts and X + tx is: 

a. Single-pan mechanical balance - less than 10

1 the range of 

the optical scale. 

 

b. Full electronic balance - less than 0.05 % of the balance 

capacity. 

 

c. Combination balance - less than 

10

1 the range of the digital 

indications. 

A sensitivity weight must be used on equal-arm balances, and is 

normally used on single-pan mechanical and electronic 

balances, to ensure that the differences determined through the 

use of the optical scale or electronic range have valid accuracy 

and traceability. (e.g., The optical scale is calibrated each time 

the procedure is used through the use of a sensitivity weight). 

The uncertainty of the sensitivity weight does not need to be 

included in calculations of uncertainty since the uncertainty 

value is distributed across the range of use. 

If a sensitivity weight will be used, select one that is (See GMP 

14, Table 2): a. Single-pan balance - between 1⁄4 and 1⁄2 the range of the 

optical scale, and at least 4 times the mass difference 

between X and S. 

b. Full electronic balance - at least 4 times the mass difference 

between X and S but not exceeding 1 % of the balance 

capacity. 

c. Combination balance - between 1⁄4 and 1⁄2 the range of the 

digital indications, and at least 4 times the mass difference 

between X and S. 

2.5.1.6. Determine whether optional sequence A or B will be used. 

Optional sequence A uses the standard on the balance for the 
February 2012 

SOP 7 Page 7 of 16 

first observation and the unknown on the balance for the second 

and third observations; this is often called the “SXX” sequence. 

Optional sequence B starts with the unknown on the balance 

first and with the standard on the balance for the second and 

third observations; this is often called the “XSS” sequence. 

2.5.1.7. Adjust the single pan balance or the combination balance so the 

first two readings of the single substitution fall in the first 

quarter of the optical scale or digital indications. The zero 

adjustment and tare adjustment may be used. Small weights 

may be placed on the balance pan to reach the desired reading 

range. These weights remain on the pan throughout the single 

substitution. Once the balance has been adjusted to the desired 

position, neither the balance dials, the zero and tare adjustments, 

nor the small weights placed on the balance pan are to be 

changed during the measurement. 2.5.1.8. If the balance is equipped with a pan arrestment mechanism, 

arrest the pan between each observation. 

2.5.2. Measurement Procedure, Optional Sequence A (SXX) 

 Table 3. Optional Sequence A. 

Measurement No. Weights on Pan Observation 

1 S + ts O1

2 X + tx O2

3 X + tx + sw O3

 

All observations should be recorded on suitable data sheets, such as those 

in the appendix. Record the laboratory ambient temperature, barometric 

pressure, and relative humidity. 

2.5.1.1. Observation 1. Place the standard weight(s), S, along with ts on 

the balance pan. If equipped with a pan arrestment mechanism, 

release the balance pan. When the pan is released, start the 

stop-watch and record observation O1 once the balance indication 

has stabilized. 

2.5.1.2. Observation 2. Remove weight(s) S and ts and replace with test 

weight X and its tare weight, tx. Release the pan, time the interval, 

and record observation O2. 

2.5.1.3. Observation 3. Add the sensitivity weight, sw, to the weights of 

observation 2. Release the pan, time the interval, and record 

observation O3. 
February 2012 

SOP 7 Page 8 of 16 

2.5.1.4. If repeated single substitutions are performed, the values between 

successive trials should not differ from one another by more than 

± 2 sd of the balance. If this difference is exceeded, reject the data 

and take a new series of measurements that agree. 

2.5.3. Measurement Procedure, Optional Sequence B (XSS) 

 Table 4. Optional Sequence B. 

Measurement No. Weights on Pan Observation 

1 X + tx O1

2 S + ts O2

3 S + ts + sw O3

Measurements for Option B are made as described in Option A except that 

X, S, tx, and ts are interchanged. 

3. Calculations 

3.1. If no air buoyancy correction is performed, calculate the conventional mass 

correction, Cx, for the test weight as follows, according to the optional sequence 

used. Incorporate an uncorrected systematic uncertainty in the uncertainty 

calculations if using this equation. In each case, the conventional mass corrections 

for the standard weight(s), Cs, the conventional mass of the tare weights, 

s CMt and x CMt , and the conventional mass of the sensitivity weight, CMsw, are 

included. The symbols Ns and Nx refer to the nominal values of S and X, 

respectively. If no tare weights, and equal nominal values are used, those terms 

may all be deleted from the equations. 

. 3.1.1. Optional Sequence A (SXX) 

  s x

3 2

sw

x s t t 2 1 N - N

O - O

CM C C CM - CM ( O - O ) s x  



 



   

3.1.2. Optional Sequence B (XSS) 

  s x

3 2

sw

x s t t 1 2 N - N

O - O

CM C C CM - CM ( O - O ) s x  



 



   
February 2012 

SOP 7 Page 9 of 16 

3.2. Mass Calculation with Air Buoyancy Correction 

3.2.1. Calculate the air density, a, as described in the Appendix to SOP No. 2. 

3.2.2. Calculate the mass of the test weight, Mx, and its mass correction Cx using 

the mass of the standard weight(s), the tare weights and the sensitivity 

weights according to the optional sequence used. 

3.2.2.1. Optional Sequence A (SXX) 

 





























































 































 

















1 - 

O - O

M 1 - 

M 1 - M 1 - - M 1 - ( O - O ) 

M 

x

a

3 2

sw

a

sw

2 1

t

a

x

t

t

a

s

t

s

a

s

x

s x





















3.2.2.2. Optional Sequence B (XSS) 

 





























































 































 

















1 - 

 O - O

M 1 - 

M 1 - M 1 - - M 1 - ( O - O ) 

M 

x

a

3 2

sw

a

sw

1 2

t

a

x

t

t

a

s

t

s

a

s

x

s x





















3.2.3. Calculate the mass correction Cx, as follows: 

where Nx is the nominal value for X. 

 

Cx = Mx - Nx
February 2012 

SOP 7 Page 10 of 16 

3.2.4. Calculate the conventional mass3 of X, CMx. It is recommended that the 

conventional mass be reported. 

3.2.4.1. Conventional mass 





 























8.0

1 - 

M 1 - 

CM n

n

x

x

x







3.2.5. If requested, the apparent mass versus the reference density of brass may 

be calculated. This value should only be provided when requested by the customer for 

use when calibrating mechanical balances that have been adjusted to this reference 

density.

3.2.5.1. Apparent mass versus brass 

4. Measurement Assurance 

4.1. Duplicate the process with a suitable check standard (See GLP 1, SOP 9, SOP 30, 

and Sec. 7.4). 

4.2. Plot the check standard value and verify that it is within established limits; a t-test 

may be incorporated to check observed value against accepted value. 

4.3. The mean of the check standard is used to evaluate bias and drift over time. 

4.4. Check standard observations are used to calculate the standard deviation of the 

measurement process, sp. 

5. Assignment of Uncertainty 

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

the mass standards used, us, estimates of the standard deviation of the measurement 

 

3 Conventional Mass: “The conventional value of the result of weighing a body in air is equal to the mass of a 

standard, of conventionally chosen density, at a conventionally chosen temperature, which balances this body at this 

reference temperature in air of conventionally chosen density.” The conventions are: reference density 8.0 g/cm3

; 

reference temperature 20 C; normal air density 0.0012 g/cm3. Conventional mass was formerly called “Apparent 

Mass versus 8.0 g/cm3

” in the United States. See OIML D28 (2004).





 























8.3909

1 - 

M 1 - 

AM n

x

n

x

x vs brass 




February 2012 

SOP 7 Page 11 of 16 

process, sp, and estimates of the effect of other components associated with this 

procedure, uo. These estimates should be combined using the root-sum-squared method 

(RSS), and the expanded uncertainty, U, reported with a coverage factor of two (k=2), to 

give us an approximate 95 % level of confidence. See SOP 29 for the complete standard 

operating procedure for calculating the uncertainty. 

5.1. The expanded uncertainty for the standard, U, 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 usually 

need to be divided by the coverage factor k. 

5.2. The value for sp is obtained from the control chart data for check standards using 

single substitution measurements. (See SOP No. 9.) 

5.3. Include an uncorrected systematic standard uncertainty if no buoyance correction 

was performed. Calculate the magnitude of the air buoyancy correction per SOP 2 

and use a rectangular distribution. 

5.3. Other standard uncertainties usually included at this calibration level include 

uncertainties associated with calculation of air density, standard uncertainties 

associated with the density of the standards used, and any noted bias that has been 

determined through analysis of control charts and round robin data. See SOP 2 

and 29 for additional details. 

5.4. .Evaluate compliance to applicable tolerances as needed by the customer. The 

expanded uncertainty, U, must be  1/3 of the applicable tolerances published in 

ASTM E617-97 and OIML R111 standards. Additionally, the mass value plus the 

expanded uncertainty must be less than the applicable tolerance to confidently 

state that mass standards are in or out of tolerance. 

6. Report 

Report results as described in SOP No. 1, Preparation of Calibration/Test Reports. 
February 2012 

SOP 7 Page 12 of 16 

Appendix 

Single Substitution Data Sheet 

(Optional Sequence A, SXX)

Laboratory data and conditions: 

Operator Before After 

Date Temperature 

Balance Pressure 

Load Relative Humidity 

Standard deviation of the process, 

from control chart, sp

Degrees of Freedom

Mass standard(s) data:

ID Nominal Mass Correction Unc: 

From cal. report 

Unc: 

k factor 

Density 

g/cm3 

S 

ts 

X

tx 

Sc 

tSc 

sw 

Mass Correction = True Mass if using buoyancy correction. Mass Correction = Conventional Mass if NOT using buoyancy 

correction. Density is used only with buoyancy corrections. 

Observations: 

Measurement # Weights Balance Observations, Units _________ 

Time: 

1 (O1) S + ts

2 (O2) X + tx 

3 (O3) X + tx + sw 

Time: 

Measurement Assurance (Duplication of the Process): 

Measurement # Weights Balance Observations, Units _________ 

Time: 

1 (O1) S + ts

2 (O2) Sc + tSc 

3 (O3) Sc + tSc + sw 

Time: 

Check Standard in Control? _____Yes_____No 
February 2012 

SOP 7 Page 13 of 16 

Appendix 

Single Substitution Data Sheet 

(Optional Sequence B, XSS)

Laboratory data and conditions: 

Operator Before After 

Date Temperature 

Balance Pressure 

Load Relative Humidity 

Standard deviation of the process, 

from control chart, sp

Degrees of Freedom

Mass standard(s) data:

ID Nominal Mass Correction Unc: 

From cal. report 

Unc: 

k factor 

Density 

g/cm3 

S 

ts 

X

tx 

Sc 

tSc 

sw 

Mass Correction = True Mass if using buoyancy correction. Mass Correction = Conventional Mass if NOT using buoyancy 

correction. Density is used only with buoyancy corrections. 

Observations: 

Measurement # Weights Balance Observations, Units _________ 

Time: 

1 (O1) X + tx

2 (O2) S + ts 

3 (O3) S + tS + sw 

Time: 

Measurement Assurance (Duplication of the Process): 

Measurement # Weights Balance Observations, Units _________ 

Time: 

1 (O1) S + ts

2 (O2) Sc + tSc 

3 (O3) Sc + tSc + sw 

Time: 

Check Standard in Control? _____Yes_____No 
February 2012 

SOP 7 Page 14 of 16 

Example: Without Buoyancy Corrections 

Single Substitution Data Sheet 

(Optional Sequence A) 

SXX

Laboratory data and conditions: 

Operator HO Before After 

Date 8/27/86 Temperature (°C) 22.6 22.4

Balance H 20 Pressure (mm Hg) 751.7 752.0

Load 50 g Relative Humidity (%) 50 48

Standard deviation of the process, 

from control chart, sp

0.018 mg Degrees of Freedom 183

Mass standard(s) data:

ID Nominal Mass Correction Unc: 

From cal. report 

Unc: 

k factor 

Density 

g/cm3 

S 50 g 0.255 mg 0.033 mg 3 8.00 

ts --- --- --- --- --- 

X 50 g TBD TBD TBD 7.95 

tx --- --- --- --- --- 

Sc 50 g 0.315 mg 0.045 mg 2 8.00 

tx --- --- --- --- --- 

sw 50 mg -0.084 mg 0.000 85 mg 2 8.5 

Mass Correction = True Mass if using buoyancy correction. Mass Correction = Conventional Mass if NOT using buoyancy 

correction. Density is used only with buoyancy corrections. 

Observations: 

Measurement # Weights Balance Observations, Units _____mg____ 

Time: 3:40 PM 

1 (O1) S + ts 12.62 

2 (O2) X + tx 12.51 

3 (O3) X + tx + sw 62.37 

Time: 3:45 PM 

Measurement Assurance (Duplication of the Process): 

Measurement # Weights Balance Observations, Units ___mg___ 

Time: 3:50 PM 

1 (O1) S + ts 12.67 

2 (O2) Sc + tSc 12.73 

3 (O3) Sc + tSc + sw 62.60 

Time: 3:55 PM 

Check Standard in Control? _____Yes_____No 
February 2012 

SOP 7 Page 15 of 16 

Calculation of Conventional Mass 

Use equation 3.1.1 for optional sequence A (SXX) without buoyancy corrections

4

: 

Since no tare weights were used and equal nominal values were used, the equation may be 

simplifies as follows: 

  



 



  

O - O

CM C C ( O - O ) 

3 2

sw

x s 2 1

  



 



  

62.37 - 12.51

49.916mg C 0.255mg (12.51 - 12.62 ) x

Cx = 0.255 mg + (-0.110 124) mg

Cx = 0.144 876 mg

Calculation of Uncertainty 

Calculate the uncertainty for the calibration: 

U  uc * 2

2

o

2

p

2

c s u  u  s  u

The uncertainty for the standard from the calibration report (or data sheet) must be divided by 

the k factor to determine the us. Refer to SOP 29 for the use of multiple standards. An additional 

uncertainty for not performing the air buoyancy correction can be determined using the 

magnitude of the air buoyancy correction from SOP 2. 

2 2 2

c u  ( 0.011 )  ( 0.018 )  ( 0.0010 )

U = 0.021 330 7 * 2 = 0.042 661 5 mg

Uncertainty Statement 

 

The uncertainty reported is the root sum square of the standard uncertainty of the standard, the 

standard deviation of the process, and an uncorrected systematic error for lack of buoyancy 

corrections, multiplied by a coverage factor of 2 (k=2) for an approximate 95 % confidence 

interval. Factors not considered in the evaluation: magnetism (weights are considered to meet 

magnetism specifications unless measurement aberrations are noted), balance eccentricity and 

 

4 Keep in mind that these equations may be truncated for the purpose of this example and minor differences 

may be seen in the ending decimal places due to the use of calculators or spreadsheets. 

uc = 0.021 330 7 mg
February 2012 

SOP 7 Page 16 of 16 

linearity (these factors are considered as a part of the measurement process when obtaining the 

standard deviation of the process). 

Compliance Evaluation 

You may need to evaluate the conventional mass correction and its uncertainty to determine if a 

weight is or is not within specified tolerances. The magnitude of the expanded uncertainty must 

be less than 1/3 of the tolerance to be able to perform that evaluation, according to ASTM 

E 617-97 and OIML R111 documentary standards. 

Load = 50 g 

ASTM E 617 OIML R111 

Class Tolerance (mg) Class Tolerance (mg) 

1 0.12 E2 0.10 

2 0.25 F1 0.30 

If we look at three times the calculated expanded uncertainty: 0.043 mg x 3 = 0.129 mg, we 

observe that the uncertainty complies with the 1/3 rule for ASTM Class 2, and OIML Class F1

but not ASTM Class 1 or OIML Class E2. This may not be the best procedure for this level of 

work. 

Next, the weight value is considered to be within tolerance when the absolute value of its error 

plus its uncertainty, do not exceed the tolerance established for the particular class of weight. 

Value and uncertainty: 0.145 mg  0.043 mg. 

0.145 mg + 0.043 mg = 0.188 mg (upper limit of the value, or error bar if graphed) 

0.145 mg – 0.043 mg = 0.102 mg (lower limit of the value, or error bar if graphed) 

We can see that the correction of 0.145 is within these limits: 0.102  Cx  0.188, therefore, the 

value also complies with ASTM Class 2 and OIML Class F1, and a compliance statement may be 

included on the calibration report. 

Reporting 

The conventional mass correction and uncertainty are reported as follows: 

Cx = 0.145 mg  0.043 mg 

 
February 2012 

SOP 8 Page 1 of 12 

SOP No. 8 

Recommended Standard Operating Procedure 

for 

Medium Accuracy Calibration of Mass Standards 

by 

Modified Substitution 

 

1. Introduction 

1.1. This SOP describes procedures to be followed for determining whether or not 

mass standards are within the tolerances specified for a particular class of 

standards (e.g., NIST Class F, ASTM Class 5, 6, 7 or OIML Classes M1, M2, and 

M3) where the uncertainty is usually much smaller than the tolerance application 

provided the laboratory maintains standards and balances used for higher 

precision work. This procedure permits the metrologist to report that the weights 

under test were compared against a reference standard with the results reported on 

the laboratory report form. The comparison is important because the built-in 

weights of a balance do not represent laboratory standards or provide 

metrological traceability unless they have been formally calibrated. If a State law 

or other regulation requires that field weights be compared against the State (or 

reference) standards, this procedure can be used to fulfill this requirement. It is 

suitable for calibration when moderate accuracy is required, and does not 

eliminate errors due to drift. The procedure does not incorporate measurement 

control steps to ensure the validity of the standards and the measurement process; 

therefore, additional precautions are described for the use of check standards. The 

expanded uncertainty with this procedure must be  1/3 of the tolerance per 

ASTM and OIML standards. If uncertainty to tolerance ratios are greater than 

required, SOP 7 (single substitution) or SOP 4 (double substitution) procedures 

are preferred. 

Note: If you use SOP 8, you are most likely using working standards that are 

equivalent or only slightly better than your customer’s weights. If this is the case, 

the working standards must have a higher level of calibration than the weights 

being calibrated. Therefore, you will need standards, balances, procedures and 

uncertainties better than the customer weights and this procedure to calibrate your 

own working standards. 

1.2. Prerequisites 

1.2.1. Verify that valid calibration values are available for the standards used in 

the test. 

 

1.2.2. Verify that the working standards to be used have sufficiently small 

standard uncertainties for the intended level of calibration. Primary 

reference standards should never be used for this level of calibration. 

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

sufficiently small process standard deviation as verified by a valid control 

chart or preliminary experiments to ascertain the performance quality 

when a new balance is put into service. 1.2.4. Verify that the operator is experienced in precision weighing techniques 

and has had specific training in SOP 7, SOP 8, SOP 29, and GMP 10. 
February 2012 

SOP 8 Page 2 of 12 

1.2.5. Verify that the laboratory facilities meet the following minimum 

conditions to meet the expected uncertainty possible with this procedure 

and to ensure that the balances are used within the operating requirements 

specified by the manufacturer. 

Table 1. Environmental conditions. 

Echelon1 Temperature Relative Humidity

(%)

III 18 C to 27 C, maximum change 2.0 C/h 40 to 60  20 / 4 h 

 

2. Methodology 

2.1. Scope, Precision, Accuracy 

This method is applicable to all lower echelon mass calibration (tolerance testing) 

provided that the uncertainty requirements can be met. The achievable precision 

using this procedure is appropriate, provided the expanded uncertainty of the 

measurement is no more than one-third of the permissible tolerance of the mass 

standard tested. The accuracy achievable with this procedure depends on the 

accuracy of the calibration of the working standards and the precision of the 

intercomparison. 

 

2.2. Summary 

The mass to be tested is compared with a calibrated working standard by a 

modified substitution procedure. The comparison may be made using a 

single-pan, an equal-arm, or a fully-electronic balance. The reference standard is 

placed on the balance to obtain a convenient reference point and a sensitivity test 

is conducted. The error (departure from nominal value) of the weight tested is 

determined by comparing its reading to the reading obtained for the reference 

standard. A weight is considered to be within tolerance when the absolute value of 

its error plus its uncertainty does not exceed the tolerance established for the 

particular class of weight. 

