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SOP 4 

Recommended Standard Operating Procedure

for

Weighing by Double Substitution

1 Introduction

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 

weighing instrument indications are minimized by using the weighing instrument 

as a comparator and by calibrating the weighing instrument indications over the 

range of use for the measurement with a sensitivity weight that has a known mass. 

Accordingly, the procedure is especially useful for calibrations of weights in 

OIML1 Classes F1 to M3, ASTM2 Classes 2 through 7, and forms the basis for 

advanced weighing designs used in SOP 5 and SOP 28 (See NISTIR 5672). The 

procedure incorporates two methods of measurement assurance: one is a 

mathematical check that evaluates the within-process repeatability and the other 

involves using a check standard to monitor the reference value(s) of the standard.

This procedure may be used for any nominal values provided adequate standards 

and equipment are available. Detailed measurement ranges, standards, equipment, 

and uncertainties for this SOP are generally compiled in a separate document in the 

laboratory.

Prerequisites

Valid calibration certificates with appropriate values and uncertainties must 

be available for all of the standards used in the calibration. All standards 

must 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.

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.

The weighing instrument must be in good operating condition with 

sufficiently small process standard deviation as verified by a valid control 

chart. When a new weighing instrument is put into service, preliminary 

 1 OIML is the International Organization for Legal Metrology. Weight classes are published in OIML R 111, 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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experiments must be conducted to ascertain initial performance quality and 

adequacy.

The operator must be trained and experienced in precision weighing 

techniques with specific training in SOP 2, SOP 4, SOP 29, GMP 4, and 

GMP 10.

Laboratory facilities must 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

(%)

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

III Lower and upper limits: 18 °C to 27 °C

Maximum changes: ± 5 °C / 12 h and ± 3 °C / h 40 to 60 ± 20 / 4 h

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

the balance before commencing measurements. It is essential for the 

difference in temperature between the weights and the air inside the mass 

comparator to be 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.

2 Methodology

Scope, Precision, Accuracy

This method is applicable for all types of weighing instruments. The achievable 

precision and expanded uncertainty depend on the resolution, sensitivity, linearity, 

and other operating characteristics of the weighing instrument 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 accuracy obtained in the measurement process.

Summary 

The weighing instrument is adjusted if necessary, to obtain measurement 

indications for all weighings that are within the range of the optical scale or digital 

 3 Echelon II corresponds to weights of Classes OIML F1 and F2, ASTM Classes 2, 3 and 4. Echelon III corresponds 

to weights of Classes OIML M1, M2, and M3, ASTM Classes 5, 6 and 7. This procedure does not provide adequate 

redundancy for OIML Classes E1 or E2, ASTM Classes 000, 00, 0, or 1 and SOP 5 or SOP 28 should be used.
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indications without changing built-in weights, if present (this includes changing 

dials or the weighing platform masses). The standard and the test weight are each 

weighed twice. A small, calibrated weight, called a sensitivity weight, is added to 

the standard and unknown test weight in the latter two weighings to provide 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 minimize effects due to drift in the measurement process.

The double substitution procedure is the same for all weighing instruments, but the 

adjustment of the weighing instrument to prepare for the intercomparison and the 

selection of the sensitivity weight differs slightly depending upon the type of 

instrument used. When steps specific to a particular instrument are required, they 

are given in subsections of the procedure identified by a, b, and c along with the 

type of instrument.

Apparatus/Equipment Required

Precision analytical balance or mass comparator with sufficient capacity 

and resolution for the calibrations planned.

Calibrated 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 SOP 34 for suitable limits).

Calibrated sensitivity weights and tare weights selected to comply with the 

guidelines in 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. Forceps and gloves must be selected to avoid damage or 

contamination of 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 weighing instrument 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.
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Calibrated barometer with sufficiently small resolution, stability, and 

uncertainty (See 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

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 weight(s) 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 nominal value of a specific weight. Subscripts s, x, are used to identify the weight.

