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SOP 5

Recommended Standard Operating Procedure

for

Using a 3-1 Weighing Design

1 Introduction

Purpose 

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

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

second standard called a check standard. (The check standard may be made up of a 

summation of weights.) The weights are compared using an equal-arm, single-pan 

mechanical, full electronic, or a combination balance utilizing built-in weights and 

a digital indication. The specific SOP for the double substitution procedure for each 

balance is to be followed. The 3-1 weighing design provides two methods of 

checking the validity of the measurement using an integrated F-test to monitor the 

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

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

calibrations for weights in OIML1 Classes E2 to F2 and ASTM2 Classes 0 and 1, in 

which it is critical to assure that the measurements are valid and well documented. 

This procedure is recommended as a minimum for precision calibration of 

laboratory working standards that are subsequently used for lower level calibrations 

and for routine calibration of precision mass standards used for balance calibration. 

For surveillance of reference standards, such as OIML E1, ASTM 00 or better, 

working standards, and for calibration of precision mass standards used to calibrate 

other mass standards, see SOP 28 for the use of higher level weighing designs. 

Prerequisites

Verify that valid calibration certificates with appropriate values and 

uncertainties are available for all reference standards used in the calibration. 

All standards must have demonstrated metrological traceability to the 

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

Institute such as NIST.

Standards must be evaluated to ensure that standard uncertainties for the 

intended level of calibration are sufficiently small. Reference standards should 

only be used to calibrate the next lower level of working standards in the 

laboratory and should not be used to routinely calibrate customer standards.

 1 OIML is the International Organization for Legal Metrology. Weight classes are published in OIML R111, which 

is freely available at http://www.oiml.org.

2 ASTM International (formerly the American Society for Testing and Materials) publishes the E617 standard for 

mass specifications and tolerances. 
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The balance that is used must be in good operating condition with sufficiently 

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

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

preliminary experiments to ascertain its performance quality when new 

balances are put into service.

The operator must be experienced in precision weighing techniques. The 

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

familiar with the concepts in GMP 10. (These procedures are published in 

NISTIR 6969, Selected Laboratory and Measurement Practices, and 

Procedures to Support Basic Mass Calibrations.)

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. Equilibration of balances and weights requires 

environmental stability of the laboratory within the stated limits for a 

minimum of 24 hours before a calibration.

Table 1. Environmental conditions.

Echelon3 Temperature Requirements During a Calibration Relative Humidity 

(%)

I

OIML E1, ASTM 000, 00, 0

Lower and upper limits: 18 °C to 23 °C

Maximum changes: ± 0.5 °C / 12 h and ± 0.3 °C / h

40 to 60 ± 5 / 4 h

OIML E2, ASTM 1

Lower and upper limits: 18 °C to 23 °C

Maximum changes: ± 1 °C / 12 h and ± 0.7 °C / h

II Lower and upper limits: 18 °C to 23 °C

Maximum changes: ± 2 °C / 12 h and ± 1.5 °C / h 40 to 60 ± 10 / 4 h

It is important that the difference in temperature between the weights and the air inside the mass 

comparator is as small as possible. Keeping the reference weight and the test weight inside, or next 

to, the mass comparator before and during the calibration can reduce this temperature difference. 

Standards and test artifacts must be allowed to reach equilibration in or near the balance before 

starting measurements.

 3 Echelon I corresponds to weights of Classes OIML E1 and E2, Echelon II corresponds to weights of Classes OIML 

F1 and F2. 
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2 Methodology

Scope, Precision, Accuracy 

This method can be performed on any type of balance or mass comparator using the 

appropriate double substitution method for the weighing instrument. Because considerable 

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

accuracy. The weighing design uses three double substitutions to calibrate a single 

unknown weight. This introduces redundancy into the measurement process and permits 

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

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

the measurement outputs to assign a value to the unknown weight. The standard deviation 

of the process depends upon the resolution of the balance and the care exercised to make 

the required weighings. The accuracy will depend upon the accuracy and uncertainty of the 

calibration of the standard weights and the precision of the comparison. 