2.3. Apparatus/Equipment 

 

2.3.1. Single-pan, equal arm, or full-electronic balance with sufficient capacity 

for the load tested and with readability equal to or less than one-tenth of 

the acceptable tolerance tested. 

 

2.3.2. Mass standards calibrated with an expanded uncertainty of one-tenth or 

less than the tolerance tested and which have demonstrated metrological 

traceability to the international system of units (SI), which may be to the 

SI through a National Metrology Institute such as NIST. 

 2.3.3. Calibrated sensitivity weights with recent calibration certificates which 

have demonstrated metrological traceability to the international system of 

units (SI), which may be to the SI through a National Metrology Institute 

such as NIST. 

 

1 Echelon III corresponds to weights of Classes OIML M, M2, and M3 or NIST Class F. Uncertainty values must 

be fully assessed if this procedure is attempted for any higher class weights.This procedure does not provide 

adequate redundancy for OIML Classes E1 or E2 and SOP 5 or SOP 28 should be used. 
February 2012 

SOP 8 Page 3 of 12 

2.3.4. Uncalibrated counterweights, T, of approximately the same mass as the 

standard weights (for option C). 

 

2.4. Procedure - Option A, Use of Single-Pan Balance 

 2.4.1. Select a reference standard of the same nominal value as the weight under 

test. Place the standard on the balance pan. Adjust the optical scale 

reading (See GMP No. 4) to approximately midscale using uncalibrated 

tare weights and the balance’s coarse and fine Zero controls. This setting 

must not be altered during a measurement sequence. Record the reading as 

O1. 

2.4.2. Add a sensitivity weight equal to approximately one-quarter full scale 

reading and record reading as O2. 2.4.3. Calculate the value of a scale division using the equation in 3.2. If it is 

within ± 2 % of nominal value (usual case) the nominal value of a division 

can be used for tolerance testing. 

2.4.4. Remove the sensitivity weight and adjust the optical scale to account for 

corrected value of standard used. Example: Suppose that the nominal range of the optical scale is 100 mg 

and that the reference standard has a correction of -2.5 mg. The optical 

scale is adjusted to read 47.5 mg when the standard is on the pan. Under 

this condition, the reading 50.0 mg represents the nominal mass of the 

reference standard. 2.4.5. Remove the Standard. 

2.4.6. Place the weight to be tested on the balance pan, read the optical scale and 

record reading as Xn. The error in the weight is the amount by which the 

indication deviates from the mid-scale reading. If the weight indication is 

more than the mid-scale value, the weight is heavy by the indicated 

difference; if the indication is less than the mid-scale value, the weight is 

light. 

2.4.7. After several weights have been tested ((no more than 10 unknown 

weights may be tested without rechecking the standard or check standard), 

put the standard on the balance pan and record the reading. The difference 

between this indication and the previous one for the standard indicates a 

balance drift. This drift will normally be very small. If the drift exceeds 10 

% of the tolerance applicable to the weights under test or affects a 

measurement result to the extent that a weight may be out of tolerance, the 

measurement should be repeated and more frequent checks of the standard 

should be made or a more appropriate procedure should be used. The 

average drift may be monitored over time to be included as a component 

of the uncertainty; otherwise 10 % of the applicable tolerance, as a 

rectangular distribution may be included in the uncertainty calculations. 

2.4.8. Readjust the optical scale at any time that a significant difference is 

observed when rechecking a standard. 
February 2012 

SOP 8 Page 4 of 12 

2.4.9. Calculate the mass correction for the unknown weights using the 

appropriate equation in Section 3. 

 

2.5. Procedure - Option A1 Use of Single-Pan Balance 

 2.5.1. Select a reference standard of the same nominal value as the weight under 

test. Place the standard on the balance pan. Adjust the optical scale 

reading (See GMP No. 4) to midscale using uncalibrated tare weights and 

the balance’s coarse and fine Zero controls. This setting must not be 

altered during a measurement sequence. Record the reading as O1. 

2.5.2. Add a sensitivity weight equal to approximately one-quarter full scale 

reading and record reading as O2. 2.5.3. Calculate the value of a scale division using the equation in 3.2. If the 

sensitivity is within ± 2 % of nominal value (usual case) of the scale 

division, the nominal value of a division may be used. 

2.5.4. Remove the sensitivity weight and re-adjust the optical scale to obtain a 

midscale indication, if the indication has changed from that set in 2.5.1. 2.5.5. Remove the Standard. 2.5.6. Place the weight to be tested on the balance pan, read the optical scale and 

record the indication as Xn. The error in the weight is the amount by which 

the indication deviates from the mid-scale reading. If the weight indication 

is more than the mid-scale value, the weight is heavier than the standard 

by the indicated difference; if the indication is less than the mid-scale 

value, the weight is lighter than the standard. 

2.5.7. After several weights have been tested (no more than 10 unknown weights 

may be tested without rechecking the standard or check standard) put the 

standard on the balance pan and record the reading. The difference 

between this indication and the previous one for the standard indicates a 

balance drift. This drift will normally be very small. If the drift exceeds 10 

% of the tolerance applicable to the weights under test or affects a 

measurement result to the extent that a weight may be out of tolerance, the 

measurement should be repeated and more frequent checks of the standard 

should be made or a more appropriate procedure should be used. The 

average drift may be monitored over time to be included as a component 

of the uncertainty; otherwise 10 % of the applicable tolerance, as a 

rectangular distribution may be included in the uncertainty calculations. 2.5.8. Readjust the optical scale at any time that a significant difference is 

observed when rechecking a standard. 

2.5.9. Calculate the correction of the unknown using the equation 

Cx = Cs + (Xn-O1). 

2.6. Procedure - Option B, Use of Full Electronic Balance 

 

2.6.1. Select a reference standard of the same nominal value as the weight under 

test. Place the standard on the pan. Zero the balance and record reading as 

O1. 
February 2012 

SOP 8 Page 5 of 12 

2.6.2. Add a calibrated sensitivity weight (sw  2 times the tolerance but not 

exceeding 1 % of the balance capacity) and record the reading as O2. 

Verify whether the nominal scale division is within ± 2 % of nominal 

value of the scale division using the equation in 3.2. If so, the nominal 

value of the scale division may be used. 2.6.3. Remove sensitivity weight and zero the balance so weight differences, d, can be read directly from the balance indications. 2.6.4. Remove all weights from the balance pan. 2.6.5. Place the weight to be tested on the balance pan. Record the reading as 

Xn. 

2.6.6. After several weights have been tested (no more than 10 unknown weights 

may be tested without rechecking the standard or check standard) recheck 

the zero as in 2.5.3 and record the reading. The difference between this 

indication and the previous one for the standard indicates a balance drift. 

This drift will normally be small. If the drift exceeds 10 % of the tolerance 

applicable to the weights under test or affects a measurement result to the 

extent that a weight may be out of tolerance, the measurement should be 

repeated and more frequent checks of the standard should be made or a 

more appropriate procedure should be used. The average drift may be 

monitored over time to be included as a component of the uncertainty; 

otherwise 10 % of the applicable tolerance, as a rectangular distribution 

may be included in the uncertainty calculations. 

2.6.7. Readjust the zero at any time that a significant difference is observed 

when rechecking a standard. 

2.6.8. Calculate the mass correction for each weight using the equation 

Cx = Cs + (Xn-O1) = Cs + Xn (when O1 is zeroed) 

2.7. Procedure - Option C, Use of Equal Arm Balance 

 

2.7.1. Select a reference standard of the same nominal value as the weight under 

test. Place the standard on the left balance pan together with small, 

calibrated weights equal to the correction required for the standard, 

provided it is light. If (and only if) the standard is heavy, do nothing 

further at this point but follow instructions in 2.6.4. Add sufficient 

counterweights to the right pan to obtain a sum of turning points of 

approximately twice midscale value. If necessary, number the graduated 

scale such that adding weights to the left pan will increase the balance 

reading. Record the sum of the turning points as 01. 

 

2.7.2. Add an appropriate calibrated sensitivity weight to the left pan and record 

the sum of the turning points as 02. Calculate the sensitivity, 

  O - O

CM sensitivity 

2 1

sw 
February 2012 

SOP 8 Page 6 of 12 

 where CMsw is the conventional mass of the sensitivity weight. 

 

2.7.3. Remove all weights from the left pan. 

 

2.7.4. Place weight to be tested on the left pan. If the standard used in 2.6.1 was 

heavy, add small correction weights to the left pan, equivalent to the 

correction required for the standard. Add small, calibrated tare weights as 

required to left or right pan to obtain an approximate balance and record 

the sum of the turning points as Xn. 2.7.5. Calculate the mass correction of the weight tested as follows. 

2.7.5.1.If added tare weights are placed on the left pan (tx). 

2.7.5.2.If added tare weights are placed on the right pan (ts) 

2.7.6. After several weights have been tested (no more than 10 unknown weights 

may be tested without rechecking the standard or check standard), recheck 

the turning point 01, as described in 2.6.1. Only a small difference should 

be observed. If the difference exceeds 2 % of the sum of the turning points 

in O1, the measurement should be repeated and more frequent checks of 

the standard should be made or a more appropriate procedure should be 

used. The average drift may be monitored over time to be included as a 

component of the uncertainty; otherwise 10 % of the applicable tolerance, 

as a rectangular distribution may be included in the uncertainty 

calculations. 

2.8.Tolerance Evaluation 

2.8.1. Compare the correction plus the expanded uncertainty of the weight tested 

with the tolerance for the class of weights to which it belongs. If the 

absolute value of the correction plus the expanded uncertainty is numerically smaller than the tolerance, the weight is considered to be within 

tolerance. If the correction is larger than permissible, the weight is 

considered to be outside the tolerance and appropriate action should be 

taken. It is recommended that weights whose absolute value of the 

correction exceeds 75 % of the tolerance limit be adjusted closer to the 

nominal value where possible. 

 

Table 2. Example of Weighing Sequence. 

Measurement No. Weights on Pan Observation 

1 S O1

2 S + sw O2

3 S O3

4 X1 O4

  xt

sw

x n - CM

 O - O

CM C ( X - O ) 



 



 

2 1

1

  st

sw

x n CM

 O - O

CM C ( X - O )  



 



 

2 1

1
February 2012 

SOP 8 Page 7 of 12 

Measurement No. Weights on Pan Observation 

5 X2 O5

6 X3 O6

7 X4 O7

8 X5 O8

9 X6 O9

10 X7 O10

11 X8 O11

12 X9 O12

13 X10 O12

14 Sc O13

15 S O14

 

3. Calculations 

3.1. Air buoyancy corrections are generally not made with the modified substitution, 

although with the use of spreadsheets and the need to record envionrmental 

conditions, there is no reason why the buoyancy correction may not be routinely 

included (mass densities must be known or assumed). Calculate the conventional 

mass correction, Cx, for the test weight as follows, according to the optional 

sequence used. In each case, the conventional mass corrections for the standard 

weight(s) are included. 

Table 3. Symbols used. 

Symbol Description 

CMi conventional mass of weight i

Ns nominal value of S

Nx nominal value of X

3.2. Evaluate the sensitivity of the balance: 

 If the sensitivity error is less than 2 % of the nominal value of a division on the 

optical scale or the electronic range of operation, proceed with the modified 

substitution. If the sensitivity error is greater, SOP 7, Single Substitution may be 

acceptable. If corrections for sensitivity are not made, an uncorrected systematic 

error may be incorporated into the uncertainty; this value is generally about 2 % 

of the applicable tolerance. 

3.3. Calculate the mass correction of each unknown weight as follows if the correction 

for the standard IS NOT used in setting a reference point on the balance: 

Cx = Cs + d 

Cx = Cs + ( Xn – O1 ) 

 Note: If an electronic balance is used and zeroed with the standard on the 

balance, O1 is “0" and d becomes the Xn balance reading. 

  O - O

CM sensitivity 

2 1

sw 
February 2012 

SOP 8 Page 8 of 12 

3.4. Calculate the mass correction of each unknown weight as follows if the correction 

for the standard IS used in setting a nominal reference point on the balance: 

Cx = ( X balance reading – Nnominal reference point ) 

 Note: In this case the standard and its correction are used to artificially set a 

perfect nominal reference point for use in comparing the unknown weights. 

3.5. If tare weights and unequal nominal values are used, use the following equation 

for modifying section 3.3: 

Cx = Cs + CMts – CMtx + ( Xn – O1 ) + Ns – Nx

4. Measurement Assurance 

4.1. Duplicate the process with suitable check standards for each nominal load or a 

designated number of check standards per balance (See GLP 1, SOP 9, SOP 30, 

and Sec. 7.4). 

4.2. Evaluate the value against the expected limits and plot the check standard value to 

monitor changes over time; a t-test may be incorporated to check observed value 

against accepted value. 

4.3. The mean and observed values of the check standard are used to evaluate bias and 

drift over time. 

4.4. Check standard observations are used to calculate the standard deviation of the 

measurement process, sp. 

5. Assignment of Uncertainty 

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

the mass standards used, us, estimates of the standard deviation of the measurement 

process, sp, and estimates of the effect of other components associated with this 

procedure, uo. These estimates should be combined using the root-sum-squared method 

(RSS), and the expanded uncertainty, U, reported with a coverage factor of two (k=2), to 

give us an approximate 95 % level of confidence. See SOP 29 for the complete standard 

operating procedure for calculating the uncertainty. 

5.1. The expanded uncertainty for the standard, U, 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 usually 

need to be divided by the coverage factor k. 

5.2. 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 modified substitution measurements. 

5.3. Uncertainty associated with allowable drift. Include the monitored average drift 

for each balance and procedure as a rectangular distribution, or include a value 

that is approximately 10 % of the applicable tolerance for weight weight, again as 

a rectangular distribution. Whenever the drift exceeds 10 %, conduct a root cause 

analysis to determine the cause; this situation may require fewer weights to be 

calibrated in the weighing series, service of the balance, better environmental 

controls, or additional training and practice by the metrologist. 
February 2012 

SOP 8 Page 9 of 12 

5.4. Uncertainty associated with allowable sensitivity error. This procedure allows for 

up to 2 % error in the optical or electronic range of use. This may result in 

calibration errors up to 2 % of the applicable tolerance. This is potentially another 

uncorrected systematic error that may be treated as a rectangular distribution. 

5.5. Uncertainty associated with uncorrected air buoyancy. When buoyancy 

corrections are not made, the resulting uncorrected systematic error may be 

treated as a rectangular distribution, with the magnitude of the buoyancy 

correction determined using equations provided in SOP 2. 

5.6. Uncertainty associated with bias that is observed in the check standards as 

determined through analysis of the control charts. When bias is observed in the 

control charts, it must be assessed according to the equations provided in SOP 29 

and may be incorporated as an uncorrected systematic error using the equations in 

SOP 29. 

5.7. The expanded uncertainty, U, must be  1/3 of the tolerance applicable as per 

ASTM E617-97, OIML R111, or NIST Class F. 

6. Report 

 Report results as described in SOP No. 1, Preparation of Calibration/Test Reports. 
February 2012 

SOP 8 Page 10 of 12 

Appendix - Modified Substitution Data Sheet 

Laboratory data and conditions: 

Operator Before After 

Date Temperature 

Balance Pressure 

Load Relative Humidity 

Standard deviation of the process, 

from control chart, sp

Degrees of Freedom

Mass standard(s) data:

ID Nominal Mass Correction* Expanded Unc: 

From cal. Report 

Unc: 

k factor 

Density 

g/cm3 

S 

tS 

X TBD TBD TBD 

tx 

Sc 

tSc 

sw 

*Mass Correction = True Mass with buoyancy correction. Mass Correction = Conventional Mass with no buoyancy correction. 

Observations:

Observation No. Weights (Insert ID of X) Balance Observations, Units 

Time: 

1 (O1) S 

2 (O2) S + sw 

Error < 2 % of optical scale or electronic range? _____Yes_____No

3 (O3) S

4 (O4) X1

5 (O5) X2

6 (O6) X3

7 (O7) X4

8 (O8) X5

9 (O9) X6

10 (O10) X7

11 (O11) X8

12 (O12) X9

13 (O13) X10

14 (O14) Sc

15 (O15) S 

Time: Drift < 1/10 Tol. ?: _____Yes_____No

Check Standard in Control? _____Yes_____No 

Up to 10 unknown weights may be checked with this procedure if the drift is less than 1/10 of the tolerance. 
February 2012 

SOP 8 Page 11 of 12 

Appendix - Modified Substitution Data Sheet 

Laboratory data and conditions: 

Operator GH Before After 

Date 9/30/96 Temperature (C): 22.5 22.8 

Balance PM 2000 MC (2300 g) Pressure (mm Hg): 747.5 747.1

Load 5 lb Relative Humidity (%) 45 43

Standard deviation of the 

process, from control chart, sp

0.85 mg Degrees of Freedom 120

Mass standard(s) data:

ID Nominal Mass Correction Expanded Unc: 

From cal. Report 

Unc: 

k factor 

Density 

g/cm3 

 Set W S 5 lb 7.5 mg 0.50 mg 3 7.85 

ts None 

X 5 lb TBD TBD TBD 7.85 

tx None 

 Set S Sc 5 lb 2.4 mg 0.73 mg 2 7.95 

tSc None 

 Set W sw 5 g 0.000 002 g 

Mass Correction = Conventional Mass if NOT using buoyancy correction. Density is used only with buoyancy corrections. 

Observations:

Observation No. Weights Balance Observations, Units 0.001 g 

Time: 11:00 am 

1 (O1) S 0.000 

2 (O2) S + sw 4.999 

Error < 2 % of optical scale or electronic range? _____Yes_____No

3 (O3) S 0.000 

4 (O4) SN: 3424-12 X1 0.205 

5 (O5) SN: 3424-22 X2 0.104 

6 (O6) SN: 3424-32 X3 -0.089 

7 (O7) Sc -0.005 

8 (O8) S 0.003 

Time: 11:20 am Drift < 1/10 Tol. ?: _____Yes_____No

Check Standard in Control? _____Yes_____No 

Up to 10 unknown weights may be checked with this procedure if the drift is less than 1/10 of the tolerance. 
February 2012 

SOP 8 Page 12 of 12 

Sensitivity guidelines indicate a sensitivity weight, greater than or equal to 2 times the tolerance 

(460 mg) but not exceeding 1 % of the balance capacity (23 g). For a 5 lb load, the Class F 

tolerance is 230 mg so a 5 g weight was selected. 

A 2 % sensitivity error in a 5 gram range would allow readings between 4.900 g and 5.100 g in 

the sensitivity test. Calculate the sensitivity: 

which deviates less than 2 % of the nominal value of a division. 

The drift observed was 3 mg and is less than 1/10 of the tolerance and 23 mg could be allowed, 

so the error for each nominal weight may be calculated as follows. 

Calculate the uncertainty for the calibration: 

U = uc * 2 

2

bias

2

sensitivity

2

drift

2

p

2

c s u  u  s  u  u  u

The uncertainty for the standard must be divided by the k factor for the standard and the tare 

weight to determine each us. The additional uncertainty for not performing the air buoyancy 

correction can be determined using the magnitude of the air buoyancy correction from SOP 2. 

Since the working standard was previously calibrated with buoyancy corrections and is the same 

density as the unknown weights, the uncertainty for the buoyancy correction drops out of the 

equation. The tolerance for the 5 lb weight is 230 mg, so 1/10 of the tolerance is 23 mg which is 

divided by the square root of three to get 13 mg. The sensitivity error is multiplied by 230 mg 

(maximum deviation that would be allowed before adjustment) to determine a potential error of 

0.04609 mg, again divided by the square root of three to get 0.0266 mg. We have not included a 

component for bias in this example. 

2 2 2 2

c u  ( 0.167 )  ( 0.85 )  (13 )  ( 0.0266 )

uc = 13.028 856 mg 

U = 13.028 856 * 2 = 26.057 713.732 38 mg = 26 mg (when rounded correctly) 

The expanded uncertainty is less than 1/3 of the tolerance (76 mg) and the absolute value of the 

combined correction and uncertainty are within the applicable tolerance, so the unknown weights 

can either be reported as “within tolerance” or the values and uncertainties can be reported. The 

value plus the uncertainty for the first unknown X1 is within 25 % of the Class F tolerance, 

therefore it is desirable to adjust the weight closer to nominal value, although based on the 

tolerances and uncertainties, and it can be claimed as “within tolerance.” 