C 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, including discussions with customers, to 

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

with inadequate cleanliness are returned without calibration and 

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

duplicate calibrations.

 4

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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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 will likely 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 with a 

clean bulb 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, 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):

Table 3. Cleaning stabilization.

Weight class F1 F2 to M3

After cleaning with alcohol 1 to 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 

equilibrated 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 cold, additional equilibration is 

required to address problems with varying surface moisture levels. 

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

ΔTa 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 24 1

100, 200, 500 kg 10 1

10, 20, 50 kg 4 1

1, 2, 5 kg 3 1

100, 200, 500 g 2 0.5

< 100 g 1 0.5

± 2 °C < 100 g to 5 000 kg 1 0.5

± 0.5 °C < 100 g to 5 000 kg 0.5 0.5

a

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 identify 

where in the weighing instrument range the readings occur, to 

determine if tare weights are required, to determine the sensitivity 

weight to be used, and to determine the time interval required for the 

weighing instrument indication to stabilize. See NISTIR 6969, SOP 

34 for specific instructions on evaluation of the need for and 

selection of tare weights and sensitivity weights. 

Tare weights are rarely needed for 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 included in the process of 

determining mass values and 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 SOP 34, Table 1). 

A sensitivity weight must be used on equal-arm balances, single-pan 

mechanical, and is usually used on electronic balances, to ensure 

that the measured differences between the standard(s) and unknown 

test items have valid accuracy and traceability (See SOP 34, Table 

 5 Consider equivalent ASTM Classes for equilibration times.
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2) (e.g., the optical/digital scale is calibrated each time the 

procedure is performed through the use of a sensitivity weight). 

If the sensitivity of the balance has been analyzed and is periodically 

evaluated and found to introduce negligible errors or uncertainties 

according to SOP 34 and measurement requirements, the equations 

in Section 3 must be modified to eliminate the sensitivity factor to 

avoid “division by zero” errors in the calculations. 

Sensitivity weights are selected to meet the criteria in SOP 34.

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 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. Option A is used in SOP 5 and SOP 28.

2.5.1.7 Adjust the single pan mechanical 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 substitution and 

calibration is not required. Once the balance has been adjusted to the 

desired scale indication, 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.

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
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All observations should be recorded on suitable data sheets such as those in 

the appendix or may be entered into computer software. Measure and record 

the laboratory ambient temperature, barometric pressure, and relative 

humidity before and after the mass measurements.

2.5.2.1 Observation 1. Place the standard weight(s), S, along with ts (if 

applicable) on the balance pan. If equipped with a pan arrestment 

mechanism, release the balance pan. When the pan is released, time 

the interval or wait for the stability indicator and record observation 

O1 once the balance indication has stabilized.

2.5.2.2 Observation 2. Remove weight(s) S and ts (if present) and replace 

with test weight X and its tare weight, tx (if applicable.) Release the 

pan, time the interval or wait for the stability indicator, and record 

observation O2.

2.5.2.3 Observation 3. Add the sensitivity weight, sw, to the weights of 

observation 2. Pick up the largest weight on the pan to ensure 

consistent stabilization timing when needed. Release the pan, time 

the interval or wait for the stability indicator, and record observation 

O3.

2.5.2.4 Observation 4. Remove weights X and tx (if present) and replace 

with S and ts (if applicable) on the balance pan. The sensitivity 

weight, sw, remains on the balance pan. Release the pan, time the 

interval or wait for the stability indicator, and record observation O4.

2.5.2.5 Compare the two differences (O2 – O1) and (O3 – O4) to evaluate 

within process repeatability; they should not differ from one another 

by a laboratory determined limit (e.g., 2 standard deviations of the 

balance or 10 balance divisions are commonly used) for this process 

and load. If this difference is exceeded, reject the data and redo the 

measurements. Investigate possible causes of excess variability 

should within process repeatability exceed the limits.