Summary 

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

intercompared in a specific order using several double substitution processes. The 

balance and the weights must be prepared according to the appropriate double 

substitution method for the balance being used. Once the balance and weights have 

been prepared, all readings must be taken from the reading scale of the balance 

without adjusting the balance or weights in any way during the process. Within a 

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

minimize effects due to instrument drift. The 3-1 weighing design includes air 

buoyancy corrections.

Apparatus/Equipment Required 

Precision analytical balance or mass comparator with sufficient capacity and 

resolution for the planned calibrations.

Calibrated reference standards or working standards, of nominally equal mass 

to the unknown mass standards being calibrated. Calibrated tare weights are 

used as needed to ensure that the standard(s) and test artifacts are of equal 

nominal mass (See NISTIR 6969, SOP 34 for suitable limits).

Calibrated sensitivity weights and tare weights selected to comply with the 

requirements of SOP 34.

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

range if needed.

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

by hand.
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Forceps and gloves must be selected to avoid damage or contamination to 

mass standards. 

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

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

consistent timing that can minimize errors due to linear drift.

Calibrated barometer with sufficiently small resolution, stability, and 

uncertainty (See NISTIR 6969, SOP 2, e.g., accurate to ± 66.5 Pa (0.5 

mmHg)) to determine barometric pressure.4

Calibrated thermometer with sufficiently small resolution, stability, and

uncertainty (see SOP 2, e.g., accurate to ± 0.10 °C) to determine air 

temperature.4

Calibrated hygrometer with sufficiently small resolution, stability, and 

uncertainty (see SOP 2, e.g., accurate to ± 10 percent) to determine relative 

humidity.4

 4

NISTIR 6969 includes SOP 2 for the calculation of air density. 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 limits specified are recommended for high precision calibration.
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Symbols

Table 2. Symbols used in this procedure.

Symbol Description

S standard reference weight

X weight to be calibrated

Sc check standard

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

ρa density of air at time of calibration

ρn density of normal air (1.2 kg/m3

)

ρ density of masses; subscripts s, x, ts, tx, sw are used to identify the weight

Procedure

Preliminary Procedure

2.5.1.1 Weights are visually inspected for cleanliness and damage. Follow 

the laboratory policy and customer contract review process to 

determine if and when standards will be cleaned, or standards with 

inadequate cleanliness are returned without calibration, and when 

“as found” and “as left” values will be obtained through duplicate 

calibrations.

2.5.1.2 If cleaning weights, it is important to clean weights before any

measurements are made, unless “as found’ data is to be measured, 

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 by blowing air across the surface or by brushing 

with a clean soft bristled brush. 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, 
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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:

Table 3. Cleaning stabilization time.

Weight class E1 E2 F1 F2 to M3

After cleaning with alcohol 7 to 10 days 3 to 6 days 1 to 2 days 1 hour

After cleaning with distilled water 4 to 6 days 1 to 3 days 1 day 1 hour

2.5.1.4 Prior to performing any calibration tests, the weights need to be 

equilibrated to the ambient conditions of the laboratory.

Specifically, 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 cold, additional equilibration is 

required to address problems with varying surface moisture. 

Weights must be completely dry prior to calibration.
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Table 4. Minimum equilibration times.5

ΔTa Nominal Mass6

OIML Class 

E1

(time in h)

OIML Class 

E2

(time in h)

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 - 70 33 4

10, 20, 50 kg 45 27 12 3

1, 2, 5 kg 18 12 6 2

100, 200, 500 g 8 5 3 1

10, 20, 50 g 2 2 1 1

< 10 g 1 1 1 0.5

± 5 °C

1 000, 2 000, 5 000 kg - - 24 1

100, 200, 500 kg - 40 10 1

10, 20, 50 kg 36 18 4 1

1, 2, 5 kg 15 8 3 1

100, 200, 500 g 6 4 2 0.5

10, 20, 50 g 2 1 1 0.5

< 10 g 0.5 0.5 0.5 0.5

± 2 °C

100 kg to 5 000 kg - 16 1 0.5

10, 20, 50 kg 27 10 1 0.5

1, 2, 5 kg 12 5 1 0.5

100, 200, 500 g 5 3 1 0.5

< 100 g 2 1 1 0.5

± 0.5 °C

100 kg to 5 000 kg - 1 0.5 0.5

10, 20, 50 kg 11 1 0.5 0.5

1, 2, 5, kg 7 1 0.5 0.5

100, 200, 500 g 3 1 0.5 0.5

< 100 g 1 0.5 0.5 0.5 a

ΔT = Initial difference between weight temperature and laboratory temperature.