1.000 200 4 g/div 4.999 div

5.000 002   g sensitivity

C 7.5 mg ( -5 mg - 0) 2.5 mg

C 7.5 mg ( - 89 mg - 0 ) - 81.5 mg

C 7.5 mg (104 mg - 0 ) 111.5 mg

C 7.5 mg ( 205 mg - 0 ) 212.5 mg

Sc

x

x

x

3

2

1

  

  

  

  
February 2012 

SOP 9 Page 1 of 7 

 SOP No. 9 

 Recommended Standard Operating Procedure 

 for 

 Control Charts for Calibration of Mass Standards

 

1. Introduction 

 

1.1. This SOP describes procedures for the development of control charts and their use 

for demonstrating attainment of statistical control of a mass calibration process. 

 

1.2. Prerequisites 

The use of this SOP requires that appropriate apparatus, methodology, and standards 

are available, and that the laboratory thoroughly understands the basic principles of 

the measurement process used and has had sufficient experience to perform the 

necessary operations required for the measurements of concern. 

2. Summary 

 

An appropriate check standard (or control standard) is incorporated into the measurement 

process and weighed at established intervals, the results are plotted on an x (X-bar) chart. 

The abscissa (x) represents the sequence of measurements and the ordinate (y) the measured 

values. A central line is drawn, indicating the mean ( x ) of the measured values and control 

limits are indicated within which the results of measurements are expected to be randomly 

distributed, based on statistical considerations. The system is considered to be in statistical 

control when the individual values are within the designated limits. The system is considered 

to be out of control if values are present outside established limits for which no reasonable 

and correctable cause have been determined and corrected, unusual trends are observed, or if 

the mean exceeds the control limits. The statistical information on which the control limits 

are based can be used to calculate confidence limits for measurements made while the 

system is demonstrated to be stable and in a state of statistical control. 

3. Procedure 

3.1. Definition of Monitored System 

The monitored system is considered to consist of the balance, the standard operating 

procedure, the laboratory environment, the check standard or control standard, the 

operator, and any other sources that contribute to the variance or bias of the 

measurement data. Any of the above that can be considered to be constant or 

negligible contributors to the variance may be consolidated and monitored by a 

single control chart. Any that cannot be so considered (for example: different 

standard, different balance, different SOP) typically require separate control charts. 
February 2012 

SOP 9 Page 2 of 7 

The variability of balance precision that is load dependent must be considered. For 

many balances, precision is a function of load, and a distinct control chart is required 

(in principle) for every load tested. This is not always feasible, except in the case of 

SOP 5 or 28 where check standards are incorporated into the measurement process. 

Hence, control charts used for measurement assurance and evaluation of 

measurement uncertainty are generally satisfactory if developed using data from 

check standards at two or three intervals for each balance appropriately spaced 

within the range of balance use, or at least with one check standard for each decade. 

On balances where few nominal values (loads) are tested, a control chart should be 

established for each load. 

3.2. Selection, Care, and Calibration of Check Standards 

3.2.1. A check standard used in high precision calibration measurement must be 

stable and normally be comparable to the primary standard or to the typical 

calibration work, depending on what is being monitored. For lower order 

calibrations, the check standard should simulate the laboratory's primary 

standards to the extent feasible. It should be calibrated with an expanded 

uncertainty equal to or better than the precision of the process being 

monitored. All check standards should be cared for in the same way as 

primary standards to prevent their damage or deterioration. Lower order 

check standards should be recalibrated at regular intervals according to Good 

Measurement Practice (GMP) 11 on Setting and Adjusting Calibration 

Intervals. 

Table 1. Recommended check standards for typical test situations. 

Balance Range of Measurement Check Standard(s) 

Echelon III (Class F) 

SOP 7, 8 

5000 lb to 0.001 lb or 

2268 kg to 1 mg 

2 to 3 values per balance OR 

1 chart per load 

Echelon II 

SOP 3, 4, 6, 7 

5000 lb to 0.001 lb 

and 1000 kg to 1 mg 

2 to 3 values per balance 

OR 1 check standard per decade

SOP 5 typically 1 kg to 1 mg 

each nominal value incorporates 

a check standard 

SOP 28 typically 1 kg to 1 mg 

1 check standard per decade 

(e.g., 1 kg, 100 g, 10 g, 1 g, 

100 mg, 10 mg, 1 mg) 

3.3. Establishing Control Chart Parameters 

3.3.1. The control chart parameters consist of the central line, the best estimate of 

the mean of measurements of the check standard, and control (or “action”) 
February 2012 

SOP 9 Page 3 of 7 

and warning limits that represent probabilistic limits for the distribution of 

results around the central line. These parameters are evaluated on the basis of 

a reasonable number of initial measurements and updated as additional 

measurement data are accumulated. 

3.3.2. Make at least seven (minimum number) and preferably 12 or more, 

independent measurements of the check standard under the same conditions 

that will be used to make routine measurements. No two measurements 

should be made on the same day and the time of day should be varied as 

would be typical during routine laboratory operations. This is necessary to 

estimate the long-term standard deviation to the extent feasible. To make 

statistically valid decisions or calculate uncertainties based on this data, 25 to 

30 points are necessary. 

Calculate the mean, x and the estimate of the standard deviation, s in the 

conventional manner. 

Establish the control chart parameters as follows: 

Central Line = x

Upper Control/Action Limit = x + 3 s

Upper Warning Limit = x + 2 s

Lower Warning Limit = x - 2 s 

Lower Control/Action Limit = x - 3 s 

Control chart parameters for Echelon III (Class F or other) may be completed 

as follows to track practical limits: 

Central Line = x

Upper Control/Action Limit = x + 1/4 tolerance 

Upper Warning Limit = x + 1/10 tolerance

Lower Warning Limit = x - 1/10 tolerance

Lower Control/Action Limit = x - 1/4 tolerance
February 2012 

SOP 9 Page 4 of 7 

3.3.3. Upgrading Control Chart Parameters 

Upgrade control chart parameters when a significant amount of additional 

data is available or when the previously determined parameters are no longer 

pertinent due to changes in the system. 

Note: Ordinarily, upgrading is merited when the amount of new data is equal 

to that already used to establish the parameters in use, or when at least seven 

 additional data points have been recorded. 

Calculate x and s for the new set of data and examine for significant 

differences from the former using the t-test and F-test, respectively. If the 

tests fail and results are significantly different, determine the reason for the 

difference, if possible, and decide whether corrective action is required. If 

data does not agree within statistical limits, establish new parameters using 

the most recent data and note the reasons for not using previous data. If no 

significant differences between the data sets are found, pool all data and 

calculate new control chart parameters based on all existing data. 

3.4. Frequency of Measurement 

The check standard should be measured and plotted with sufficient frequency to 

minimize the risk of loss of data during the period from last-known-in to firstknown-out of control condition. It is good practice to measure the check standard at 

least once during each period when a set of test measurements is made. For critical 

calibrations or those of highest accuracy, it is desirable to alternate measurements of 

test items and check standards, but for real-time evaluation it is preferable to 

incorporate the check standard in the calibration design as in SOP 5 or SOP 28. 

Whenever there has been a long period of inactivity, it is good practice to make a 

series of measurements of the check standard and to plot the results on a control chart 

to demonstrate attainment of statistical control prior to resuming measurements with 

that specific calibration system. 

Control charts should be updated as close to real time as feasible to effectively 

monitor the measurement process to prevent the possible release of questionable data 

resulting in recall. 

4. Use of Control Charts 

4.1. Monitoring a Measurement Process 

Use the following criteria to interpret control chart results. 

4.1.1. If plotted points are stable and randomly distributed within the warning 

limits, decide that the system is in control. If the process is in control (and the 
February 2012 

SOP 9 Page 5 of 7 

process statistic is normal), 99.73 % of the points will fall within the 

control/action limits. Observations outside the limits, or systematic patterns 

within, should be investigated. 

4.1.2. If a plotted point is outside the warning limits but within the control limits, 

investigate the presence of calculation errors. If none were made, re-measure 

the check standard. The re-measured value must be within the warning limits 

to merit the decision of "in control". If the results are not within limits, 

consider the measurement process "out of control". Reject all data obtained 

since last "in-control" measurement and take corrective action (hence 

“action” limit). Accept no further data until the system is demonstrated to be 

in-control as indicated by at least two successive measurements of the check 

standard within the warning limits. 

 If a plotted point is outside the control limits and arithmetically correct, the 

system is considered to be out of control. Data are rejected, corrective actions 

must be taken and re-attainment of statistical control demonstrated, as above, 

before data may be accepted. 

4.1.3. Additional guidelines for the evaluation of control charts based on 

probability statistics should be used to evaluate the presence of drift, shifts, 

and possible bias. Examples for further evaluation include: 

4.1.3.1. Any single point or series of points outside of three standard 

deviations (keeping in mind the probability that a three standard 

deviation limit could reasonably expect to allow three points out of 

one thousand to be outside these limits); 

4.1.3.2. Two of the last three points above (or below) two standard 

deviations; 

4.1.3.3. Four of the last five points above (or below) one standard 

deviation; 

4.1.3.4. Eight consecutive points on one side of the mean or reference 

value; 

4.1.3.5. Six points in a row trending up (or down); and 

4.1.3.6. Fourteen points alternating up and down. 

4.2 Monitoring a Reference Value 

4.2.1. When a Reference Value for the check standard is outside of one standard 

deviation of the mean value, it may necessitate obtaining an updated 

calibration. The deviation or offset must be considered with respect to the 

reported uncertainty for the reference value as well as the measurement 

process being used to evaluate the value. In some cases, such as with very 

large tolerances, a measurement process might be quite small compared to 

the offset, in which case the offset can be used as an uncorrected systematic 
February 2012 

SOP 9 Page 6 of 7 

error in the uncertainty calculations. Where tolerances are small or 

uncertainty requirements stringent, updated calibrations may be required. 

Note: the reference value may exceed acceptable limits even when a measurement 

process appears statistically in control. When tolerances are large compared to 

process variability, reference value offsets may simply be noted. 

4.3. Transfer of Measurement Statistics 

4.3.1. Absence of a significant difference between the central line and the accepted 

value for the check standard may be considered as evidence of insignificant 

bias at the level of confidence of the statistical test used. This conclusion is 

valid, as long as the system remains in control. On occasion, small 

differences (less than one standard deviation) from unknown sources will be 

obvious over time and the value observed for the bias should be incorporated 

into the uncertainty per SOP 29. 

4.3.2. The estimate of the standard deviation of the process, sp, used to establish the 

control limits may be used to calculate confidence intervals for all pertinent 

measurements made while the system is in control. However, see SOP 29 for 

calculation of measurement uncertainty using the process variability, sp. The 

value of the test weight is said to be within the limits 

where y represents the mean of the measurements on the test weight and 

2 2 2 *

S P O U  k u  s  u , with the value of k determined by the confidence level 

required for the interval. 

y  U
February 2012 

SOP 9 Page 7 of 7 

Appendix A - Checklist for Creation of Control Charts 

Control Charts cover entire scope (are available for each measurement parameter). 

 

Control charts have titles (or otherwise include) 

 Laboratory name or other identifying information 

 SOP(s) 

 Balance/Other equipment 

 Standard ID, Check standard ID 

 Nominal load/value 

 Dates/chronology/time periods if not used on the x axis 

 Legends if multiple series and extra information are plotted; to avoid confusion 

 

Control charts have x and y axis with labels and 

 All measurement values have units of measure associated with them 

 

Control charts have 

 Mean value (and units) 

 Standard deviation (and units) 

 Degrees of freedom or number of points noted (if not obvious or if small number) 

 Alternative summaries of this information and suitable references (e.g., tables) 

 Printed summary reports of control data with the charts 

 Reference values and source and bias if appropriate/available 

 

Control charts have limits that are based on 

 Statistical controls of 

 Warning limits (i.e., two standard deviations) and 

 Action/control limits (i.e., three standard deviations) 

 Or specification limits (e.g., tolerances or smaller ratios of tolerances) 

 

Good Items (on chart or in spreadsheet or database table summaries). Control charts have (when 

applicable and meaningful if not otherwise noted, e.g., in a table) 

 Tolerances: when applicable 

 Uncertainties: for reference value, check standard, and the process output 

 Equipment information: device readability, configuration (stability settings/timing) 

 Standard information: calibration date and interval information 

 Responsible staff: need on chart or in database 

 Status of control: in control, out of control with latest date of review 

 History: previous limits and history of the chart/data with F-test and/or t-test results 
This page intentionally made blank. 
February 2012

SOP 29 Page 1 of 10 

SOP No. 29 

Standard Operating Procedure 

for the 

Assignment of Uncertainty 

1. Introduction 

1.1. Purpose 

Laboratories performing calibrations that meet ISO/IEC Guide 25, ISO/IEC 

17025, or ANSI/NCSL Z 540-1-1994 must report uncertainties in conformance 

with the ISO Guide to the Expression of Uncertainty in Measurement (hereafter 

called the GUM). This SOP provides instruction for the laboratory to meet this 

requirement. 

1.2. Prerequisites 

1.2.1. Calibration certificates with valid uncertainties must be available for all 

standards. 

1.2.2. Statistical data regarding the calibration measurement process must be 

available; preferably from measurement control programs within the 

laboratory. 

1.2.3. Knowledge of the technical basis for the measurement is critical for 

completeness in uncertainty evaluation. This can be obtained through 

reference papers, reference procedures, brainstorming, experimentation, 

interlaboratory comparisons, cause and effect diagrams and the like. Flow 

charts at the end of this SOP show a number of common factors affecting 

measurements in the areas of mass, length, and volume. 

2. Methodology 

2.1. Scope, Precision, Accuracy 

Each measurement made in a laboratory has a corresponding uncertainty assigned 

to the calibration value. The uncertainty is directly related to the measurement 

parameter (scope), range of the measurement, the equipment or measurement 

process being used (affecting precision), and the standards available with 

associated uncertainties. 

2.2. Summary 

This uncertainty analysis process follows the following eight steps: 

1) Specify the measurement process; 

2) Identify and characterize uncertainty components; 
February 2012

SOP 29 Page 2 of 10 

3) Quantify uncertainty components in applicable measurement units; 

4) Convert uncertainty components to standard uncertainties in units of the 

measurement result; 

5) Calculate the combined uncertainty; 

6) Expand the combined uncertainty using an appropriate coverage factor; 

7) Evaluate the expanded uncertainty against appropriate tolerances, user requirements, 

and laboratory capabilities; and 

8) Report correctly rounded uncertainties with associated measurement results. 

Special methods for handling bias/errors are addressed as well. 

3. The Process of Measurement Uncertainty Estimation 

3.1. Step 1. Specify the process. 

Clearly specify the measurement process in question, including the item being measured 

and the input quantities upon which it depends. This will usually require a quantitative 

expression related to the process (i.e., a measurement equation). Where possible, you 

may reference an SOP or other method description along with the specific standards and 

measurement assurance process that is used to adequately complete this step. 

3.2. Step 2. Identify and characterize uncertainty sources. 

Identify all possible sources of uncertainty in a comprehensive list, characterizing them 

based on the evaluation method that will be used to quantify them (Type A, statistical 

methods or Type B, scientific judgment) and to categorize them based on their 

relatedness with something such as an uncertainty budget table or a cause and effect 

diagram. Using the measurement equation identified in 3.1 provides a good starting 

point. All of the parameters in this expression may have an uncertainty associated with 

them. When there are discrete steps in the measurement process, additional uncertainties 

may be associated with each. 

What follows are the most common uncertainties associated with metrological 

measurements. Keep in mind that this list is not exhaustive. Each item listed below is 

identified as a standard uncertainty u, when determined using Type B methods of 

evaluation (technical judgment, theoretical assessments, reference items) and a standard 

uncertainty s, when determined with Type A methods of evaluation (statistical methods). 

Each standard uncertainty is further defined by an arbitrary subscript related to the 

source for ease in remembering that source. 

3.2.1.Standard uncertainty from the measurement process, sp, (Type A evaluation). 

3.2.1.1. Standard deviation from a measurement assurance chart or control chart. 

The value for sp is obtained from the control chart limits and the current 

knowledge that the measurements are in a state of statistical control. This 
February 2012

SOP 29 Page 3 of 10 

will have to be ascertained by measuring at least one check standard 

during the course of the current measurements combined with data 

accumulated using the same process in previous measurements. The 

control chart data must reflect the measurement process being performed. 

3.2.1.2. Standard deviation from a series of replicate measurements. 

Measure a stable test object at least seven times, no two measurements of 

which may be made on the same day. Calculate the standard deviation in 

the conventional manner. The latter is the value of sp, keeping in mind 

that it does not fully represent the measurement process under all 

typically encountered conditions. 

Note: Repetitive measurements made on the same day estimate the 

short-term standard deviation of the process and may underestimate 

actual measurement variability. 

3.2.1.3. Standard deviation when a process standard deviation is “zero”. 

When the standard deviation is less than the resolution of the measuring 

instrument, and a check standard is not available, the standard deviation 

must be assessed. Generally, the smallest standard deviation to be used in 

this instance may be estimated as 0.577 times the instrument resolution 

from the equation noted below, with d being the instrument resolution. 

Assess the standard deviation and use the larger of it or the value from 

this equation. 

3

d

s p 

3.2.2.Standard uncertainty for the standards, us (Type B evaluation). 

3.2.2.1. When using standards calibrated by another laboratory. 

The information for the standards comes from the calibration report, 

generally reported as an expanded uncertainty with its coverage factor 

(k). The expanded uncertainty is simply divided by the stated k value to 

obtain the combined uncertainty for the standard, uc, which represents 

the us when used in your laboratory. When k is not equal to two, 

additional factors associated with the degrees of freedom must be 

included. See Appendix A to determine the degrees of freedom 

associated with coverage factors. Keep track of the degrees of freedom 

for later use. 

3.2.2.2. When using a standard calibrated in your laboratory (Type B evaluation). 
February 2012

SOP 29 Page 4 of 10 

If the standard was calibrated in your own laboratory, calculate the 

combined standard uncertainty, uc, at k=1 and use that as the standard 

uncertainty for the standard, us. 

3.2.2.3. When using more than one standard (Type B evaluation). 

When more than one standard is used in a calibration, the standard 

uncertainty for each, us1, us2, us3, etc., is included in the RSS equation if 

the standards have had independent calibrations. When calibrations are 

performed at the same time, or when a standard is used multiple times, 

the standards are likely considered dependent, so the standard 

uncertainties are added (us1 + us2) to determine a value to represent us. 

(This will be the case with two 1 kg standards that were calibrated at the 

same time using a weighing design and subsequently used together as 

standards (restraints) in a weighing design. It will also be the case of a 25 

gal standard used four times to calibrate a 100 gal standard.) Standards 

with independent calibrations are combined by standard root sum square 

methods. 

3.2.3.Standard uncertainty due to other factors, uo (Type B evaluation.) 

These are factors related to the measurement equation, but distinct from the 

standard uncertainties associated with the process and the standards. These items 

are often much smaller in a well-controlled process than the standard uncertainties 

associated with the process and the standards. Examples are given in each 

calibration SOP. 

The laboratory must carefully consider any other components of uncertainty that 

might contribute to the uncertainty of the measurement. In mass measurements, 

these might include effects from the magnitude of air buoyancy corrections as 

discussed in SOP 2 when these corrections are not made, or the uncertainty from 

the calculation of air density when buoyancy corrections are made (also discussed 

in SOP 2). In volume, this might include uncertainties associated with reading a 

meniscus, as described in GMP 3. 

Additionally, the laboratory should include any other components that are not 

negligible, such as the uncertainty associated with the density of the standards, the 

density of the unknown test items, effects due to lack of thermal equilibrium 

during the test, or known and quantified effects due to magnetized mass standards. 