2.5.2.6 Repeat 2.5.2.1 through 2.5.2.5 with a check standard, Sc. It is 

acceptable to begin with the check standard to ensure that the 

standards and process are in control, followed by the steps with the 

unknown standard, X.
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Measurement Procedure, Optional Sequence B (XSSX)

Table 6. 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

If no air buoyancy correction is performed, calculate the conventional mass 

correction, CSc, for the check standard using the appropriate equation from 3.1.1 or 

3.1.2, depending on the sequence used, and incorporate an uncorrected systematic 

uncertainty in the uncertainty calculations when using these equations due to the 

lack of performing air buoyancy corrections. Use the equations provided in SOP 2 

to calculate the magnitude of the air buoyancy correction and treat it as a rectangular 

distribution. Calculate the value for the test weight correction using the same 

equations, but substitute X for the Sc subscript for the unknown in equations 3.1.1. 

and 3.1.2. as appropriate.

Optional Sequence A (SXXS)

( ) ( ) s Sc

21 34 sw

Sc s t t s Sc

3 2

O O O O CM C C CM CM N N

2 OO

  −+−  

=+ − +     + −

  −  

Eqn. (1)

Optional Sequence B (XSSX)

( ) ( ) C sS c c

12 43 sw

S s t t s S

3 2

O O O O CM C C CM CM N N

2 OO

  −+−  

=+ − +     + −

  −  

Eqn. (2)

Mass Calculation with Air Buoyancy Correction

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

Option B.

Calculate the mass MSc of the check standard using the mass of the standard 

weight(s), the tare weights, and the sensitivity weights, according to the 

optional sequence used.
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3.2.2.1 Optional Sequence A (SXXS).

( 21 34 ) ( )

3 2

1

111

2

1

S Sc

s Sc

a

sw

aaa sw

Stt

Stt

Sc

a

Sc

ρ M 

ρρρ O O O O ρ M M M ρ ρ ρ O O

M 

ρ

ρ

        −       −+ −       − + −− − +         −           =     −

 

Eqn. (3)

3.2.2.2 Optional Sequence B (XSSX).

( 12 43 ) ( )

3 2

1

111

2

1

S Sc

s Sc

a

sw

aaa sw

Stt

Stt

Sc

a

Sc

ρ M 

ρρρ O O O O ρ M M M ρ ρ ρ O O

M 

ρ

ρ

        −       −+ −       − + −− − +         −           =     −

 

Eqn. (4)

Calculate the Conventional Mass6 of Sc, CMSc.

1

1

8.0

c

n

S

Sc

Sc

n

M

CM

ρ

ρ

ρ

    −

  =     −  

Eqn. (5)

Calculate the Mass Correction of Sc.

 CMN Sc Sc Sc = − Eqn. (6)

where NSc is the nominal value for Sc and ensure it is within statistical

control or take suitable corrective action. The mass or conventional mass 

(depending on laboratory control charts) determined for the check standard 

should be 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 

 6 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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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.

Calculate the mass of the test weight, Mx, and its mass correction, Cx, using 

the equations in 3.2.2.1 or 3.2.2.2 where Sc is replaced by X. Calculate the 

mass correction Cx, as follows:

 CMN x xx = − Eqn. (7)

where Nx is the nominal value for X.

Calculate the Conventional Mass of X, CMx. The conventional mass should 

be reported. 

1

1

8.0

n

x

x

x

n

M

CM

ρ

ρ

ρ

    −

  =     −  

Eqn. (8)

Calculate the 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.)

 

1

1

8.3909

n

x

x

x vs brass

n

M

AM

ρ

ρ

ρ

    −

  =     −  

Eqn. (9)

4 Measurement Assurance

The process integrates a suitable check standard (See GLP 1, SOP 9, and SOP 30).

The check standard value is calculated and immediately entered on the control chart 

to verify that the mass is within established limits. All values must be entered in the 

control chart, even if failing this statistic, unless a mistake (i.e., typographical error) 

is identified and corrected, to ensure the variability obtained for the process is not 

unduly reduced over time. 