Weighing Design Matrix

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

design, in a matrix format as shown in NBS Technical Note 952, Designs 

for the Calibration of Standards of Mass, J. M. Cameron, M. C. Croarkin, 

and R. C. Raybold, 1977:

 5 Consider equivalent ASTM Classes for equilibration times.

6 Nominal masses in the 1, 2, 5 combinations include intermediate values such as 3.
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Table 5. Weighing Design Matrix.

Weight ID

Comparison S X Sc

a1 + -

a2 + -

a3 + -

Standard (Restraint) +

Check Standard +

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

the exception that the design order is reversed and Restraint B is used7. The 

restraint is another name for the “standard” used in the comparison that may 

be found in NBS Technical Note 952. The following table may be useful 

for data reduction purposes if using the NIST Mass Code. When creating a 

data file for this design, the restraint, check, and design vectors will appear 

in the order and as the numbers in the following table:

Table 6. Weighing Design Vector for Mass Code.

Row in Mass Code Design First Entry Second Entry Third Entry

Restraint 1 0 0

Check 0 0 1

Following series sum 0 0 0

Report 0 1 1

1st double sub 1 -1 0

2nd double sub 1 0 -1

3rd double sub 0 1 -1

Measurement Procedure 

Perform the measurement as a series of 3 double substitutions. Review SOP 

4 (NISTIR 6969) if needed. Record the pertinent information for the 

standard, S, unknown, X, and check standard, Sc, as indicated on a suitable 

data sheet such as the one in the Appendix of this SOP. Measure the 

laboratory ambient temperature, barometric pressure, and relative humidity, 

before and after the set of three double substitutions (i.e., before and after 

all twelve observations). If performing buoyancy corrections on each

double substitution within the design, record environmental data before 

observations 1, 5, 9 and after measurement 12. Perform the mass 

measurements in the order shown in the following table.

 7 Additional 3-1 weighing designs are published. This procedure provides the calculations and solutions for the 

position of mass standards/artifacts in this design only. 
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Table 7. 12 Measurement Observations.

Double

Substitution

Measurement

Number

Weights

on Pan Observation

a1: S vs X 1 S + ts O1

2 X + tx O2

3 X + tx + sw O3

4 S + ts + sw O4

a2: S vs Sc 5 S + ts O1

6 Sc + tsc O2

7 Sc + tsc + sw O3

8 S + ts + sw O4

a3: X vs Sc 9 X + tx O1

10 Sc + tsc O2

11 Sc + tsc + sw O3

12 X + tx + sw O4

3 Calculations

Note: As the NIST Mass Code software can be used to perform the calculations for this 

process, the balance of this section is provided as reference to be used for manual or 

computerized calculations using other software packages.

Calculate the average air density, ρA, as described in the Appendix to NISTIR 6969, 

SOP No. 2, Option B.

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

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

with the calculations used in SOP 4, NISTIR 6969; the signs will be opposite SOP 

4, Option A.):

A sw

1243 sw

x

3 2

M 1 

( + ) OOOO

a = 

2 O O

ρ

ρ

    − − −  

− Eqn. (1)

Calculate the within process standard deviation, sw, for the 3-1 weighing design. 

This standard deviation has one degree of freedom. 

s aaa w = 0.577( 123 − + ) Eqn. (2)
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Calculate the F statistic which compares the observed within process standard 

deviation, sw, to the pooled (accepted) within process standard deviation. (See 

NISTIR 6969, Sections 8.4, 8.5, and 8.9.2, for a discussion of the statistics used in 

weighing designs.)

2

Accepted

2

Observed

w

w

s

s F = Eqn. (3)

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

95 % confidence level (Table 9.12, NISTIR 6969) to be acceptable. The F-value is 

obtained from the F-table for numerator degrees of freedom equal one, and 

denominator degrees of freedom equal to the number of degrees of freedom in the 

pooled within process standard deviation. If the data fails the F-test and the source 

of the error cannot be determined conclusively, the measurement must be repeated.