Each component that is considered should be included as an additional standard 

uncertainty uo1, uo2, uo3, etc., and included in the RSS equation when data shows 

these factors to be significant. Note that components of uncertainty are not limited 

to those mentioned. Documentation of each component evaluated should be 

maintained to complete the documentation required by the specific calibration 

procedure and ISO/IEC 17025. 
February 2012

SOP 29 Page 5 of 10 

3.2.4.Standard uncertainty due to factors unrelated to the measurement process, uu.

These are factors that may be related to characteristics of the items being testing 

or of the standard and are usually minimized in well-known and controlled 

measurement processes. 

3.2.5.Special uncertainties from other sources (Type B evaluations). Includes bias or 

unidentified errors. 

It is a general requirement of the GUM that corrections be applied for all 

recognized and significant systematic effects and potential errors. Where a 

correction is applied based on a bias, an estimate of the associated uncertainty 

must then be included in the uncertainty analysis. Due to the various approaches 

present in the metrology system, several examples and possible approaches are 

presented in the section on calculating the combined or expanded uncertainties. At 

this stage, a determination must be made with regard to 1) identifying cause and 

2) level of significance. 

3.2.5.1. Identifying bias (offsets) and cause. 

Bias or measurement offsets from reference values are often noted on 

control charts when an independent reference value is available and from 

results in proficiency testing. Control chart values are obtained over 

time and usually have more degrees of freedom than results from one or 

a few proficiency testing results. Therefore, noted offsets on control 

charts are more reliable estimates of bias when independent reference 

values are available. Evaluation of data from multiple sources of data 

should be considered whenever available. 

If the cause can be identified, it is usually corrected or applied to the 

measurement equation. In some cases, it is not possible to unarguably 

define the cause without exhaustive studies that provide little benefit. In 

those cases, the significance level must be evaluated before incorporating 

this type of an uncorrected systematic error in the calibration 

uncertainties. 

3.2.5.2. Significance level. 

When there is little to be gained from exhaustive studies on the 

measurement process to identify bias or potential errors, a test of 

significance may be conducted to determine alternative approaches for 

incorporating the bias into the uncertainty calculations. 

In deciding whether a known bias can reasonably be included in the 

uncertainty, the following approach may be followed: 
February 2012

SOP 29 Page 6 of 10 

3.2.5.2.1. Estimate the combined uncertainty without considering the 

relevant bias. 

3.2.5.2.2. Evaluate whether or not the bias is less than two times the 

combined uncertainty (i.e., 2 + s + u

2

o

2

p

2 bias  us ). 

3.2.5.2.3. If the bias is less than twice the combined uncertainty, it may 

be included in the uncertainty using one of several 

approaches that must clearly be communicated in the report. 

3.2.5.2.4. If the bias is larger than twice the combined uncertainty, the 

error must be investigated further prior to providing 

calibration data. 

If the deviations show that a standard is out of control, it 

should not be used for calibration until corrective action has 

been taken and the value for the standard is verified as being 

within criteria limits. 

 

lab uc criteria limit : x  xref  2

If these differences are smaller than the criteria limits, 

investigation and corrective action may be unrealistic. If the 

deviations are less than the surveillance limits shown above, 

and corrective action is not taken, the deviations may be 

included in the uncertainty statement following one of 

several options given in the following section. In ALL cases, 

the method used to incorporate bias must be clearly reported. 

3.2.5.3. Adding the bias to the expanded uncertainty (e.g., used in PMAP 

software). In this case, the bias is simply added to the expanded 

uncertainty and is reported as such. 

3.2.5.4. Example when uncertainties for the laboratory data and the reference 

data are considered equivalent (e.g., laboratory data is compared to data 

from another laboratory having equivalent precision). 

In this case, a rectangular distribution is considered where the value 

might possibly be anywhere within the range shown for each laboratory 

data point. This method is referenced in section 4.6 of NIST Technical 

Note 1297. This approach may also be used in the case where a standard 

U  bias  u *k  bias c
February 2012

SOP 29 Page 7 of 10 

is predictably drifting with use over time. In this case, a mid-range value 

is chosen and ud (uncertainty for differences) is calculated as follows: 

 or more simply : 0.29 d, where d is the bias

3

1

2

bias  d

u

3.2.5.5. Example when uncertainties for the laboratory data are considered 

secondary to a reference value (e.g., the difference between the 

laboratory data and NIST data). This method was originally published in 

1994. 

In this case, a reference value is given precedence over the laboratory 

data and a mid-range value is not chosen. The extreme value is more 

probable. In this case, the following equation may be used: 

 or more simply : 0.577 d, where d is the bias

3

bias  d

u

3.3. Step 3. Quantify uncertainty estimates 

All uncertainty estimates identified in the previous step must be quantified in units that 

represent the measured values. Type A methods of evaluation usually provide quantified 

estimates in the units of interest. 

Type B methods of evaluation may be conducted with spreadsheets using the basic 

expression identified in the SOP or identified when the process was specified. Scenario 

testing can be done to determine the impact and quantify specific variables on the final 

measured quantity. The knowledge gained in this step often proves useful in identifying 

potential areas of improvement especially if contributing factors are graphed in a 

histogram. 

 

3.4. Step 4. Convert all factors to standard uncertainties 

In those cases where the uncertainty factors were determined statistically (Type A 

methods), the standard deviation is used to represent the standard uncertainty. In other 

cases, estimates must be made to ensure that the quantified uncertainties represent “onestandard-deviation” values or a k=1 coverage level. 

The appropriate distribution factor must be used when converting estimated uncertainty 

values to standard uncertainties. According to NIST Technical Note 1297, a rectangular 

distribution is generally used when detailed information about the distribution is 

unknown. 

3

value

un 
February 2012

SOP 29 Page 8 of 10 

3.5. Step 5. Calculate the combined uncertainty 

The combined standard uncertainty, uc, includes the standard uncertainty reported for 

the standards used, us, the standard uncertainty of the measurement process, sp, the 

standard uncertainty from other sources, uo, which includes all other factors the 

laboratory considers significant, the standard uncertainty due to factors related to the 

measured item but unrelated to the measurement process, uu, and finally, the standard 

uncertainty due to bias or differences, ud, when ud is included. The standard uncertainties 

are usually combined using the root-sum-of-the-squares (RSS) method as follows: 

Table 1. Symbol descriptions. 

Symbol Description 

U Expanded uncertainty 

uc combined standard uncertainty 

sp standard uncertainty (standard deviation) of the “process” 

us standard uncertainty of the “standard” 

uo standard uncertainty of “other factors” 

uu

standard uncertainty of factors “unrelated” to the measurement 

process 

ud

standard uncertainty of “differences” (may be treated in different 

ways) 

k coverage factor 

3.6. Step 6. Calculate the expanded uncertainty 

The combined standard uncertainty is then multiplied by a coverage factor, k, based on 

the degrees of freedom, to provide a level of confidence of approximately 95 % or 99 %, 

respectively (depending on what is required by the customer, but most often 95 %). The 

equation used to determine the expanded uncertainty is as follows: 

where k = 2 a value from the table in Appendix A, taken from the column 95.45. When 

there are a small number of degrees of freedom, the coverage factor must be determined 

from a statistical table such as provided in the Guide to the Expression of Uncertainty in 

Measurement or NIST Technical Note 1297 (See Appendix A). 

Situations where the degrees of freedom associated with the value for a standard in the 

traceability chain require special care to ensure that an appropriate coverage factor is 

used for calculating the uncertainty of subsequent reported values. When there are a 

U u k  c *

2 2 2 2 2

c p us uo uu ud u = s    
February 2012

SOP 29 Page 9 of 10 

small number of degrees of freedom used to determine the standard deviation of the 

process and/or the k value from the calibration report represents a small number of 

degrees of freedom, the effective degrees of freedom must be used to determine an 

appropriate coverage factor. Use the equation provided in Section 8 or the WelchSatterthwaite equation provided in NIST Technical Note 1297 or the Guide to the 

Expression of Uncertainty in Measurement. 

3.7. Step 7. Evaluate the expanded uncertainty 

The expanded uncertainty may be evaluated against established criteria such as 

tolerance limits, customer requirements, and/or calibration and measurement capabilities 

on the laboratory scope. For example, the specifications for mass standards clearly state 

that the expanded uncertainty must be less than 1/3 of the tolerance. 

3.8. Step 8. Report the uncertainty 

Once the uncertainty has been calculated, round the value according to GLP 9, report 

the value, and include a statement such as the following is reported: 

The combined standard uncertainty includes the standard uncertainty reported for the standard, 

for the measurement process, and for any observed deviations from reference values (e.g., 

from NIST), which are less than surveillance limits. The combined standard uncertainty is 

multiplied by k, a coverage factor of (insert value used) to give the expanded uncertainty 

which defines an interval with an approximate 95 % level of confidence. 

The value selected for the coverage factor must be appropriate for the degrees of freedom 

available and the desired level of confidence associated with the uncertainty as noted in Step 

7. 
February 2012

SOP 29 Page 10 of 10 

Appendix A 

Value of tp() from the t-distribution for degrees of freedom that defines an interval tp() to +tp() that 

encompasses the fraction p of the distribution. Note: Table is taken from NIST Technical Note 1297.

Degrees of 

freedom 

ν

Fraction p in percent 

68.27(a) 90 95 95.45(a) 99 99.73(a) 

1 1.84 6.31 12.71 13.97 63.66 235.80 

2 1.32 2.92 4.30 4.53 9.92 19.21 

3 1.20 2.35 3.18 3.31 5.84 9.22 

4 1.14 2.13 2.78 2.87 4.60 6.62 

5 1.11 2.02 2.57 2.65 4.03 5.51 

6 1.09 1.94 2.45 2.52 3.71 4.90 

7 1.08 1.89 2.36 2.43 3.50 4.53 

8 1.07 1.86 2.31 2.37 3.36 4.28 

9 1.06 1.83 2.26 2.32 3.25 4.09 

10 1.05 1.81 2.23 2.28 3.17 3.96 

11 1.05 1.80 2.20 2.25 3.11 3.85 

12 1.04 1.78 2.18 2.23 3.05 3.76 

13 1.04 1.77 2.16 2.21 3.01 3.69 

14 1.04 1.76 2.14 2.20 2.98 3.64 

15 1.03 1.75 2.13 2.18 2.95 3.59 

16 1.03 1.75 2.12 2.17 2.92 3.54 

17 1.03 1.74 2.11 2.16 2.90 3.51 

18 1.03 1.73 2.10 2.15 2.88 3.48 

19 1.03 1.73 2.09 2.14 2.86 3.45 

20 1.03 1.72 2.09 2.13 2.85 3.42 

25 1.02 1.71 2.06 2.11 2.79 3.33 

30 1.02 1.70 2.04 2.09 2.75 3.27 

35 1.01 1.70 2.03 2.07 2.72 3.23 

40 1.01 1.68 2.02 2.06 2.70 3.20 

45 1.01 1.68 2.01 2.06 2.69 3.18 

50 1.01 1.68 2.01 2.05 2.68 3.16 

100 1.005 1.660 1.984 2.025 2.626 3.077 

 1.000 1.645 1.960 2.000 2.576 3.000 

(a)For a quantity z described by a normal distribution with expectation μz and standard deviation σ, 

the interval μz ± k σ encompasses p = 68.27, 95.45, and 99.73 percent of the distribution for k = 1, 2, 

and 3, respectively. 
February 2012 

SOP 30 Page 1 of 9 

 

SOP 30 

Recommended Standard Operating Procedure 

for a 

Process Measurement Assurance Program 

1. Introduction 

1.1. Purpose 

The Process Measurement Assurance Program (PMAP) is used for the control 

and surveillance of measurement performance using check standards in 

measurement and calibration procedures. Incorporation of these measurement 

control steps ensures the validity of the measurement process and the standards 

used. The variables used in calculation and assignment of uncertainty can be 

determined and controlled by the use of this Standard Operating Procedure 

(SOP). 

 

1.2. Prerequisites 

1.2.1. A Standard Operating Procedure that describes and provides rigor and 

consistency in the calibration process. 

1.2.2. Check standards that represent the items to be calibrated. These check 

standards should be stable and their values should be established with 

accuracy, since they will be used to control the uncertainty in the 

calibration process. Check standard values are to be determined by an 

independent laboratory at a higher level of calibration than the 

procedure for which the standard will be used. As a minimum, the 

check standard may be calibrated within the laboratory if they are 

qualified to work at the next higher level and use a procedure one 

level higher than the calibration process to be controlled. 

1.2.3. The operator must be experienced in the calibration process and the 

standard operating procedure for the calibration to which this method 

is applied. The operator should also have had specific training on SOP 

29 (uncertainty identification and calculations). 

1.2.4. A calculating system for statistical control that performs standard 

deviations, control limits, “F” and “t” tests, root-sum-of-the-squares 

(RSS), and control charting. It is recommended that a computer 

program be used for efficiency, consistency, and uniformity. 
February 2012 

SOP 30 Page 2 of 9 

2. Methodology 

2.1. Scope, Precision, Accuracy 

This method can be used on any measurement or calibration process where a 

check standard can be substituted for, or measured as, the item being 

calibrated. The method duplicates the standard operating procedure with the 

check standard serving as a surrogate for the item being calibrated. The results 

of these check standard measurements are recorded, charted, and analyzed to 

establish the measurement capability and to set process control limits, called 

reference limits. The reference limits are used to establish process uncertainties 

and to control future measurement performance. 

2.2. Summary 

Process Measurement Assurance Program (PMAP) is used for determining and 

controlling the measurement process uncertainty as the calibration is 

performed. The uncertainty includes effects of the measurement instrument, 

the operator, the procedure, the standards, and the environment. Each 

application is modeled to meet the following objective for determining and 

controlling uncertainties 1) in the measurement process; 2) in the calibration 

standards; or 3) in both the process and the standards. 

The check standard is selected to evaluate the standard deviation of the process 

(sp), other process uncertainties (uo) such as material density or air density 

accuracy, and possible bias (ud) of the process (see SOP 29). Reference 

measurements for the check standard are performed after calibrating the 

calibration standards and after the servicing of the measurement instrument. 

Control measurements of the check standard are graphed for visual 

examination of process performance and are evaluated against control 

reference limits. 

Statistical tools, ‘F’ and ‘t’ tests, determine if there are process changes in 

variability and bias (systematic error). These tests are used when process 

performance is questionable; when current data is evaluated to establish new 

reference control limits; and when evaluating uncertainty statements. 
February 2012 

SOP 30 Page 3 of 9 

2.3. Apparatus/Equipment Required 

2.3.1. A calibration process that meets the requirements of the standard 

operating procedure (SOP). 

2.3.2. A check standard that has been selected to evaluate the established 

objectives of monitoring: the 1) measurement process uncertainty; 2) 

the calibration standards; or 3) both the process and the standards. 

2.3.3. A data input, statistical calculating, and control charting system to 

provide analysis of measurement control data. (i.e., special software 

package or spreadsheet). 

Table 1. Symbols description

Symbol Description 

Sc Control measurement of check standard 

Scs Accepted value of check standard 

U Expanded Uncertainty (of the process) 

uc Combined standard uncertainty 

us Standard uncertainty of the standard 

uo Standard uncertainty of other factors 

ud Standard uncertainty of differences 

sp Standard deviation of the process 

k Coverage factor 

2.4. Procedure 

2.4.1. Preliminary Procedure 

2.4.1.1. Model the Calibration Process 

Set objectives for the PMAP application. These objectives will 

establish the value of the check standard, the check standard 

measurement procedure, and influence the frequency of 

control measurements of the check standard. Objectives may 

be: 1) Determine the standard deviation of the process (sp); 2) 

Determine the Expanded Uncertainty (U); 3) Measure the 

value of the calibration standard uncertainty (us). The model 

may allow any one objective or a combination of the 

objectives to be established with a single PMAP application 

and PMAP control chart. 

Diagram the process to clarify the measurement steps and 

determine the approach that will achieve the established 

objectives. When determining and controlling the expanded 

uncertainty (U), evaluate the range of use of the process to 
February 2012 

SOP 30 Page 4 of 9 

ensure that the check standard values will determine the 

variability and the maximum bias that result from the process 

and the calibration standard. Determining and controlling 

expanded uncertainty requires duplicating the calibration 

process and determining where in the process to insert the 

control measurement using the check standard. When the 

objective of PMAP is to control the uncertainty of the 

calibration standard (us), the PMAP measurement may vary 

from the calibration process to allow inserting the check 

standard measurement close to the measurement of the 

calibration standard (see appendix, Model SOP 8). Diagram 

the calibration process to establish how many check standards 

and PMAP applications are required to meet objectives. Also, 

diagram the process to determine where in the process to 

insert the control measurements of the check standard. 

2.4.1.2. Select and Calibrate Check Standard 

For calibration process uncertainty determination and control, 

select a check standard that approximates the item to be 

calibrated. The selected check standard should be selected to 

evaluate maximum random variation (sp) and bias of the 

process. The check standard selected should also be used to 

evaluate other objectives (uo) of the specific calibration 

process. For multiple ranges of use, a check standard and 

PMAP application will be required for various portions of the 

range. For example, Double Substitution (SOP 4) will require, 

at a minimum, a check standard for each decade (1000 g, 100 

g, 10 g, 1 g, 100 mg, and 10 mg) on each balance used. The 

selected check standard should be calibrated to establish its 

accepted value (Sc) with an uncertainty level sufficient to 

control the calibration process uncertainty. The calibration of 

the check standard must be completed using a standard which 

is independent of the calibration standard that the PMAP 

process is designed to control. 

For control of the calibration standard, use a check standard 

that is not part of routine measurement and that will evaluate 

the changes in the calibration standard’s accepted value, not 

the maximum random variation of the process. The check 

standard used to control the calibration standard should be 

used less frequently (less than 1⁄4 as often) than the calibration 

standard. For example, in (SOP 28), using Design A.1.2 (a 4-1 

weighing design) at 1000 g, requires the selection of a 1000 g 

check standard that is measured less frequently than the item 

under test in order to evaluate the two 1000 g calibration 

standards. The selected check standard should be calibrated 
February 2012 

SOP 30 Page 5 of 9 

using a calibration standard other than the calibration 

standard(s) it will be controlling. 

2.4.2. Establish Control Measurements for Reference Limits 

2.4.2.1. Control measurements to establish initial reference limits may 

be made at any time to verify current measurement process 

performance. But to achieve control of calibration standards 

accuracy and measurement instrument capability, the control 

measurements should be made just after calibration of the 

calibration standards and servicing of the measurement 

instrument. Any significant change in the calibration status 

can then be detected by the performance change in the 

reference limits data. 

2.4.2.2. Make the control measurement by duplicating the calibration 

process with the check standard substituted for the calibrated 

item. Make an observation of the check standard (Sc) and 

determine its measured value by completing calculations as 

described in the calibration SOP. 

2.4.2.3. Record the measured value (Sc) and plot it on the control chart 

and evaluate it with reference to the accepted value of the 

check standard. Record date, time, and information tags with 

data. 

2.4.2.4. Evaluate the bias (difference) between the mean of the 

measured Sc values and the accepted value for Sc from its 

calibration report. 

Bias (deviation) of check standard = Observed mean of Sc - Accepted Sc

2.4.2.5. Repeat the control measurement at various intervals to sample 

environmental change and other factors than can affect 

measurement performance. Although a control chart and some 

statistical control can be established with as few as seven to 

12 measurements, a minimum of twenty-five is recommended 

for estimates and control of uncertainties. 

2.4.3.Create and Prepare Control Charts 

2.4.3.1. Construct a graph with the deviation of the check standard 

measurements on the y-axis and chronological date and time 

(or observation number) on the x-axis. The accepted value of 

the check standard (Sc) is identified near the center of the 

chart. The y-axis of the control chart should extend plus and 
February 2012 

SOP 30 Page 6 of 9 

minus three standard deviations from the mean ( x ) of the 

control measurements. Control measurements may be charted 

as deviation from the accepted value of the check standard 

(Sc) or as the observed measurement result, or as a calculated 

correction. 

2.4.3.2. Establish control chart parameters by calculating the mean 

and the estimate of the standard deviation (sp) of the check 

standard reference measurements (Sc). 