A t-test may be incorporated to check the observed value of the check standard 

against the mean value using the following equation and a 95 % confidence level. 

The t-statistic is calculated for stability analysis. 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 
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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. (10)

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 their applicable expanded 

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

investigated and followed with suitable corrective action. See SOP 9, Section 4.2 

for assessment methodology.

Where SOP 9 is followed, and check standards are used for 2 to 3 nominal values 

per balance, more frequent calibration intervals may be needed to monitor the 

working standards and the larger standard deviation of the nominal values 

bracketing the nominal value should be used. If check standards were already 

checked on a given day and found to be in control, additional check standard 

measurements are not required.

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 SOP 29 for the complete standard operating procedure for calculating 

the uncertainty. 
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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. Tare weights used in determining the 

mass of unknown weights are treated as additional standards in the measurement 

process. 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 %).

Check standards are not used to assign values to unknown weights; however, the 

uncertainty of calibration of the check standard may be used to assess the control 

status. 

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

double substitution measurements. Where the standard deviation is less than the 

balance resolution, the laboratory may round up to the value of the balance division 

or use one of the equations in 5.2.1 to determine the smallest standard deviation to 

be included in the uncertainty calculations. (See SOP No. 9.)

Where the standard deviation of the measurement process from the control 

chart is less than the resolution of the balance being used, 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 more 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. (11) 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. As the most conservative approach, the 

laboratory may also round up to the value of the balance division as 

noted earlier.

0.6

3 p

d s d = ≈ Eqn. (11)
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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 control chart sp

and the result from Eqn. (12) may 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 0.03 mg. If the laboratory 

calculated standard deviation is 0.075 mg, then 0.075 mg is used. 

Use of Eqn. (12) is appropriate as the degrees of freedom approaches 

30; the check standard calculated standard deviation is less than is 

calculated using Eqn. (11).

0.3

2 3 p

d s d = ≈ Eqn. (12)

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

following options in priority preference for calculating and uncertainty associated 

with air buoyancy. 

Option 1, preferred. Use the formulae provided in OIML R111 (2004), 

C.6.3-1, C.6.3-2, and C.6.3-3.

Option 2. Calculate the uncertainty by quantify 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.7

Option 3. If buoyancy corrections are not performed, include an uncorrected 

systematic standard uncertainty due to the buoyancy effect using the 

equations for the magnitude of the air buoyancy correction per SOP 2 and 

use a rectangular distribution. This approach is not recommended for 

precision calibrations; however, if the resulting value from this approach is 

evaluated and does not significantly affect the expanded uncertainty, it may 

be adequate. 

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 

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

2 and 29 for additional details.

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

the following table.

Table 7. Example uncertainty budget table.

Uncertainty Component 

Description Symbol Source Typical Distribution

Uncertainty of the standard 

mass(es) (5.1) us Calibration certificate Normal divided by coverage factor

Accepted standard deviation of 

the process (5.2) sp

Control chart, standard 

deviation chart

OR estimates when sp is smaller 

than balance division

Normal

Uncertainty of the air 

buoyancy correction (5.3) ub OIML R111 Rectangular

Air temperature (for air 

density) ut SOP 2 or R111 Rectangular

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

Air relative humidity (for air 

density) uRH SOP 2 or R111 Rectangular

Air density (formula) uρa SOP 2 or 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 See SOP 29

Draft a suitable uncertainty statement for the certificate. For example: 

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 % confidence interval. Factors not considered in the evaluation: 

magnetism (weights not evaluated for 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.
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6 Certificate

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

Report the mass (if requested by the customer), conventional mass, environmental 

conditions during the calibrations, mass density used (reported, measured, or 

assumed), 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 E617 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.
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Appendix A

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. certificate

Unc:

k factor

Density

g/cm3

Unc 

Density

(k = 1)

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 (See Section 4.5):

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
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Appendix B

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. certificate

Unc:

k factor

Density

g/cm3

Unc 

Density

(k=1)

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 (See Section 4.5):

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