Calculate the least-squares measured difference dsc for Sc.

1 23 2

Sc

a aa d 

3

−− − = Eqn. (4)

Calculate the observed mass of Sc, Msc.

s sc

s Sc

A AA s Sc t t

s t t

Sc

A

Sc

M MM 1 + + 1 1 d

M =

1 

ρ ρρ

ρ ρρ

ρ

ρ

    

  −    −− −

    

    −

 

Eqn. (5)

Calculate the Conventional Mass of Sc, CMSc:

Sc

Sc

Sc

0.0012

M 1 

CM = 0.999850

ρ

    −

  Eqn. (6)

Evaluate the mass MSc, or conventional mass CMSc, of Sc.

The observed mass or conventional mass is evaluated with a t-test in the procedure 

based on comparison with the accepted mass or conventional mass value

determined from the mean of the control chart. The mass or conventional mass 

(depending on what is tracked in the laboratory) is also plotted on the control chart 

and must lie within the control limits (See NISTIR 6969, SOP 9). If the value is not

within limits, and the source of error cannot be found, measurement must be 

stopped until suitable corrective action is taken. Corrective action is demonstrated 

through evaluation of additional measurement results that are within limits. 
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Calculate the least-squares measured difference, dx, for X.

123

x

2a a +a d =

3

− − Eqn. (7)

Calculate the mass of X, Mx.

1

s x

s x

A AA s x t t

s t t

x

A

x

M MM 1 ++ 1 1 d

M

ρ ρρ

ρ ρρ

ρ

ρ

    

  −    −− −

     =     −

 

Eqn. (8)

Calculate the conventional mass8 of X, CMx. The conventional mass should be 

reported. 

x

x

x

0.0012

M 1

CM 0.999850

ρ

    −

  = Eqn. (9)

Calculate apparent mass versus brass only if requested. 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. (This is rare.) Apparent 

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

x

x

x

0.0012

M 1

AM vs. 8.4

0.0012 1

8.3909

ρ

    −

  =     −  

Eqn. (10)

4 Measurement Assurance

The within process standard deviation is incorporated into the NIST Mass Code 

and is used to conduct an F-test of the observed standard deviation versus the 

pooled/accepted standard deviation of the process at a 95 % confidence level. If 

calculations are performed manually or using other software, follow the process 

described in section 3 for F-test evaluation.

 8 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: artifact 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)).
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SOP 5 weighing design integrates a suitable check standard (See NISTIR 6969, 

GLP 1, SOP 9, and SOP 30).

The check standard value is calculated and immediately evaluated on the control 

chart to verify that the mass is within established limits. A t-test may be 

incorporated to check the observed value of the check standard against the accepted 

value. It is evaluated using the following equation and a 95 % confidence level. All 

values must be entered in the control chart, even if failing this statistic to ensure the 

variability obtained for the process is truly representative of the process and not 

unduly reduced over time. The observed value of the check standard is compared 

to the accepted mean value of the check standard and divided by the standard 

deviation for the check standard observations over time. This equation monitors 

stability over time but should not be used to assess for bias. A calculated t-value 

less than two is within the warning limits of the process. A calculated t-value 

between two and three represents a value between the warning limits and 

control/action limits. A calculated t-value exceeding three represents a value 

outside of the control/action limits and suitable action must be taken. Calculated 

values of the t-statistic may also be monitored over time to determine the presence 

of drift.

( ) p

c c

s

S S

t − = Eqn. (11)

Check standard measurement results obtained over time are used to calculate the 

standard deviation of the measurement process, sp.

The mean value of the check standard over time is also compared to an appropriate

reference value of the check standard with respect to applicable expanded 

uncertainties to evaluate bias and drift over time. Excessive drift or bias must be 

investigated and followed with suitable corrective action. (See NISTIR 6969, SOP 

9, Section 4.2 for assessment methodology.)