 

c c

p

p

c

c

Process Bias = Mean of Observed S - AcceptedS

2s Warning Limits = x + 2 (s )

3s Action Limits = x + 3 (s )

Mean Line = x of ObservedS data

Reference Line = AcceptedS

Control chart parameters are asfollows:

 

2.4.4. Establish Reference Limits and Process Uncertainty 

2.4.4.1. Establish reference control limits (as described in 2.4.3.2) by 

calculating control limits and process bias using the control 

measurements obtained when calibration standards and 

measurement instruments are calibrated. These limits are to be 

stored and used as a reference for future control 

measurements. Future control measurements should be 

control-charted and tested against these limits for “in” or 

“out” of control status. This reference data will also be 

statistically used to periodically evaluate process and 

calibration standard performance for change from the 

calibrated reference status. 

2.4.4.2. Establish the Expanded Uncertainty (U) by using the 

reference data, calibration standard uncertainty (us), and any 

other uncertainty (uo) not covered by the reference data. 

c

c p s o d

U k u

u s u u u

*

( ) ( ) ( ) ( ) 2 2 2 2



   

According to the PMAP model, sp, uo, and ud are evaluated by 

comparison with the reference data. The uncertainty for the 

standard, us, and, in some situations, additional uo are included 

in PMAP calculation of the process expanded uncertainty (U). 
February 2012 

SOP 30 Page 7 of 9 

2.4.5. Make Control Measurements 

2.4.5.1. Control measurements of the check standard should be made 

periodically to ensure that the current measurement 

performance remains in control of the established reference 

limits. Control measurements should be tested for “in” or 

“out” of control status and charted on the control chart. The 

frequency of the control measurements is dependent on the 

objective of the application. 

When the objective is to determine and control the calibration 

process uncertainties, control measurements should be made 

during the calibration process to ensure the calibration results 

are accurate and within reference uncertainty statements. 

Control measurements should be made prior to returning 

calibrated items to the customer. A minimum of 25 control 

measurements are required within the calibration period or 

interval assigned to the calibration standards and the 

calibration process. 

When the objective is to control calibration standards, 

working standards, or primary standards, the frequency of 

control measurements should be less than 1⁄4 the use of the 

standard being controlled. The reduction in measurements 

ensures that the check standard receives less use and wear 

than the standard being controlled. Control measurements are 

charted and the Process Bias (observed mean value – 

accepted) is evaluated to detect any significant change in the 

calibration standard being controlled. 

2.4.6 Evaluate Process Performance at Specific Intervals 

2.4.6.1. Examine each control measurement data point as it is charted. 

Evaluate each data point for its control status and investigate 

causes for out of control data. Analyze the measurement 

process uncertainty, including the process standard deviation 

(sp), process bias, and other uncertainties quarterly or every 

five to ten data points to ensure that significant changes in 

uncertainties do not occur. 

2.4.6.2. Evaluate current performance using the control measurements 

described in 2.4.5 to establish the current standard deviation 

of the process (sp), process bias and other uncertainties. 

Perform each evaluation at specific calibration intervals 

established by the calibration of standards and service of the 

measurement instrument. This evaluation, referred to as 

“calibration” of the process, is performed at intervals that will 
February 2012 

SOP 30 Page 8 of 9 

ensure detection of changes in the calibration uncertainty 

statements. 

2.4.6.3. Use statistical tools at specific calibration intervals to evaluate 

current data performance to reference data that was 

established at calibration of standards and at the service of the 

measurement instrument. This evaluation will assist in 

deciding when to recalibrate calibration standards and service 

the measurement instrument. 

Use the F test to evaluate if a significant change in (sp) 

performance has occurred. 

 2

small

2

large

( )

( )

p

p

s

s

F test 

The current and previously established reference standard 

deviations are compared and evaluated using F test table 

values based on degrees of freedom in the measurements. 

Use the t-test to evaluate if a significant change in 

measurement process bias has occurred. 

2

old

old

new

2

new

new old

n 

( ) ( )

( ) ( )

p p s

n

s

Bias Bias

t test



 

The current (new) and previous (old) reference bias is 

compared and evaluated using t test table values based on the 

degrees of freedom in the measurements. 

2.4.6.4 Take action based on the results of statistical evaluation. If F 

and t tests reveal no significant change in process 

performance, use the current data analysis to establish new 

process reference limits, control chart and uncertainty 

statements (as described in 2.4.3 and 2.4.4). If the process has 

a stable history, it is permissible to pool the current data with 

previous reference limits to establish new reference limits, 

control chart, and uncertainty statements. Continue control 

measurements until the next calibration interval. 

If F and t tests reveal significant change in the measurement 

process, investigate the specific cause. If the cause for the 

change cannot be identified and corrected, then collect new 

reference data to establish new reference limits and process 

uncertainties. If a specific cause is found and corrected, and 

subsequent control measurements indicate an “in-control” 

status, continue collecting control measurements and test 
February 2012 

SOP 30 Page 9 of 9 

against established reference limits until the next calibration 

interval. 

2.4.7. Continue the measurement assurance procedure as described in 2.4.2 

through 2.4.6 to determine and control the measurement process 

capability. 

Significant changes in the measurement process capability can result 

from the following: 

 measurement procedure change; 

 measurement instrument change; 

 calibration standards change; and/or 

 location change. 

These changes can require repeating the procedure (from 2.4.2 

through 2.4.6). 
This page intentionally made blank. 
 February 2012 

SOP 33 Page 1 of 15 

SOP No. 33 

Standard Operating Procedure 

for 

Calibration of Weight Carts 

1. Introduction 

1.1. This Standard Operation Procedure (SOP) describes the procedure to be 

followed for the calibration of weight carts used to test livestock and 

vehicle scales. 

1.2. Prerequisites 

1.2.1. Facility. 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 

(%)

III 18 C to 25 C, maximum change 1.0 C/h 40 to 60  10 

1.2.2. Balance/scale. Verify that the balance, scale, or load cell is in good 

operating condition with sufficiently small process standard 

deviation as verified by a valid control chart or preliminary 

experiments to ascertain its performance quality. 

1.2.2.1.If a scale or load cell is used for the calibration that is not a 

permanent piece of equipment in the calibration laboratory, 

appropriate verification and repeatability statistics must be 

obtained prior to a calibration to determine suitability and 

acceptability for calibration. Records must be maintained 

of this verification. Minimum verification includes an 

increasing and decreasing load test to at least the capacity 

of the weight cart, a shift test, a sensitivity test at the test 

load, and evaluation of the repeatability based on a 

minimum of 7 repeated weighings (following the same 

SOP to be used to test the weight cart). 

1.2.2.2.The expanded uncertainty (k=2) must be less than one-third 

of the applicable tolerance and evaluated prior to 

calibration. 
 February 2012 

SOP 33 Page 2 of 15 

1.2.3. Standards. Ensure the availability of suitable working standards or 

recently calibrated (and unused) field standards in quantities up to 

the capacity of the weight cart(s) that will be calibrated. All 

standards must have known conventional mass values. 

1.2.4. Staff. Verify that the operator is experienced in precision weighing 

techniques and has had specific training in SOP 8, SOP 4, SOP 29, 

and GMP 10. Safety precaution: weight carts should always be 

calibrated with at least two people present. 

2. Methodology 

2.1. Scope, Precision, Accuracy 

This method is applicable to lower echelon mass calibration (tolerance 

testing) of weight carts, provided that the uncertainty requirements can be 

met. The achievable precision using this procedure is appropriate, 

provided the expanded uncertainty of the measurement is no more than 

one-third of the permissible tolerance of the weight cart. The accuracy 

achievable with this procedure depends on the accuracy of the calibration 

of the working standards and the precision of the comparison 

equipment/devices chosen and the procedure selected (e.g., SOP 8 or SOP 

4). 

2.2. Summary 

A weight cart is calibrated by substitution with standards having known 

values, using weighing equipment that has been evaluated as having 

sufficient resolution and repeatability, to maintain uncertainty to tolerance 

ratios needed in legal metrology. Modified or double substitution methods 

are used (SOP 8, SOP 4) and are modified to obtain the replicate 

observations needed for measurement assurance and determination of the 

calibration uncertainty. 

2.3. Equipment /Apparatus/Assistance 

2.3.1. A balance, scale, or load cell with suitable resolution and 

repeatability, in good repair. 

2.3.1.1.General guidelines regarding selection of weighing 

equipment must include consideration of the device 

repeatability to determine the achievable standard deviation 

of the measurement process. To ensure that appropriate 

uncertainty to tolerance ratios are maintained, the following 

guidance is provided: 
 February 2012 

SOP 33 Page 3 of 15 

2.3.1.1.1. When using Modified SOP 8:  For a 2000 lb cart, the maximum division size should 

be 0.01 lb or less.  For a 5000 lb cart, the maximum division size should 

be 0.05 lb or less. 

2.3.1.1.2. When using Modified SOP 4:  For a 2000 lb cart, the maximum division size should 

be 0.05 lb or less.  For a 5000 lb cart, the maximum division size should 

be 0.1 lb or less. 

2.3.1.2.Some balances will require suitable supports to be mounted 

on the scale deck (platform) to receive the weight cart 

dimensions without damage to the scale or weight cart. 

Supports must be appropriately stable and sturdy. 

2.3.2. Standards and correction weights, meeting NIST Class F 

tolerances with current Calibration Reports and known values. 

2.3.3. Hand tools: 

 wrenches 

 screw driver; 

 wire cutters; and 

 lead and wire seals and seal press. 

2.3.4. Means to move the weight cart safely. Weight carts are large, 

bulky items and are awkward to move safely. Appropriate safety 

precautions must be followed in the laboratory. Movement of 

liquid powered fuel carts is usually not done using the power 

system of the carts because exhaust from gasoline or diesel 

powered carts is considered unsafe in the laboratory; if no other 

options exist, suitable air handling must be available. Use of 

forklifts to move the carts should only be considered when other 

options are unavailable and extreme care must be taken. Use of a 

hoist may provide adequate means to move the cart, however, 

adequate care must be taken to prevent tipping the cart or bumping 

into other items in the laboratory as the cart is lifted and moved. 

2.3.5. A basic knowledge of weight carts, including the braking system 

and operating controls is required. Only qualified personnel should 

operate and inspect the weight cart. Verification that the brakes are 

operative prior to movement of the cart is essential. 

2.3.6. Suitable adjustment material must be available for placement in the 

adjusting compartments; lead or steel shot is recommended. 

2.3.7. Inspection checklist (See Appendix A). Weight carts are not 

considered Class F standards due to the extent of motorized and 

mechanical parts and fluids; therefore, special care must be taken 

in their calibration and use. An inspection checklist must be 
 February 2012 

SOP 33 Page 4 of 15 

completed for each new cart prior to placing it into service. The 

checklist becomes a permanent part of the records for each weight 

cart and must be available for review prior to each use or 

calibration. 

2.3.8. For safety reasons, at least two people are required to conduct a 

calibration of a weight cart. 

3. Fundamental Considerations 

3.1. Weight carts must be properly cleaned and painted, with all repairs and 

maintenance completed (as needed) prior to calibration. Obtaining actual 

“as found” values for reverse traceability is not possible with this 

procedure. Obtaining “as found” values may be achieved by performing a 

calibration before and after repairs and maintenance are completed. 

Alternatively, see the “Recommended Method for Intermediate 

Verification of Mass Values for Weight Cart Users” included as Appendix 

B. 

3.2. Weight carts must be completely dry, with no ice or moisture on the 

surface or under carriage of the cart. 

Experience has shown that a light coating of frost can cause errors in 

excess of the tolerance for the cart. In heated facilities, the cart must be 

allowed sufficient time to warm to prevent condensation. 

Table 2. Stabilization Time.

Temperature Range 

22 oC (lab temp) 

Temperature Range 

72 oF (lab temp) 

Minimum 

Equilibration Time* 

± 5 oC, 17 oC to 28 oC ± 10 oF, 62 oF to 82 oF 4 hours 

< 17 oC and > 28 oC < 62 oF and > 82 oF 24 hours/overnight 

* deviation is allowed if the known cart temperature is at the lab 

temperature ± 2.5 oC (5 oF). 

3.3. Weight carts that repeatedly demonstrate excessive “as found” out-oftolerance conditions from the previous calibration date, may be required to 

be submitted on a more frequent basis; with every consideration being 

given to determine the cause of the variance. 

4. Pretest Weight Cart Inspection 

4.1. Complete the Weight Cart Inspection Checklist that is included in 

Appendix A. The Inspection Checklist is an integral part of the Calibration 

Report of a weight cart and must be transported with each weight cart and 

verified prior to each use or calibration. Note and ensure completion of 

any needed repairs or maintenance prior to calibration. 
 February 2012 

SOP 33 Page 5 of 15 

5. Procedure 

5.1. With the exception of items modified and clarified in this SOP, follow 

SOP 8 or SOP 4 to determine the conventional mass value of the weight 

cart. Follow good laboratory practices such as properly exercising the 

balance prior to calibration. 

5.2. SOP 8 Modifications. 

5.2.1. Repeat the intermediate weighing of the weight cart as if multiple 

weights were being verified. Obtain at least 3 observations that are 

taken with a release/removal of the weight from the sensing 

device. Determine the calculated differences per SOP 8 and 

calculate the range between the largest and smallest numerical 

mass value differences. Record the range values in a suitable 

control chart for weight cart calibrations. 

5.2.2. Mass adjustments to the cart may be made at the end of exercising 

the balance or after the first observation. The cart must be adjusted 

as close as possible to zero error. The first reading then becomes a 

“zero” difference when compared to the standards. Adjustments 

must be made quickly to avoid errors due to balance drift or 

shifting. In the event that balance drift or shifts are noticed, the 3 

replicate observations should be made after the adjustment is 

completed. 

5.2.3. Care must be taken to ensure that the center of the mass of the 

weight cart is centered on the balance pan. By rotating the cart by 

180 degrees and remeasuring it, off-center loading factors may be 

considered as a part of standard deviation. 

5.3. SOP 4 Modifications. 

5.3.1. Repeat the double substitution procedure 3 times. Determine the 

calculated differences per SOP 4 and calculate the range between 

the largest and smallest differences. Record the range information 

in a suitable control chart for weight cart calibration range values. 

5.3.2. Mass adjustments to the cart may be made at the end of exercising 

the balance or after the first double substitution. The cart must be 

adjusted as close as possible to zero error. The first reading then 

becomes a “zero” difference when compared to the standards. 

Adjustments must be made quickly to avoid errors due to balance 

drift or shifting. In the event that balance drift or shifts are noticed, 

the 3 replicate double substitutions should be made after the 

adjustment is completed. 
 February 2012 

SOP 33 Page 6 of 15 

5.3.3. Care must be taken to ensure that the center of the mass of the 

weight cart is centered on the balance pan. By rotating the cart by 

180 degrees and remeasuring it, off-center loading factors may be 

considered as a part of standard deviation. 

6. Calculations 

6.1. Follow calculations as described in SOP 4 or SOP 8 as appropriate. 

7. Tolerance 

7.1. Tolerances are established in NIST Handbook 105-8. Weight carts must 

be adjusted to zero error during calibration. The expanded (k=2) combined 

uncertainty of the calibration must be less than one-third of the tolerance 

specified in Handbook 105-8. 

7.2. The values of weight carts and their tolerance status are issued and 

evaluated on the basis of Conventional Mass (previously called Apparent 

Mass versus a reference density of 8.0 g/cm3

 at 20 °C). If a reference 

density other than 8.0 g/cm3 is used, the reference density must be 

reported. No correction for the effect of air buoyancy is general made 

unless stated on the test report. 

7.3. Tolerances for weight carts (See also NIST Handbook 105-8). 

Nominal Mass (lb) Tolerance (± lb) 

2 000, 2 500 0.50 

3 000, 3 500 1.00 

4 000, 4 500 1.25 

5 000, 5 500 1.50 

6 000 + 2.00 

8. Measurement Assurance 

8.1. Adequate measurement assurance is a required component of every 

quality system and uncertainty analysis, but maintaining a surrogate 

weight cart for use in the laboratory as a check standard is impractical. 

8.2. Three repeated observations are made and the range of values (largest 

mass difference – smallest mass difference) is entered into a control chart. 

The pooled standard deviation of the process is used or the pooled mean 

range value is used to estimate a standard deviation of the process. When 

plotting each new range value, verify that it is within previously 

established limits. 
 February 2012 

SOP 33 Page 7 of 15 

8.3. Alternatively, 7 to 12 repeated observations are made (following the same 

SOP to be used to test the weight cart) and a short-term standard deviation 

of the process is calculated each time a weight cart is calibrated. A 

minimum of 7 replicate weighings may also be used in the event that a 

balance is used that is not permanent equipment of the laboratory. 

9. Assignment of Uncertainty 

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

uncertainty of the mass standards used, us, and estimates of the standard deviation 

of the measurement process, sp. These estimates should be combined using the 

root-sum-squared method (RSS), and the expanded uncertainty, U, reported with 

a coverage factor of two (k=2), to give an approximate 95 percent level of 

confidence. See SOP 29 for the complete standard operating procedure for 

calculating the uncertainty. 

9.1. The expanded uncertainty for the standards, U, are obtained from the 

calibration report. The combined standard uncertainty, uc, is used and not 

the expanded uncertainty, U, therefore the reported uncertainty for the 

standards will usually need to be divided by the coverage factor k. 

9.1.1. A combination or summation of standards is typically needed for 

the calibration of weight carts. The dependence or independence of 

the calibration of such standards must be determined prior to 

combining the uncertainties associated with the standards. In the 

case of dependent standards (calibrated against the same working 

standard), the us values are summed by addition prior to including 

them in a root sum square calculation: (us1 + us2 + ... usn). In the 

case of independent standards, the us values are summed by root 

sum square: (us1

2

+ us2

2

+...usn2)

0.5. All us values must represent one 

standard deviation prior to combination. 

9.2. Standard deviation of the measurement process, sp, is obtained by using 

the pooled standard deviation of similar size carts over time or by 

calculating the estimated standard deviation of the process based on the 

mean range value calculated from applicable range charts. 

9.3. Buoyancy corrections are not generally needed at the uncertainty level this 

procedure is designed for. An uncorrected systematic error should be 

included as a rectangular distribution. 

9.4. Additional components due to balance drift or sensitivity errors may need 

to be included per SOP 8. 

9.5. The expanded uncertainty, U, must be  1/3 of the tolerance applicable as 

per NIST Handbook 105-8. 
 February 2012 

SOP 33 Page 8 of 15 

10. Calibration Report 

Prepare a Calibration Report according to SOP 1. A sample calibration report is 

included in Appendix A. Please note that the Inspection Checklist is an integral 

part of the calibration report. 
 February 2012 

SOP 33 Page 9 of 15 

Appendix A 

Sample Report of Calibration 

For 

Weight Cart 

Test Number: Date Issued: Date of 

Calibration: 

Submitted by: 

 

Manufacturer: Date of Manufacture: 

 

Model Number: ID/SN Number: 

Calibration Values: 

As Found 

Conventional Mass* (lb) 

As Left 

Conventional Mass (lb) Uncertainty (lb) 

 

* The Conventional Mass is the mass determined/calculated by weighing in air of reference 

density 1.2 mg/cm3

 against reference weights of density 8.0 g/cm3. Buoyancy corrections are 

considered negligible and were not made unless otherwise stated. 

Conditions of Test & Traceability: 

The above weight cart was compared with standards of the State of XXXX, which are traceable to 

the National Institute of Standards and Technology (NIST) and have known values. The assigned 

test number shown above provides documented evidence for measurement traceability. 

Test method: {Enter SOP 4 or 8 as applicable} NISTIR 6969, Selected Procedures for Mass 

Calibrations. 

Balance: 

Temperature: 

Pressure: 

Relative Humidity: 

Components of Uncertainty: 

The uncertainty is reported as an expanded uncertainty (k=2) and calculated according to SOP 29 

which is compliant with the ISO/IEC Guide to the Expression of Uncertainty in Measurement. 

Components included in the calibration uncertainty are the uncertainty associated with the 

standards and the standard deviation of the measurement process as determined through repeated 

observations or other suitable range chart for this measurement process. Factors included on the 

inspection checklist have not been included in the calibration uncertainty. However, factors on 

the checklist may contribute measurement errors that are significant if not properly maintained 

during use. 
 February 2012 

SOP 33 Page 10 of 15 

Inspection Checklist: 

The weight cart was cleaned and painted and allowed to come to thermal and environmental 

equilibrium in the laboratory prior to calibration. The weight cart was adjusted (as needed and 

noted above) as close as possible to zero error. All fluid levels were adjusted as close as possible 

to the full/reference marks and sealed. Liquid levels must be maintained as close to reference 

levels as possible during use. The fluid levels include: 

_____Engine Oil; 

_____Hydraulic Fluid (sealed); 

_____Battery Water Level (sealed battery); and 

_____Liquid Fuel {Enter gasoline or diesel here} (note reference mark). 

The attached Inspection Checklist is an integral component of this Calibration Report and a copy 

must be maintained with the cart and reviewed prior to use. 

Any maintenance, repairs, replacement of parts, or damage to weight cart or its components will 

likely result in an out-of -tolerance condition; therefore, maintenance or replacement of 

components such as batteries, tires, filters, or other items listed on the checklist, require 

calibration of the weight cart prior to subsequent use. 

Tolerance: 

Reference: 2001 Draft NIST Handbook 105-8, Specifications and Tolerances for Weight Carts. 

The applicable tolerances are as follows: 

Nominal Mass (lb) Tolerance ± (lb) 

2 000, 2 500 0.50 

3 000, 3 500 1.00 

4 000, 4 500 1.25 

5 000, 5 500 1.50 

6 000 + 2.00 

For carts intermediate between these values, the next smaller tolerance is applied. 

 

 

Authorized Signature 

 
 February 2012 

SOP 33 Page 11 of 15 

Inspection Checklist for Weight Cart 

Test Number: Date of Inspection: 

Manufacturer: Date of Manufacture: 

 

Model Number: ID/SN Number: 

_____Nominal Mass of Weight Cart: __________ Suitably marked: Yes/No 

_____Powered by: Electric/Generator__________ Diesel__________ Gasoline __________ 

_____Fluid Levels: Engine Oil __________ 

 Hydraulic Fluid __________ Sealed: Yes/No 

 Battery __________ Sealed: Yes/No 

 Liquid Fuel __________ Reference line present: Yes/No 

_____Fluid drain tubes extend beyond the body of the cart: Yes/No 

_____Number of axles: __________ 

_____Number/Size of Tires: __________ 

_____Sealed wheel bearings: Yes/No 

_____Drain holes present in locations where water may accumulate: Yes/No 

_____Weight restraint railing permanently fixed and solid: Yes/No 

_____Adjusting cavity accessible: Yes/No 

_____Adjusting cavity approximate capacity: ______lb 

_____Adjusting cavity sealed: Yes/No 

_____Service brakes functioning properly: Yes/No 

_____Parking brakes functioning properly: Yes/No 

_____Remote control functioning properly: Yes/No 

_____General condition at time of calibration (note any accumulated dirt/debris, damage, loose parts, or 

evidence of tampering or unauthorized entry of seals). 

_____List and report any repair and maintenance performed, parts replaced, etc., Leaks repaired, new 

battery, carburetor, exhaust system, wheels changed, welding performed, etc.. Include any comments or 

changes since the last calibration. 
 February 2012 

SOP 33 Page 12 of 15 

Appendix B 

Recommended Method 

for 

Intermediate Verification of Mass Values 

for Weight Cart Users 

1. Introduction 

1.1. This Recommended Method is intended to provide an intermediate 

verification of weight cart mass values to obtain a rough estimate of an “as 

found” value to provide adequate evidence for or against the need for 

corrective action on scales that have been tested. 

1.2. This method may be performed as often as needed during a calibration 

cycle as a preventive action due to the inherent instability of weight cart 

mass values. It should also be performed prior to repairs or maintenance to 

obtain and record adequate evidence of mass values and calibration status 

prior to the change of mass values resulting from repair and maintenance. 

1.3. Records of all such intermediate tests should be retained with the 

Inspection Checklist and should be submitted with the completed 

Inspection Checklist when the weight cart is submitted for calibration. 

1.4. This method may NOT be used as a substitute for a proper laboratory 

calibration according to SOP 33 and does NOT provide a traceable mass 

value with an appropriate calibration uncertainty. Recall that the error on 

the standard must be less than one third of the applicable tolerance applied 

to a scale under test. This level of verification is difficult and may be 

impossible to determine under field conditions. Only errors in excess of 

Handbook 44 requirements may be determined using this method. 

2. Prerequisites 

2.1. Facility. Intermediate verification tests are normally conducted under field 

or under maintenance shop/garage conditions rather than in a controlled 

laboratory environment. As such, the avoidance of extreme conditions of 

temperature, wind, and precipitation are critical. A rough estimate of 

conditions at the time of verification must be recorded when data is 

obtained. 

2.2. Equipment. A suitable scale must be selected. This typically needs to be of 

better resolution than the scales being tested with the weight cart or as a 

minimum, a scale with the smallest resolution on which the cart may be 

used. A repeatability test, with a minimum of 3 repetitions is needed to 
 February 2012 

SOP 33 Page 13 of 15 

ensure that the mass values will repeat to within one-half of a scale 

division at the load of the weight cart. 

2.3. Standards. Class F field standards are used as long as they are within the 

prescribed calibration cycle. A small Class F weight kit is used as 

correction weights to determine small errors and partial division values. 

2.4. Staff. Staff must be trained in substitution calibration methods and be able 

to perform calculations associated with determining mass values. 

2.5. The weight cart fuel levels, where liquid fuels are used, must be set at the 

reference mark on the fuel sight gauge. 

3. Fundamental Considerations 

Weight carts must be completely clean and dry, with no mud, ice or moisture on 

the surface or under carriage of the cart. Tires must be cleaned as well. (Values 

obtained under actual operating conditions may provide some insight to the user 

regarding the impact of maintenance and cleanliness of the weight cart). 

4. Pretest Weight Cart Inspection 

Complete the Weight Cart Inspection Checklist that is included in Appendix A. 

5. Procedure 

5.1. Establish scale repeatability with a minimum of three repetitions such that 

the scale repeats to within one-half of a division at the weight cart 

capacity. 

5.2. Load Class F field standards onto the scale in the same location(s) where 

the weight cart will be placed to the same load as the stated conventional 

mass of the weight cart. Assign a mass value to this summation of 

standards based on the nominal values of the weights. The added mass 

values of these weights is the value for your standards. With the weights 

loaded on the scale deck, use small error weights to determine the break 

point of the nearest larger scale division. 

5.3. Remove the large Class F field standards and leave the small error weights 

on the scale. 

5.4. Load the weight cart onto the scale in the same location(s) where the 

weights were loaded. Add or remove the small error weights until the 

scale responds in the same way as the previously loaded mass standards to 

determine the break point of the same scale division. 
 February 2012 

SOP 33 Page 14 of 15 

6. Calculations 

Calculate the difference between the error weights used with the Class F field 

standards and the error weights used with the weight cart. Because this is not an 

accepted calibration method, the observed difference in these values only 

represents an estimated mass value. (Error weights used with Class F standards – 

Error weights used with the weight cart = Observed Estimated Error). 

7. Tolerance & Evaluation 

The observed estimated error must be less than one third of the smallest tolerance 

applied to a scale on which the weight cart might be used. Corrective action, 

possibly including retests of recent scale tests, may be needed depending on the 

number of intermediate verifications and records regarding observed estimated 

errors. 

8. Measurement Assurance 

Measurement assurance is not a component of this intermediate verification. 

9. Assignment of Uncertainty 

An uncertainty is not calculated, estimated, or assigned with this intermediate 

verification method. 

10. Calibration Report 

A calibration report is not prepared for an intermediate verification. 
 February 2012 

SOP 33 Page 15 of 15 

Example of Intermediate Verification Procedure and Calculations 

A. A 2000 lb weight cart is to be tested on a scale with 10 lb divisions. 

B. When 2) 1000 lb Class F field standards are placed on the scale, the scale reading 

repeats to within one-half of a division when 3 replicate weighings are performed 

(weights must be moved on and off the scale or the scale arrested and released to 

determine this repeatability). The scale reads 2000 lb for each test. 

C. Error weights are slowly placed on the scale deck in increments of 0.1 lb and the 

addition of 5.5 lb causes the scale to read the next larger division of 2010 lb. 

D. The 2) 1000 lb Class F field standards are removed and the 2000 lb weight cart is 

placed on the scale deck, leaving the previous 5.5 lb error weights. Error weights 

of 6.5 lb are needed for the scale to break to 2010 lb with the weight cart on the 

scale deck (an additional 1.0 lb was added). 

E. Error weights used with Class F standards = 5.5 lb 

F. Error weights used with the weight cart = 6.5 lb. 

G. Error weights used with Class F standards – Error weights used with the weight 

cart = Observed Estimated Error; 5.5 lb – 6.5 lb = - 1 lb. 

H. Evaluation: the smallest tolerance on the scale will be one-half of a division, or 5 

lb. One third of 5 lb is 1.7 lb. The error on standards used on the scale must be 

less than this level. The observed error on the cart of – 1.0 lb is less than 1.7 lb 

and corrective action for previous scale tests is not needed at this time. The –1 lb 

value is recorded on the Inspection Checklist as an Intermediate Verification. 

I. If the cart has been damaged or if repairs to the cart are needed, calibration is 

required at this time. 
This page intentionally made blank. 
February 2012 

Statistical Techniques Page 1 of 10 

8. STATISTICAL TECHNIQUES 

Statistics are used in metrology to summarize experimental data, to provide the basis for 

assessing its quality, and to provide a basis for making probabilistic decisions in its use. The 

essential basic statistical information for describing a simple data set is: 

 

The mean of the set, x 

The standard deviation of the set, s 

The number of individuals in the set, n 

 

If the set is a random sample of the population from which it was derived, if the measurement 

process is in statistical control, and if all of the observations are independent of one another, then 

s is an estimate of the population standard deviation,  , and x is an unbiased estimate of the 

mean, μ. 

 

The population consists of all possible measurements that could have been made under the test 

conditions for a stable test sample. In this regard, the metrologist must be aware that any changes 

in the measurement system (known or unknown) could possibly result in significant changes in 

its operational characteristics, and, hence the values of the mean and standard deviation. 

Whenever there is doubt, statistical tests should be made to determine the significance of any 

apparent differences before statistics are combined. 

 

The following discussion reviews some useful statistical techniques for interpreting 

measurement data. In presenting this information, it is assumed that the reader is already familiar 

with basic statistical concepts. For a detailed discussion of the following techniques and others 

not presented here, it is recommended that the reader consult NIST Handbook 91 - Experimental 

Statistics, by Mary G. Natrella [19]. That handbook also contains comprehensive statistical 

tables from which the tables contained in Chapter 9 of this publication were taken. 

 

8.1 Estimation of Standard Deviation from a Series 

of Measurements on a Given Object 

Given n measurements x1, x2, x3, ......,xn

 

Mean,  

n

x x x x

x

    n  ... 1 2 3

Standard deviation estimate,  

1

2



   nx x

s n 

The estimate, s, is based on n-1 degrees of freedom 
February 2012 

Statistical Techniques Page 2 of 10 

8.2 Estimation of Standard Deviation from the Differences 

of k Sets of Duplicate Measurements 

Given k differences of duplicate measurements, d1, d2, d3, ..., dk, a useful formula for estimating 

the standard deviation is: 

k

d

s i

d 2

2   where sd is based on k degrees of freedom. 

Note that i i d  x  x 1 , for example. 

 

The values d1, d2 etc., may be differences of duplicate measurements of the same sample (or 

object) at various times, or they may be the differences of duplicate measurements of several 

similar samples (or objects). 

 

8.3 Estimation of Standard Deviation from the Average 

 Range of Several Sets of Measurements 

 

The range, R, is defined as the difference between the largest and smallest values in a set of 

measurements. 

 

Given R1, R2, R3, ..., Rk

Mean,  

k

R R R R R     k  ... 1 2 3

Standard deviation can be estimated by the formula, *

2 d

R

sR 

The value of * d2 will depend on the number of sets of measurements used to calculate Rs , and on 

the number of measurements in each set, i.e., 2 for duplicates, 3 for triplicates, etc. Consult a 

table such as Table 9.1 for the appropriate value of * d2 to use. The effective number of degrees 

of freedom for Rs is in the table. Note: It is recommended to calculate the standard deviation for 

each set of replicates and then to calculate a pooled standard deviation whenever possible. 

8.4 Pooling Estimates of Standard Deviations 

 

Estimates of the standard deviation obtained at several times may be combined (pooled) to 

obtain a better estimate based upon more degrees of freedom. The following equation may be 

used for this purpose: 

     

      1 1 1 ... 1

1 1 1 ... 1

1 2 3

2 2

3 3

2

2 2

2

1 1

       

        

k

k k

p n n n n

n s n s n s n s

s where 

p s will be based on    1 1  1 ...  1 n1   n2   n3    nk  degrees of freedom. 
February 2012 

Statistical Techniques Page 3 of 10 

8.5 "Within" and "Between" Standard Deviation 

 

Estimation of the within-series, sw, and between-series, sb standard deviation, (also referred to as 

short-term and long-term standard deviations in the applications described here) is an important 

way to characterize a measurement process. The former provides guidance as to how many 

repetitions of a measurement are required to obtain a result on a single occasion with a given 

precision, while the latter is a better estimate of the precision of replication (reproducibility) of a 

result on various occasions and is a more realistic evaluation of measurement variability. 

 

To estimate these standard deviations, sets of measurements may be made on several occasions. 

To simplify the calculations, each set should consist of the same number of measurements. For 

most measurements, it is recommended that duplicate measurements be made on at least 12 

separate occasions when estimating sw and sb. 

 

Given k sets of duplicate measurements made on k occasions the following table and calculation 

can be made. 

Table 8.1

Occasion Measured Values Range Mean 

1 1 x 1 x R1

 

1

1 1

2

x

x x    

2 2 x 2 x R2

 

2

2 2

2

x

x x    

3 3 x 3 x R3

 

3

3 3

2

x

x x    

· · · · · 

· · · · · 

· · · · · 

k k x k x Rk

 

k

k k x

x x    

2

 

 

1. Calculate  

k

R R R R R     k  ... 1 2 3

2. Calculate *

2 d

R

s s w  R  

Note: One may use the procedure of 8.2 to calculate sw if preferred. 
February 2012 

Statistical Techniques Page 4 of 10 

3. Calculate sx as follows: 

 

k

x x x x

x

    k  ... 1 2 3

 

1

2



   kx x

s k

x

4. Calculate sb (for the case of duplicate measurements) 

2

2

2 w

b x

s

s  s 

Note that sb is an estimate of the long term component of the standard deviation of a single 

measurement. The long term standard deviation of the mean of n measurements taken at a single 

occasion is estimated by: 

n

s

s x s w

b n b

2

2 ( )  

Important note: Do not use this approach for handling within-series and between-time standard 

deviations with the Mass Code. Separate formulas are available for that application. 

8.6 Confidence Interval for the Mean 

 

The estimation of the confidence interval for the mean of n measurements is one of the most 

frequently used statistical calculations. The formula used will depend on whether the population 

standard deviation,, is known or whether it is estimated on the basis of measurements of a 

sample(s) of the population. 

 

Using Population Standard Deviation,  

 

Strictly speaking, , is never known for a measurement process. However, the formula for use in 

such a case is: 

n

z

x

 

Variable Description 

x sample mean 

s known standard deviation 

n number of measurements of sample 

z standard normal variate, depending on the confidence level desired 

For 95 % confidence z = 1.960; for 99.7 % confidence z = 3.0. 

For other confidence levels, see Table 9.2 
February 2012 

Statistical Techniques Page 5 of 10 

Using Estimate of Standard Deviation, s 

 

In the usual situation, s is known, based on  degrees of freedom and the formula for use is: 

n

ts

x 

 

Variable Description 

x sample mean 

s estimate of standard deviation 

n number of measurements on which the mean is based 

t Student's t value, based on the confidence level desired and 

the  degrees of freedom associated with s (see Table 9.3). 

Note that t z as n. For many practical purposes, the standard deviation may be 

considered as known when estimated by at least 30 degrees of freedom. 

 

8.7 Confidence Interval for  

The standard deviation,  , is ordinarily not known but is, instead, an estimated value based on a 

limited number of measurements, using procedures such as have been described above. Such 

estimates may be pooled, as appropriate, to obtain better estimates. In any case, the uncertainty 

of the estimated value of the standard deviation may be of interest and can be expressed in the 

form of a confidence interval, computed as indicated below. 

 

The interval is asymmetrical because the standard deviation is ordinarily underestimated when 

small numbers of measurements are involved due to the fact that large deviations occur 

infrequently in a limited measurement process. Indeed, it is the general experience of 

metrologists that a few measurements appear to be more precise than they really are. 

 

The basic information required to compute the interval is an estimate of the standard deviation, s, 

and the number of degrees of freedom on which the estimate is based. The relationships to use 

are: 

 

Lower limit BLs 

 

Upper limit BUs 

 

Interval BLs to BUs 

 

The values for BL and BU depend upon the confidence level and degrees of freedom associated 

with s. Values for use in calculating the confidence level are given in Table 9.7. A more 

extensive table (Table A-20) is available in NIST Handbook 91 [19]. 
February 2012 

Statistical Techniques Page 6 of 10 

8.8 Statistical Tolerance Intervals 

 

Statistical tolerance intervals define the bounds within which a percentage of the population is 

expected to lie with a given level of confidence. For example, one may wish to define the limits 

within which 95 % of measurements would be expected to lie with a 95 % confidence of being 

correct. The interval is symmetrical and is computed using the expression 

x  ks

where k depends on three things 

 

Variable Description 

p the proportion or percentage of the individual measurements to be 

included 

 the confidence coefficient to be associated with the interval 

n the number of measurements on which the estimate, s, is based 

Table 9.6 may be used to obtain values for k for frequently desired values of  and p. A more 

extensive table is Table A-6 found in NIST Handbook 91 [19]. 

8.9 Comparing Estimates of a Standard Deviation 

(F Test) 

 

The F-test may be used to decide whether there is sufficient reason to believe that two estimates 

of a standard deviation differ significantly. The ratio of the variances (square of the standard 

deviation) is calculated and compared with tabulated values. Unless the computed ratio is larger 

than the tabulated value, there is no reason to believe that the observed standard deviations are 

significantly different. 

 

The F ratio is calculated using the equation 

2

2

S

L

s

s F 

where sL is the numerically larger value and sS is the smaller value of the two estimates under 

consideration. 

 

The critical value of F depends on the significance level chosen for the decision (test) and the 

number of degrees of freedom associated with sL and sS, respectively. 

Table 9.4 contains critical values for F at the 95 % level of confidence. The tabulated values of F 

are not expected to be exceeded with 95 % confidence on the basis of chance alone. As an 

example, if both the numerator and the denominator values for s were each based on 9 degrees of 

freedom, an F value no larger than 4.03 is expected with 95 % confidence, due to the 

uncertainties of the s values, themselves. Table A-5 of NIST Handbook 91 [19] contains values 

for F for other confidence levels. 
February 2012 

Statistical Techniques Page 7 of 10 

 

The F-test is useful for comparing the precision of methods, equipment, laboratories, or 

metrologists, for example. An inspection of Table 9.4 shows that when either of the values of s is 

based on a small number of degrees of freedom, the F value is large. Consequently, the 

significance of decisions based on small changes in precision can be supported statistically only 

by a relatively large number of measurements. If such changes are suspected, but the data 

requirement is difficult to meet, the decision may need to be made on the basis of information 

about the measurement process itself. 

The F-test is also useful for deciding whether estimates of the standard deviation made at various 

times differ significantly. Such questions need to be answered when deciding on whether to 

revise control limits of a control chart, for example.