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 to be determined based on the degrees of 

freedom, which if large enough will be 2, (k = 2), to give an approximate 95 percent level 

of confidence. (See NISTIR 6969, SOP 29, “Standard Operating Procedures for the 

Assignment of Uncertainty”, for the complete standard operating procedure for calculating 

the uncertainty. 

When the 3-1 weighing design is used in conjunction with specialized software for data 

reduction, see SOP 28, “Recommended Standard Operating Procedure for Using Advanced 
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Weighing Designs”, for detailed instructions on calculating the uncertainty components which 

are required.

The expanded uncertainty for the standard, U, is obtained from the calibration 

certificate. 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. When multiple standards are used, see SOP 

29 for evaluation of dependent and independent conditions and combining methods 

for the standard uncertainty of the standard. Usually only one standard is used as 

the restraint for the 3-1 weighing design, the uncertainty of the check standard is 

not included in assigning an uncertainty to the unknown mass. Where the coverage 

factor or confidence interval is not given, the laboratory should either contact the 

calibration provider to obtain the correct divisor or use a value of k = 2, assuming 

that the expanded uncertainty was reported with an approximate 95 % confidence 

interval (95.45 %).

The value for the standard deviation of the measurement process sp is obtained from 

the control chart data for check standards using only 3-1 weighing design 

measurements (see SOP No. 9.) The within-process standard deviation, sw, is only 

used as a part of the process variability evaluation using the F-test unless between 

time components are also determined. In that case, the standard deviation of the 

process, sp, is treated as st (standard deviation over time) and see SOP 28 for details. 

Statistical control must be verified by the measurement of the check standard in the 

3-1 design.

Where the standard deviation of the measurement process from the control 

chart is less than the resolution of the balance being used, the laboratory 

may round up to the nearest balance division to represent the standard 

deviation, or use the larger of the standard deviation of the process and one 

of the following estimates for repeatability is used to represent the standard 

deviation of the process:

5.2.1.1 If the laboratory prefers a conservative approach, or when the 

current and representative degrees of freedom are less than 30, the 

larger of the control chart sp and the result from Eqn. 12 should be 

used, where d is the smallest balance division. For example, if the 

balance division is 0.1 mg, the smallest standard deviation may be 

0.06 mg. If the laboratory calculated standard deviation is 0.075 mg, 

then 0.075 mg is used. 

0.6

3 p

d s d = ≈ Eqn. (12)
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5.2.1.2 When the laboratory has the confidence associated with a well 

characterized measurement process and has 30 or more degrees of 

freedom to represent the process, the larger of the observed sp or 

Eqn. 13 is used. For example, if the balance division is 0.1 mg, the 

smallest standard deviation may be 0.03 mg. If the laboratory 

observed standard deviation is larger than 0.03 mg, then that is the 

value to be used.

0.3

2 3 p

d s d = ≈ Eqn. (13)

Uncertainty due to air buoyancy corrections and air density. Select one of the 

following options in priority preference for calculating the uncertainty associated 

with air buoyancy. 

Option 1, preferred. Use the formulae provided in OIML R111, C.6.3-1, 

C.6.3-2, and C.6.3-3.

Option 2. Calculate the uncertainty by quantifying estimated impacts 

associated with the uncertainties of the air temperature, barometric pressure, 

relative humidity, and the air density formula based on laboratory 

uncertainties and calculations given in NISTIR 6969, SOP 2 and the SOP 

being used. Note: this may be done using a simplified baseline “what if” 

approach or a Kragten analysis.9

Uncertainty associated with the density of the standards and the unknown test 

weights, uρ. Uncertainties associated with the density of the standards used in the 

calibration may be incorporated into the estimated calculations in section 5.3. 

Uncertainty associated with bias, ud. Any noted bias that has been determined 

through analysis of control charts and round robin data must be less than limits 

provided in SOP 29 and may be included if corrective action is not taken. (See 

NISTIR 6969, SOP 29 for additional details.)

Example components to be considered for an uncertainty budget table are shown in 

the following table.

 9 A baseline “what if” approach calculates the estimated impact of each variable in the final measurement result by 

individually changing each variable of interest by the uncertainty quantity. (See the EURACHEM/CITAC 

Quantitative Guide to Uncertainties in Analytical Methods (QUAM, 2012) for a discussion of Kragten 

spreadsheets). 
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Table 8. Example Uncertainty Budget Table.