 

8.10 Comparing a Set of Measurements with a Given Value 

 

The question may arise as to whether a measured value agrees or significantly disagrees with a 

stated value for the measured object. The evaluation can be based on whether or not the 

confidence interval for the measured value encompasses the stated value. The confidence 

interval is calculated using the expression 

n

ts

x 

as previously described in Section 8.6. In using this expression, n represents the number of 

measurements used to calculate the mean, x , and t depends on the degrees of freedom, , 

associated with s and the confidence level needed when making the decision. Note that one may 

use historical data for estimating s, such as a control chart for example, in which case  will 

represent the degrees of freedom associated with establishment of the control limits and may be 

considerably larger than n 1. 

 

8.11 Comparing Two Sets of Measurements with Regard to Their Means 

 

This discussion is concerned with deciding whether the means of two measured values, A and B, 

are in agreement. The data sets used for this purpose may consist of the following: 

 

A x B x

A s Bs

A n B n

The first question to be resolved is whether A s and Bs can be considered to be different 

estimates of the same standard deviation or whether they do, indeed, differ. An F test may be 

used for this purpose. However, it will be recalled that this is not sensitive to small real 

differences, so the decision may need to be based on physical considerations, such as the known 

stability of the measurement process, for example. 

Case I
February 2012 

Statistical Techniques Page 8 of 10 

 

Confirming (or assuming) that A s and Bs are not significantly different, they are pooled, as 

already described (but repeated here for convenience) and used to calculate a confidence interval 

for the difference of the means. If this is larger than the observed difference, there is no reason to 

believe that the means differ. The steps to follow when making the calculation described above 

are: 

 

 Step 1. Choose  , the level of significance for the test 

 

 Step 2. Calculate the pooled estimate of the standard deviation, sp 

   

   1 1

1 1 2 2

  

   

A B

A A B B

p n n

n s n s

s

sp will be estimated with nA + nB - 2 degrees of freedom 

 

 Step 3. Calculate the respective variances of the means 

A

A

A

n

s

v

2

 and 

B

B

B

n

s

v

2



 Step 4. Calculate the uncertainty of X A - XB  

  VA VB U  t 

using a value for t based on 

2

 and   2 A B  n n . 

Step 5. Compare U with  

If U   , there is no reason to believe that  is significant at the level of 

confidence chosen. 

 

Case II 

 

Confirming (or assuming) that A s and Bs are significantly different, their individual values are 

used to calculate U as outlined below. 

 

 Step 1. Choose  , the level of significance for the test. 

 Step 2. Calculate the respective variances of the means. 

 Step 3. Calculate the uncertainty of XA - XB   

A

A

A

n

s

v

2

 and

B

B

B

n

s

v

2


February 2012 

Statistical Techniques Page 9 of 10 

  A B U  t  

using a value for t based on 

2

 and f , the effective number of degrees of freedom 

calculated as described in Step 4. 

 Step 4. Calculate f, the effective number of degrees of freedom using the equation below 

or the Welch-Satterthwaite formula given in NIST Technical Note 1297 or the 

Guide to the Expression of Uncertainty in Measurement: 

 































 

B

B

A

A

f

V

f

V

V V

f A B

2 2

2 2

Step 5. Compare U with . 

 If U   , there is no reason to believe that  is significant at the level of 

confidence chosen. 

 

8.12 Use of Random Numbers 

 

Conducting operations in random sequences can avert problems of bias that might stem from a 

particular order of operations. For example, in the measurement of a series of items, it might be 

difficult to determine whether systematic trends in the measured values were due to differences 

in the items or to measurement system drift unless the items were measured in random order. 

 

Use of tables of random numbers is a convenient means for randomizing measurement 

operations. The operations, test objects, and other matters requiring randomization may be 

assigned serial numbers. The order of selection is then determined by use of a random number 

table, as described below. When the number of operations or test items is less than 100, a table 

such as Table 9.11, reproduced from NIST Handbook 91 [19], may be used conveniently. One 

may start from any arbitrarily selected position in the table and proceed from it in any 

pre-determined arbitrary manner. If the first number encountered is not that of one of any item, 

ignore it and proceed until a valid match is encountered. This becomes the first item in the 

sequence. Continuing in the same manner, items are selected in the sequence in which their 

serial numbers are encountered ignoring the repetition of previously identified items. The 

procedure is continued until all items have been randomly selected. 

 

As an example, select 10 specimens (numbered 01 to 10) in random order. Start from a randomly 

selected place, say column 2, row 5 of Table 9.11. Proceed from this point along the table as one 

would read a book. The starting number is 14, which is not usable. The first useful number 

encountered is 08, the next 03, and so on. Using the procedure described above, the following 

random order was found: 
February 2012 

Statistical Techniques Page 10 of 10 

Specimen No. 

08 03 09 05 06 02 07 10 04 01 

Order 

1 2 3 4 5 6 7 8 9 10 
February 2012 

Reference Tables Page 1 of 13 

9. Reference Tables 

Table 9.1 Use of Range to Estimate Standard Deviation 

Table 9.2 Z Factors for Two-sided Confidence Intervals 

Table 9.3 Student t-Variate 

 

Table 9.4 Critical Values for the F-test, F0.975

Table 9.5 Critical Values for the F-test, F0.99

Table 9.6 Two-sided Tolerance Interval Factors 

Table 9.7 Factors for Computing Two-sided Confidence Intervals for 

Table 9.8 Density of Air-free Water 

Table 9.9 Density of Air 

Table 9.10 Coefficient of Linear and Cubical Expansion for Materials Commonly Used 

in Laboratory Apparatus and Standards 

Table 9.11 A Short Table of Random Numbers 
February 2012 

Reference Tables Page 2 of 13 

Table 9.1 Use of Range to Estimate Standard Deviation 

Number of 

Sets of 

Replicates 

k 

Factor 

Degrees of 

Freedom 

Number of Replicates in a Set 

2 3 4 5 6 

1 

* d2 

 

1.41 

1.00 

1.91 

1.98 

2.24 

2.93 

2.48 

3.83 

2.67 

4.68 

3 

* d2 

 

1.23 

2.83 

1.77 

5.86 

2.12 

8.44 

2.38 

11.1 

2.58 

13.6 

5 

* d2 

 

1.19 

4.59 

1.74 

9.31 

2.10 

13.9 

2.36 

18.4 

2.56 

22.6 

10 

* d2 

 

1.16 

8.99 

1.72 

18.4 

2.08 

27.6 

2.34 

36.5 

2.55 

44.9 

15 

* d2 

 

1.15 

13.4 

1.71 

27.5 

2.07 

41.3 

2.34 

54.6 

2.54 

67.2 

20 

* d2 

 

1.14 

17.8 

1.70 

36.5 

2.07 

55.0 

2.33 

72.7 

2.54 

89.6 

∞ * d2 1.13 1.69 2.06 2.33 2.53 

*

2 d

R

sp 

Intermediate values for d2

*

 and ν may be obtained by interpolation, or from the reference from which this 

table was adapted. Example: When using a Range chart for volume calibrations, based on two runs for each 

unknown test item, the d2

*

 value is 1.41. If testing 10 points on a tape measure, using 3 runs, the average 

range value is calculated for all points and d2

*

 is 1.72. 

Adapted from Lloyd S. Nelson, J. Qual. Tech. 7 No. 1, January 1975. ©American Society for Quality 

Control, Reprinted by permission. 
February 2012 

Reference Tables Page 3 of 13 

Table 9.2 Z-Factors for Two-sided Confidence Intervals 

for the Normal Distribution 

 

Confidence Level. % Z Factor

50 0.68

67 1.00

75 1.15

90 1.645

95 1.960

95.28 2.000

99.00 2.575

99.74 3

99.993 4 4

99.999 95 5

100, 10-9 6

100, 10-12 7

100, 10-15 8

100, 10-18.9 9

100, 10-23 10
February 2012 

Reference Tables Page 4 of 13 

Table 9.3 Student t-Variate 

* 80% 90% 95% 98% 99% 99.73%

df t0.90 t0.95 t0.975 t0.99 t0.995 t0.9985

1 3.078 6.314 12.706 31.821 63.657 235.80

2 1.886 2.920 4.303 6.965 9.925 19.207

3 1.638 2.353 3.182 4.541 5.841 9.219

4 1.533 2.132 2.776 3.747 4.604 6.620

5 1.476 2.015 2.571 3.365 4.032 5.507

6 1.440 1.943 2.447 3.143 3.707 4.904

7 1.415 1.895 2.365 2.998 3.499 4.530

8 1.397 1.860 2.306 2.896 3.355 4.277

9 1.383 1.833 2.262 2.821 3.250 4.094

10 1.372 1.812 2.228 2.764 3.169 3.957

11 1.363 1.796 2.201 2.718 3.106 3.850

12 1.356 1.782 2.179 2.681 3.055 3.764

13 1.350 1.771 2.160 2.650 3.012 3.694

14 1.345 1.761 2.145 2.624 2.977 3.636

15 1.341 1.753 2.131 2.602 2.947 3.586

16 1.337 1.746 2.120 2.583 2.921 3.544

17 1.333 1.740 2.110 2.567 2.898 3.507

18 1.330 1.734 2.101 2.552 2.878 3.475

19 1.328 1.729 2.093 2.539 2.861 3.447

20 1.325 1.725 2.086 2.528 2.845 3.422

25 1.316 1.708 2.060 2.485 2.787 3.330

30 1.310 1.697 2.042 2.457 2.750 3.270

40 1.303 1.684 2.021 2.423 2.704 3.199

60 1.296 1.671 2.000 2.390 2.660 3.130

∞ 1.282 1.645 1.960 2.326 2.576 3.00

 * Columns to be used in calculating corresponding two-sided confidence interval. From: NIST Handbook 91 

p. T-5; Last column from B.J. Joiner, J. Research NIST. 
February 2012 

Reference Tables Page 5 of 13 

Table 9.4 Critical values for the F-Test, F0.975

n1 = degrees of freedom for numerator n2 = degrees of freedom for denominator

 n1

 

n2 

1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 

1 647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.7 963.3 968.6 976.7 984.9 993.1 997.2 1001 1006 

2 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.41 39.43 39.45 39.46 39.46 39.47 

3 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 14.42 14.34 14.25 14.17 14.12 14.08 14.04 

4 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84 8.75 8.66 8.98 8.51 8.46 8.41 

5 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 6.62 6.52 6.43 6.76 6.28 6.23 6.18 

6 8.81 7.26 6.60 6.23 5.99 5.83 5.70 5.60 5.52 5.46 5.37 5.27 5.60 5.12 5.07 5.01 

7 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82 4.76 4.67 4.57 4.90 4.42 4.36 4.31 

8 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.30 4.20 4.10 4.43 3.95 3.89 3.84 

9 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03 3.96 3.87 3.77 4.10 3.61 3.56 3.51 

10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72 3.62 3.52 3.85 3.37 3.31 3.26 

11 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59 3.53 3.43 3.33 3.66 3.17 3.12 3.06 

12 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44 3.37 3.28 3.18 3.51 3.02 2.96 2.91 

13 6.41 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.31 3.25 3.15 3.05 3.39 2.89 2.84 2.78 

14 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.21 3.15 3.05 2.95 3.29 2.79 2.73 2.67 

15 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12 3.06 2.96 2.86 3.20 2.70 2.64 2.59 

16 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 3.05 2.99 2.89 2.79 3.12 2.63 2.57 2.51 

17 6.04 4.62 4.01 3.66 3.44 3.28 3.16 3.06 2.98 2.92 2.82 2.72 3.06 2.56 2.50 2.44 

18 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.93 2.87 2.77 2.67 3.01 2.50 2.44 2.38 

19 5.92 4.51 3.90 3.56 3.33 3.17 3.05 2.96 2.88 2.82 2.72 2.62 2.96 2.45 2.39 2.33 

20 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84 2.77 2.68 2.57 2.91 2.41 2.35 2.29 

21 5.83 4.42 3.82 3.48 3.25 3.09 2.97 2.87 2.80 2.73 2.64 2.53 2.87 2.37 2.31 2.25 

22 5.79 4.38 3.78 3.44 3.22 3.05 2.93 2.84 2.76 2.70 2.60 2.50 2.84 2.33 2.27 2.21 

23 5.75 4.35 3.75 3.41 3.18 3.02 2.90 2.81 2.73 2.67 2.57 2.47 2.81 2.30 2.24 2.18 

24 5.72 4.32 3.72 3.38 3.15 2.99 2.87 2.78 2.70 2.64 2.54 2.44 2.78 2.27 2.21 2.15 

25 5.69 4.29 3.69 3.35 3.13 2.97 2.85 2.75 2.68 2.61 2.51 2.41 2.75 2.24 2.18 2.12 

26 5.66 4.27 3.67 3.33 3.10 2.94 2.82 2.73 2.65 2.59 2.49 2.39 2.73 2.22 2.16 2.09 

27 5.63 4.24 3.65 3.31 3.08 2.92 2.80 2.71 2.63 2.57 2.47 2.36 2.71 2.19 2.13 2.07 

28 5.61 4.22 3.63 3.29 3.06 2.90 2.78 2.69 2.61 2.55 2.45 2.34 2.69 2.17 2.11 2.05 

29 5.59 4.20 3.61 3.27 3.04 2.88 2.76 2.67 2.59 2.53 2.43 2.32 2.67 2.15 2.09 2.03 

30 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.57 2.51 2.41 2.31 2.65 2.14 2.07 2.01 

40 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.45 2.39 2.29 2.18 2.53 2.01 1.94 1.88 

60 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.33 2.27 2.17 2.06 2.41 1.88 1.82 1.74 

120 5.15 3.80 3.23 2.89 2.67 2.52 2.39 2.30 2.22 2.16 2.05 1.94 2.30 1.76 1.69 1.61 

 5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 2.11 2.05 1.94 1.83 2.19 1.64 1.57 1.48 

For use for a one-tailed test of equality of standard deviation estimate at 2.5 % level of confidence, or for a 

two-tailed test at 5 % level of confidence. 
February 2012 

Reference Tables Page 6 of 13 

Table 9.5 Critical values for the F-Test, F0.99

1 = degrees of freedom for numerator 2 = degrees of freedom for denominator

 1

  2 1 2 3 4 5 6 7 8 9 10 

10 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 

11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 

12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 

13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 4.10 

14 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03 3.94 

15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 

16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 

17 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68 3.59 

18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51 

19 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3.43 

20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 

22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26 

24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3.17 

26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18 3.09 

28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12 3.03 

30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98 

35 7.42 5.27 4.40 3.91 3.59 3.37 3.20 3.07 2.96 2.88 

40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80 

45 7.23 5.11 4.25 3.77 3.45 3.23 3.07 2.94 2.83 2.74 

50 7.17 5.06 4.20 3.72 3.41 3.19 3.02 2.89 2.78 2.70 

55 7.12 5.01 4.16 3.68 3.37 3.15 2.98 2.85 2.75 2.66 

60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63 

65 7.04 4.95 4.10 3.62 3.31 3.09 2.93 2.80 2.69 2.61 

70 7.01 4.92 4.07 3.60 3.29 3.07 2.91 2.78 2.67 2.59 

75 6.99 4.90 4.05 3.58 3.27 3.05 2.89 2.76 2.65 2.57 

80 6.96 4.88 4.04 3.56 3.25 3.04 2.87 2.74 2.64 2.55 

85 6.94 4.86 4.02 3.55 3.24 3.02 2.86 2.73 2.62 2.54 

90 6.93 4.85 4.01 3.53 3.23 3.01 2.84 2.72 2.61 2.52 

95 6.91 4.84 3.99 3.52 3.22 3.00 2.83 2.70 2.60 2.51 

100 6.90 4.82 3.98 3.51 3.21 2.99 2.82 2.69 2.59 2.50 

105 6.88 4.81 3.97 3.50 3.20 2.98 2.81 2.69 2.58 2.49 

110 6.87 4.80 3.96 3.49 3.19 2.97 2.81 2.68 2.57 2.49 

115 6.86 4.79 3.96 3.49 3.18 2.96 2.80 2.67 2.57 2.48 

120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 2.47 

 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 

For use for a one-tailed test of equality of standard deviation estimate at 1 % level of confidence, or for a twotailed test at 2 % level of confidence. 

 
February 2012 

Reference Tables Page 7 of 13 

Table 9.6 Factors for Two-sided Tolerance Intervals 

for the Normal Distribution 

 = 0.95 = 0.99 

 P 

 n 

 0.75 0.90 0.95 0.99 0.999 0.75 0.90 0.95 0.99 0.999 

2 22.858 32.019 37.674 48.430 60.573 114.363 160.193 188.491 242.300 303.054 

3 5.922 8.380 9.916 12.861 16.208 13.378 18.930 22.401 29.055 36.616 

4 3.779 5.369 6.370 8.299 10.502 6.614 9.398 11.150 14.527 18.383 

5 3.002 4.275 5.079 6.634 8.415 4.643 6.612 7.855 10.260 13.015 

6 2.604 3.712 4.414 5.775 7.337 3.743 5.337 6.345 8.301 10.548 

7 2.361 3.369 4.007 5.248 6.676 3.233 4.613 5.488 7.187 9.142 

8 2.197 3.136 3.732 4.891 6.226 2.905 4.147 4.936 6.468 8.234 

9 2.078 2.967 3.532 4.631 5.899 2.677 3.822 4.550 5.966 7.600 

10 1.987 2.839 3.379 4.433 5.649 2.508 3.582 4.265 5.594 7.129 

11 1.916 2.737 3.259 4.277 5.452 2.378 3.397 4.045 5.308 6.766 

12 1.858 2.655 3.162 4.150 5.291 2.274 3.250 3.870 5.079 6.477 

13 1.810 2.587 3.081 4.044 5.158 2.190 3.130 3.727 4.893 6.240 

14 1.770 2.529 3.012 3.955 5.045 2.120 3.029 3.608 4.737 6.043 

15 1.735 2.480 2.954 3.878 4.949 2.060 2.945 3.507 4.605 5.876 

16 1.705 2.437 2.903 3.812 4.865 2.009 2.872 3.421 4.492 5.732 

17 1.679 2.400 2.858 3.754 4.791 1.965 2.808 3.345 4.393 5.607 

18 1.655 2.366 2.819 3.702 4.725 1.926 2.753 3.279 4.307 5.497 

19 1.635 2.337 2.784 3.656 4.667 1.891 2.703 3.221 4.230 5.399 

20 1.616 2.310 2.752 3.612 4.614 1.860 2.659 3.168 4.161 5.312 

21 1.599 2.286 2.723 3.577 4.567 1.833 2.620 3.121 4.100 5.234 

22 1.584 2.264 2.697 3.543 4.523 1.808 2.584 3.078 4.044 5.163 

23 1.570 2.244 2.673 3.512 4.484 1.785 2.551 3.040 3.993 5.098 

24 1.557 2.225 2.651 3.483 4.447 1.764 2.522 3.004 3.947 5.039 

25 1.545 2.208 2.631 3.457 4.413 1.745 2.494 2.972 3.904 4.985 

26 1.534 2.193 2.612 3.432 4.382 1.727 2.469 2.941 3.865 4.935 

27 1.523 2.178 2.595 3.409 4.353 1.711 2.446 2.914 3.828 4.888 

 

From: NIST Handbook 91, p T-11 

 
February 2012 

Reference Tables Page 8 of 13 

Table 9.7 Factors for Computing Two-sided Confidence Limits for  (Part 1) 

Degrees of 

Freedom  = 0.05  = 0.01  = 0.001

SU SL SU SL SU SL

1 17.79 0.3576 86.31 0.2969 844.4 0.2480

2 4.859 0.4581 10.70 0.3879 33.29 0.3291

3 3.183 0.5178 5.449 0.4453 11.65 0.3824

4 2.567 0.5590 3.892 0.4865 6.938 0.4218

5 2.248 0.5899 3.175 0.5182 5.085 0.4529

6 2.052 0.6143 2.764 0.5437 4.128 0.4784

7 1.918 0.6344 2.498 0.5650 3.551 0.5000

8 1.820 0.6513 2.311 0.5830 3.167 0.5186

9 1.746 0.6657 2.173 0.5987 2.894 0.5348

10 1.686 0.6784 2.065 0.6125 2.689 0.5492

11 1.638 0.6896 1.980 0.6248 2.530 0.5621

12 1.598 0.6995 1.909 0.6358 2.402 0.5738

13 1.564 0.7084 1.851 0.6458 2.298 0.5845

14 1.534 0.7166 1.801 0.6549 2.210 0.5942

15 1.509 0.7240 1.758 0.6632 2.136 0.6032

16 1.486 0.7808 1.721 0.6710 2.073 0.6116

17 1.466 0.7372 1.688 0.6781 2.017 0.6193

18 1.448 0.7430 1.658 0.6848 1.968 0.6266

19 1.432 0.7484 1.632 0.6909 1.925 0.6333

20 1.417 0.7535 1.609 0.6968 1.886 0.6397

21 1.404 0.7582 1.587 0.7022 1.851 0.6457

22 1.391 0.7627 1.568 0.7074 1.820 0.6514

23 1.380 0.7669 1.550 0.7122 1.791 0.6568

24 1.370 0.7709 1.533 0.7169 1.765 0.6619

25 1.360 0.7747 1.518 0.7212 1.741 0.6668

 

From: NIST Handbook 91, p T-34 
February 2012 

Reference Tables Page 9 of 13 

Table 9.7 Factors for Computing Two-sided Confidence Limits for  (Part 2) 

Degrees of 

Freedom  = 0.05  = 0.01  = 0.001

SU SL SU SL SU SL

26 1.351 0.7783 1.504 0.7253 1.719 0.7130

27 1.343 0.7817 1.491 0.7293 1.698 0.6758

28 1.335 0.7849 1.479 0.7331 1.679 0.6800

29 1.327 0.7880 1.467 0.7367 1.661 0.6841

30 1.321 0.7909 1.457 0.7401 1.645 0.6880

31 1.314 0.7937 1.447 0.7434 1.629 0.6917

32 1.308 0.7964 1.437 0.7467 1.615 0.6953

33 1.302 0.7990 1.428 0.7497 1.601 0.6987

34 1.296 0.8015 1.420 0.7526 1.588 0.7020

35 1.291 0.8089 1.412 0.7554 1.576 0.7052

36 1.286 0.8062 1.404 0.7582 1.564 0.7083

37 1.281 0.8085 1.397 0.7608 1.553 0.7113

38 1.277 0.8106 1.390 0.7633 1.543 0.7141

39 1.272 0.8126 1.383 0.7658 1.533 0.7169

40 1.268 0.8146 1.377 0.7681 1.523 0.7197

41 1.264 0.8166 1.371 0.7705 1.515 0.7223

42 1.260 0.8184 1.365 0.7727 1.506 0.7248

43 1.257 0.8202 1.360 0.7748 1.498 0.7273

44 1.253 0.8220 1.355 0.7769 1.490 0.7297

45 1.249 0.8237 1.349 0.7789 1.482 0.7320

46 1.246 0.8253 1.345 0.7809 1.475 0.7342

47 1.243 0.8269 1.340 0.7828 1.468 0.7364

48 1.240 0.8285 1.335 0.7847 1.462 0.7386

49 1.237 0.8300 1.331 0.7864 1.455 0.7407

50 1.234 0.8314 1.327 0.7882 1.449 0.7427

 From: NIST Handbook 91, p. T-34 

 
February 2012 

Reference Tables Page 10 of 13 

Table 9.8 Density of Air-free Water in g/cm3 as a 

Function of Celsius Temperature

T 0.0 °C 0.1 °C 0.2 °C 0.3 °C 0.4 °C 0.5 °C 0.6 °C 0.7 °C 0.8 °C 0.9 °C 

0.0 0.999 840 0.999 846 0.999 853 0.999 859 0.999 865 0.999 871 0.999 877 0.999 883 0.999 888 0.999 893

1.0 0.999 899 0.999 903 0.999 908 0.999 913 0.999 917 0.999 921 0.999 925 0.999 929 0.999 933 0.999 937

2.0 0.999 940 0.999 943 0.999 946 0.999 949 0.999 952 0.999 954 0.999 956 0.999 959 0.999 961 0.999 963

3.0 0.999 964 0.999 966 0.999 967 0.999 968 0.999 969 0.999 970 0.999 971 0.999 971 0.999 972 0.999 972

4.0 0.999 972 0.999 972 0.999 972 0.999 971 0.999 971 0.999 970 0.999 969 0.999 968 0.999 967 0.999 965

5.0 0.999 964 0.999 962 0.999 960 0.999 958 0.999 956 0.999 954 0.999 951 0.999 949 0.999 946 0.999 943

6.0 0.999 940 0.999 937 0.999 933 0.999 930 0.999 926 0.999 922 0.999 918 0.999 914 0.999 910 0.999 906

7.0 0.999 901 0.999 896 0.999 892 0.999 887 0.999 881 0.999 876 0.999 871 0.999 865 0.999 860 0.999 854

8.0 0.999 848 0.999 842 0.999 835 0.999 829 0.999 822 0.999 816 0.999 809 0.999 802 0.999 795 0.999 787

9.0 0.999 780 0.999 773 0.999 765 0.999 757 0.999 749 0.999 741 0.999 733 0.999 725 0.999 716 0.999 707

10.0 0.999 699 0.999 690 0.999 681 0.999 672 0.999 662 0.999 653 0.999 643 0.999 634 0.999 624 0.999 614

11.0 0.999 604 0.999 594 0.999 583 0.999 573 0.999 562 0.999 552 0.999 541 0.999 530 0.999 519 0.999 507

12.0 0.999 496 0.999 485 0.999 473 0.999 461 0.999 449 0.999 437 0.999 425 0.999 413 0.999 401 0.999 388

13.0 0.999 376 0.999 363 0.999 350 0.999 337 0.999 324 0.999 311 0.999 297 0.999 284 0.999 270 0.999 256

14.0 0.999 243 0.999 229 0.999 215 0.999 200 0.999 186 0.999 172 0.999 157 0.999 142 0.999 128 0.999 113

15.0 0.999 098 0.999 083 0.999 067 0.999 052 0.999 036 0.999 021 0.999 005 0.998 989 0.998 973 0.998 957

16.0 0.998 941 0.998 925 0.998 908 0.998 892 0.998 875 0.998 858 0.998 841 0.998 824 0.998 807 0.997 890

17.0 0.998 773 0.998 755 0.998 738 0.998 720 0.998 702 0.998 684 0.998 666 0.998 648 0.998 630 0.998 612

18.0 0.998 593 0.998 575 0.998 556 0.998 537 0.998 519 0.998 500 0.998 480 0.998 461 0.998 442 0.998 422

19.0 0.998 403 0.998 383 0.998 364 0.998 344 0.998 324 0.998 304 0.998 284 0.998 263 0.998 243 0.998 222

20.0 0.998 202 0.998 181 0.998 160 0.998 139 0.998 118 0.998 097 0.998 076 0.998 055 0.998 033 0.998 012

21.0 0.997 990 0.997 968 0.997 947 0.997 925 0.997 903 0.997 881 0.997 858 0.997 836 0.997 814 0.997 791

22.0 0.997 768 0.997 746 0.997 723 0.997 700 0.997 677 0.997 654 0.997 630 0.997 607 0.997 584 0.997 560

23.0 0.997 536 0.997 513 0.997 489 0.997 465 0.997 441 0.997 417 0.997 392 0.997 368 0.997 344 0.997 319

24.0 0.997 294 0.997 270 0.997 245 0.997 220 0.997 195 0.997 170 0.997 145 0.997 119 0.997 094 0.997 068

25.0 0.997 043 0.997 017 0.996 991 0.996 966 0.996 940 0.996 913 0.996 887 0.996 861 0.996 835 0.996 808

26.0 0.996 782 0.996 755 0.996 728 0.996 702 0.996 675 0.996 648 0.996 621 0.996 593 0.996 566 0.996 539

27.0 0.996 511 0.996 484 0.996 456 0.996 428 0.996 401 0.996 373 0.996 345 0.996 316 0.996 288 0.996 260

28.0 0.996 232 0.996 203 0.996 175 0.996 146 0.996 117 0.996 088 0.996 060 0.996 031 0.996 001 0.995 972

29.0 0.995 943 0.995 914 0.995 884 0.995 855 0.995 825 0.995 795 0.995 765 0.995 736 0.995 706 0.995 676

30.0 0.995 645 0.995 615 0.995 585 0.995 554 0.995 524 0.995 493 0.995 463 0.995 432 0.995 401 0.995 370

31.0 0.995 339 0.995 308 0.995 277 0.995 246 0.995 214 0.995 183 0.995 151 0.995 120 0.995 088 0.995 056

32.0 0.995 024 0.994 992 0.994 960 0.994 928 0.994 896 0.994 864 0.994 831 0.994 799 0.994 766 0.994 734

33.0 0.994 701 0.994 668 0.994 635 0.994 602 0.994 569 0.994 536 0.994 503 0.994 470 0.994 436 0.994 403

34.0 0.994 369 0.994 336 0.994 302 0.994 268 0.994 234 0.994 201 0.994 167 0.994 132 0.994 098 0.994 064

35.0 0.994 030 0.993 995 0.993 961 0.993 926 0.993 891 0.993 857 0.993 822 0.993 787 0.993 752 0.993 717

36.0 0.993 682 0.993 647 0.993 611 0.993 576 0.993 541 0.993 505 0.993 469 0.993 434 0.993 398 0.993 362

37.0 0.993 326 0.993 290 0.993 254 0.993 218 0.993 182 0.993 146 0.993 109 0.993 073 0.993 036 0.993 000

38.0 0.992 963 0.992 926 0.992 889 0.992 852 0.992 815 0.992 778 0.992 741 0.992 704 0.992 667 0.992 629

39.0 0.992 592 0.992 554 0.992 517 0.992 479 0.992 442 0.992 404 0.992 366 0.992 328 0.992 290 0.992 252

 

*Based on the work of H. Wagenbreth and W. Blanke, PTB - Mitteilingen 6-71. 

Reference equations for Air-Saturated distilled water should be used and are provided in GLP 10 as published in NISTIR 

7383, see latest edition. 
February 2012 

Reference Tables Page 11 of 13 

Table 9.9 Density of Air, g/cm3, as Function of Temperature and Air Pressure*

Barometric Pressure Temperature, °C 

kPa mbar mm Hg 18 20 22 24 26 28 

 77.33 773 580 0.000 92 0.000 92 0.000 91 0.000 91 0.000 90 0.000 90 

 79.99 800 600 0.000 95 0.000 95 0.000 94 0.000 93 0.000 93 0.000 92 

 82.66 827 620 0.000 99 0.000 98 0.000 97 0.000 96 0.000 96 0.000 95 

 85.33 853 640 0.001 02 0.001 01 0.001 00 0.001 00 0.000 99 0.000 98 

 87.99 880 660 0.001 05 0.001 04 0.001 03 0.001 03 0.001 02 0.001 01 

 90.66 907 680 0.001 08 0.001 07 0.001 07 0.001 06 0.001 05 0.001 04 

 93.33 933 700 0.001 11 0.001 10 0.001 10 0.001 09 0.001 08 0.001 07 

 95.99 960 720 0.001 14 0.001 14 0.001 13 0.001 12 0.001 12 0.001 10 

 98.66 987 740 0.001 18 0.001 17 0.001 16 0.001 15 0.001 14 0.001 13 

 101.32 1013 760 0.001 21 0.001 20 0.001 19 0.001 18 0.001 17 0.001 17 

 103.99 1040 780 0.001 24 0.001 23 0.001 22 0.001 21 0.001 21 0.001 20 

 106.66 1067 800 0.001 27 0.001 26 0.001 25 0.001 24 0.001 24 0.001 23 

Computed for air at 50 % relative humidity using Option A in SOP 2. 

*Equations to compute the density of air at any pressure, temperature, and relative humidity are 

given in Section 8 of the Appendix to SOP No. 2 and Option B is preferred for all precision mass 

and volume calibrations.
February 2012 

Reference Tables Page 12 of 13 

Table 9.10 Density and Coefficient of Expansion 

Material Density Coefficient of Expansion 

 (g/cm3) Linear (/°C) Cubical (/°C) Cubical (/°F) 

Aluminum 2.7 0.000 024 0.000 069 

0.000 038 

(not suitable for 

provers) 

Brass 8.4 at 0 C 

8.3909 at 20 C 0.000 019 0.000 054 

Cast iron 7.0 

7.2 0.000 010 0.000 030 

Copper 0.000 017 0.000 050 

Diamond 0.000 001 18 0.000 003 5 

Fused silica (quartz) 0.000 000 5 0.000 001 6 

Glass, borosilicate (T1CA) 0.000 010 

Glass, borosilicate (T1CB) 0.000 015 

Glass, soda-lime 0.000 025 

Gold 18.0 0.000 043 

Invar 0.000 000 4 0.000 001 2 

Lead 11.34 0.000 029 0.000 087 

Length bench 0.000 010 6 

Mercury 0.000 018 

Nichrome 8.39 0.000 039 

Nichrome V 8.5 0.000 039 

Platinum 0.000 011 0.000 027 

Polycarbonate plastic 0.000 45 

Polypropylene plastic 0.000 24 

Polystyrene plastic 0.000 21 

Steel, stainless (mass) 7.84, 7.95, 8.0 0.000 045 

Steel, stainless (provers) 0.000 018 0.000 047 7 0.000 026 5 

Steel, tape, mild 0.000 0116 0.000 033 5 

Steel, pressure vessel, low 

carbon 0.000 016 

Steel, prover, low carbon 0.000 012 0.000 033 5 0.000 018 6 

Steel, terne plate 0.000 012 0.000 035 0.000 019 5 

Tantalum 16.6 0.000 020 

Water (20 °C) 0.000 21 

Reference values for cubical coefficient of expansion given in per F are provided as reference for prover calibrations 

used for petroleum products where the reference temperature is 60 F. 
February 2012 

Reference Tables Page 13 of 13 

Table 9.11 A Short Table of Random Numbers 

46 96 85 77 27 92 86 26 45 21 89 91 71 42 64 64 58 22 75 81 74 91 48 46 18 

44 19 15 32 63 55 87 77 33 29 45 00 31 34 84 05 72 90 44 27 78 22 07 62 17 

34 39 80 62 24 33 81 67 28 11 34 79 26 35 34 23 09 94 00 80 55 31 63 27 91 

74 97 80 30 65 07 71 30 01 84 47 45 89 70 74 13 04 90 51 27 61 34 63 87 44 

22 14 61 60 86 38 33 71 13 33 72 08 16 13 50 56 48 51 29 48 30 93 45 66 29 

40 03 96 40 03 47 24 60 09 21 21 18 00 05 86 52 85 40 73 73 57 68 36 33 91 

52 33 76 44 56 15 47 75 78 73 78 19 87 06 98 47 48 02 62 03 42 05 32 55 02 

37 59 20 40 93 17 82 24 19 90 80 87 32 74 59 84 24 49 79 17 23 75 83 42 00 

11 02 55 47 48 84 74 36 22 67 19 20 15 92 53 37 13 75 54 89 56 73 23 39 07 

10 33 79 26 34 54 71 33 89 74 68 48 23 17 49 18 81 05 52 85 70 05 73 11 17 

67 59 28 25 47 89 11 65 65 20 42 23 96 41 64 20 30 89 87 64 37 93 36 96 35 

93 50 75 20 09 18 54 34 68 02 54 87 23 05 43 36 98 29 97 93 87 08 30 92 98 

24 43 23 72 80 64 34 27 23 46 15 36 10 63 21 59 69 76 02 62 31 62 47 60 34 

39 91 63 18 38 27 10 78 88 84 42 32 00 97 92 00 04 94 50 05 75 82 70 80 35 

74 62 19 67 54 18 28 92 33 69 98 96 74 35 72 11 68 25 08 95 31 79 11 79 54 

91 03 35 60 81 16 61 97 25 14 78 21 22 05 25 47 26 37 80 39 19 06 41 02 00 

42 57 66 76 72 91 03 63 48 46 44 01 33 53 62 28 80 59 55 05 02 16 13 17 54 

06 36 63 06 15 03 72 38 01 58 25 37 66 48 56 19 56 41 29 28 76 49 74 39 50 

92 70 96 70 89 80 87 14 25 49 25 94 62 78 26 15 41 39 48 75 64 69 61 06 38 

91 08 88 53 52 13 04 82 23 00 26 36 47 44 04 08 84 80 07 44 76 51 52 41 59 

68 85 97 74 47 53 90 05 90 84 87 48 25 01 11 05 45 11 43 15 60 40 31 84 59 

59 54 13 09 13 80 42 29 63 03 24 64 12 43 28 10 01 65 62 07 79 83 05 59 61 

39 18 32 69 33 46 58 19 34 03 59 28 97 31 02 65 47 47 70 39 74 17 30 22 65 

67 43 31 09 12 60 19 57 63 78 11 80 10 97 15 70 04 89 81 78 54 84 87 83 42 

61 75 37 19 56 90 75 39 03 56 49 92 72 95 27 52 87 47 12 52 54 62 43 23 13 

78 10 91 11 00 63 19 63 74 58 69 03 51 38 60 36 53 56 77 06 69 03 89 91 24 

93 23 71 58 09 78 08 03 07 71 79 32 25 19 61 04 40 33 12 06 78 91 97 88 95 

37 55 48 82 63 89 92 59 14 72 19 17 22 51 90 20 03 64 96 60 48 01 95 44 84 

62 13 11 71 17 23 29 25 13 85 33 35 07 69 25 68 57 92 57 11 84 44 01 33 66 

29 89 97 47 03 13 20 86 22 45 59 98 64 53 89 64 94 81 55 87 73 81 58 46 42 

16 94 85 82 89 07 17 30 29 89 89 80 98 36 25 36 53 02 49 14 34 03 52 09 20 

04 93 10 59 75 12 98 84 60 93 68 16 87 60 11 50 46 56 58 45 88 72 50 46 11 

95 71 43 68 97 18 85 17 13 08 00 50 77 50 46 92 45 26 97 21 48 22 23 08 32 

86 05 39 14 35 48 68 18 36 57 09 62 40 28 87 08 74 79 91 08 27 12 43 32 03 

59 30 60 10 41 31 00 69 63 77 01 89 94 60 19 02 70 88 72 33 38 88 20 60 86 

05 45 35 40 54 03 98 96 76 27 77 84 80 08 64 60 44 34 54 24 85 20 85 77 32 

71 85 17 74 66 27 85 19 55 56 51 36 48 92 32 44 40 47 10 38 22 52 42 29 96 

80 20 32 80 98 00 40 92 57 51 52 83 14 55 31 99 73 23 40 07 64 54 44 99 21 

13 50 78 02 73 39 66 82 01 28 67 51 75 66 33 97 47 58 42 44 88 09 28 58 06 

67 92 65 41 45 36 77 96 46 21 14 39 56 36 70 15 74 43 62 69 82 30 77 28 77 

72 56 73 44 26 04 62 81 15 35 79 26 99 57 28 22 25 94 80 62 95 48 98 23 86 

28 86 85 64 94 11 58 78 45 36 34 45 91 38 51 10 68 36 87 81 16 77 30 19 36 

69 57 40 80 44 94 60 82 94 93 98 01 48 50 57 69 60 77 69 60 74 22 05 77 17 

71 20 03 30 79 25 74 17 78 34 54 45 04 77 42 59 75 78 64 99 37 03 18 03 36 

89 98 55 98 22 45 12 49 82 71 57 33 28 69 50 59 15 09 25 79 39 42 84 18 70 

58 74 82 81 14 02 01 05 77 94 65 57 70 39 42 48 56 84 31 59 18 70 41 74 60 

50 54 73 81 91 07 81 26 25 45 49 61 22 88 41 20 00 15 59 93 51 60 65 65 63 

49 33 72 90 10 20 65 28 44 63 95 86 75 78 69 24 41 65 86 10 34 10 32 00 93 

11 85 01 43 65 02 85 69 56 88 34 29 64 35 48 15 70 11 77 83 01 34 82 91 04 

34 22 46 41 84 74 27 02 57 77 47 93 72 02 95 63 75 74 69 69 61 34 31 92 13 

From: NIST Handbook 91, p T-82 
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