Uncertainty Component 

Description Symbol Source Typical Distribution

Uncertainty of the standard 

mass(es) (5.1) us Calibration certificate

Expanded U 

divided by 

coverage factor

Accepted standard deviation of 

the process (5.2, 5.2.1) sp

Control chart, standard deviation 

chart

OR estimates when sp is smaller 

than balance division

Normal

OR

Rectangular, onehalf rectangular

Uncertainty of the air buoyancy 

correction (5.3) ub OIML R111 Rectangular

Air temperature (for air density) ut SOP 2 or OIML R111 Rectangular

Air pressure (for air density) up SOP 2 or OIML R111 Rectangular

Air relative humidity (for air 

density) uRH SOP 2 or OIML R111 Rectangular

Air density (formula) uρa SOP 2 or OIML R111 Rectangular

Mass densities (5.4) uρm

Measured and reported value

OIML R111 Table B.7

Typically, 0.03 g/cm3 to 

0.05 g/cm3

Rectangular

Uncertainty associated with bias 

(5.5) ud Control chart, proficiency tests NISTIR 6969 SOP 29

Draft a suitable uncertainty statement for the certificate (e.g.,)

The uncertainty reported is the root sum square of the standard uncertainty of the 

standard, the standard deviation of the process, and the uncertainty associated with 

the 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 when using a check standard with adequate 

degrees of freedom.

NOTE: Where inadequate degrees of freedom are available, k, is determined using 

the appropriate degrees of freedom and the 95.45 % column in the table from 

Appendix A of NISTIR 6969, SOP 29.
SOP 5 – 2019 Page 16 of 18

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6 Certificate

Report results as described in NISTIR 6969, SOP No. 1, Preparation of Calibration 

Certificates. Report the mass, conventional mass, environmental conditions during the 

calibrations, mass density used (reported, measured, or assumed (specify which 

applies)), and calculated expanded uncertainties with coverage factor(s).

Conformity assessments.

Evaluate compliance to applicable tolerances as needed or required by the customer

or by legal metrology requirements. Decision criteria for uncertainty and tolerance 

evaluations include two components: 1) the expanded uncertainty, U, must be < 1/3 

of the applicable tolerances published in ASTM E 617 and OIML R111 

documentary standards and 2) the absolute value of the conventional mass 

correction value plus the expanded uncertainty must be less than the applicable 

tolerance to confidently state that mass standards are in or out of tolerance.

Compliance assessments must note the applicable documentary standard and which 

portions of the standard were or were not evaluated.
SOP 5 – 2019 Page 17 of 18

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Appendix A

Observation Sheet for 

3-1 Weighing Design When Tare Weights Are Used

Laboratory data and conditions:

Operator Before After

Date Temperature °C

Balance Barometric Pressure mmHg

Load Relative Humidity %

Pooled within process s.d., sw= Degrees of freedom for process s.d.

Check standard s.d., sp = Degrees of freedom from control 

chart

Mass standard(s) data:

ID 

(Insert Set or SN) Nominal Mass 

Correction

Expanded Unc:

from cal. 

certificate

Unc:

k factor

Density

g/cm3

Unc 

Density

k=1

S

ts

X

tx

Sc

tSc

sw

Laboratory observations:

Balance Observations, Units_______

Time: 

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

S + ts S + ts X + tx

X + tx Sc + tsc Sc + tsc

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

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

Time: 

a1 = a2 = a3 =
SOP 5 – 2019 Page 18 of 18

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Appendix B – Flow Chart of the 3-1 Weighing Design Process

Calculate

dx, Mx, CMx

Uncertainty

Prepare Certificate

Pass F-test?

Gather Data:

Standards

Balance

Procedures

Calculate

Air Density

a1, a2, a3

sw

F-ratio

Measurements

T, P, RH

12 Balance 

Observations

Plot MSc

Calculate t-value

Calculate

dSc

MSc

Pass t-test?

Measurement 

Control?

YES

YES

NO

NO

Figure 1. Appendix B – Flow Chart of the 3-1 Weighing Design Process.