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SOP 29

Standard Operating Procedure

for the 

Assignment of Uncertainty

1 Introduction

Purpose

Laboratories performing calibrations that meet ISO/IEC 17025 must report 

uncertainties in conformance with the ISO Guide to the Expression of Uncertainty 

in Measurement (hereafter called the GUM). This SOP provides instructions for the 

laboratory to meet this requirement.

Prerequisites

Calibration certificates with valid uncertainties must be available for all 

standards.

Statistical data regarding the calibration measurement process must be 

available; preferably from measurement control programs within the 

laboratory.

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. Each 

NIST SOP published in NISTIR 6969, 5672, and 7383 includes detailed 

uncertainty budget tables that may be used.

2 Methodology

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. 

Summary

This uncertainty analysis process follows the following eight steps: 

1) Specify the measurement process; 

2) Identify and characterize uncertainty components;

3) Quantify uncertainty components in applicable measurement units;
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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 and uncertainties associated with the use 

of multiple standards are addressed as well.

3 The Process of Measurement Uncertainty Estimation

Step 1. Specify the process. 

Clearly define the measurand(s) and 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. 

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. 

Table 1. Example uncertainty budget table.

Uncertainty Component 

Description Symbol Source

Type

(A or 

B)

Typical 

Distribution

Standard uncertainty from the 

measurement process

sp Process A Normal

Standard uncertainty for the 

standards us

Calibration

Certificate B See Certificate

Standard uncertainty due to other 

factors uo

Analysis or 

References B Varies

Using the measurement equation that was identified in 3.1 provides a good starting 

point as do the detailed uncertainty budget tables provided in many procedures. All

the parameters in measurement equations may have an uncertainty associated with 

them. When there are discrete steps in the measurement process, additional 

uncertainties may be associated with each. Identify actual degrees of freedom for 

each component that is based on statistical data. 

What follows are the most common uncertainties associated with metrological 

measurements. Keep in mind that this list is not exhaustive. Each item listed below 
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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 represented by a lower case 

variable and further defined by a subscript that is arbitrarily assigned and usually 

related to the source for ease in remembering that source.

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 data and the 

current knowledge that the measurements are in a state of statistical 

control. This must be ascertained by measuring at least one check 

standard during 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. Actual degrees of freedom will be identified from the 

control chart.

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 should be made on the same day. Calculate 

the standard deviation in the conventional manner to obtain the 

standard deviation of the process, sp, keeping in mind that it does 

not fully represent the measurement process under all typically 

encountered conditions and that additional uncertainty values may 

need to be addressed more fully. Actual degrees of freedom will be 

identified from the control chart.

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 of a measurement process is less than 

the resolution of the measuring instrument, d, the standard deviation 

must be assessed. Generally, the smallest standard deviation to be 

used in this instance may be estimated as 0.6 times the instrument 

resolution from the equation noted below, with d being the 

instrument resolution. Assess the standard deviation and use the

larger of the calculated standard deviation or the value from this 

equation. (Some procedures will provide for this value to be reduced 

further through repetitive measurements (e.g., SOP 4 and SOP 5); 
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however, adequate degrees of freedom and analysis of laboratory 

measurement assurance check standard data must be performed and 

be acceptable, to reduce the standard deviation of the measurement 

process further.) Actual degrees of freedom may still be identified 

from the control chart. The laboratory may also round up to the 

nearest full division of the measuring instrument.

0.6

3 p

d s d = ≈ Eqn. (1)

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).

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. Actual degrees of freedom 

will be identified during the laboratory calibration.

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. 

Standards with independent calibrations are combined by standard 

root sum square methods.

When calibrations are performed at the same time with the same 

reference standards, 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 is 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 is also the 

case when a 25 gal standard is used four times to calibrate a 100 gal 
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standard.) There are also circumstances where both approaches are 

used to combine uncertainties from multiple combinations of 

standards. Actual degrees of freedom for each standard should be 

identified during the laboratory calibration.

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 the uncertainty budget tables in each calibration SOP. Each 

component that is considered is included as an additional standard 

uncertainty uo1, uo2, uo3, etc., and included in the RSS equation when data 

shows these factors to be significant. Documentation of the assessment of 

each component must be maintained to complete the documentation 

required by the specific calibration procedure and ISO/IEC 17025.

Additionally, the laboratory should include any other components that are 

considered significant. An estimated standard uncertainty that impacts the 

second significant digit in the combined uncertainty is considered 

significant. Additional uncertainties are addressed in the uncertainty budget 

tables provided in the SOPs. Estimates of degrees of freedom for each 

component in this category is subjective but should be estimated based on 

the technical knowledge and confidence associated with each component.

Standard uncertainty due to factors unrelated to the measurement process

per se, uu. 

These are factors that may be related to characteristics of the items being 

calibrated or of the standard and are usually minimized in well-known and 

controlled measurement processes. Review the uncertainty budget tables in 

each SOP or applicable international/national procedures or reference 

papers for more information. Estimates of degrees of freedom for each 

component in this category is subjective but should be estimated based on 

the technical knowledge and confidence associated with each component.

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 community, 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 regarding 1) identifying cause and 2) level of significance.
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Uncertainty components associated with observed bias are not required to 

be included in the uncertainty; these components should be small relative to 

other components and may be included in cases where the laboratory needs 

to increase their uncertainty to ensure enough overlap of reference values 

or proficiency tests. Estimates of degrees of freedom for this component 

should be based on associated data from the analysis of bias and confidence 

regarding its relative contribution to the uncertainty.

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 should be considered whenever available

(e.g., a recent calibration, a proficiency testing result, and control 

chart data assessed as a part of an integrated evaluation). 

If the cause of bias or error 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, and the 

laboratory wants to include these components, a test of significance 

shall be conducted to determine suitability. Alternative approaches 

for incorporating the bias into the uncertainty calculations are 

presented.

In deciding whether a known bias can reasonably be included in the 

uncertainty, the following steps may be followed:
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3.2.5.2.1 Estimate the combined uncertainty without considering 

the relevant bias.

3.2.5.2.2 Evaluate whether the bias is less than the combined 

uncertainty (i.e., 22 2 bias + + < usp o s u ).

3.2.5.2.3 If the bias is less than the specified limit, it may be 

included in the uncertainty using one of several 

approaches that must be clearly communicated in the 

certificate.

3.2.5.2.4 If the bias is larger than the specified limit, the error 

must be investigated further and corrected 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.

Criteria Limit = lab ref c xx u − < Eqn. (2)

If bias differences are smaller than the criteria limits, 

investigation and corrective action may be unrealistic. If 

the bias is less than the criteria limits shown above, and 

corrective action is not taken, the deviations, may be 

included in the uncertainty. Follow one of several 

options given in the following sections, provided that the 

resulting uncertainty remains sufficiently small to meet 

the needs of the customer or application. In all cases, the 

method used to incorporate bias must be clearly reported.

3.2.5.3 Option 1. Adding the bias to the expanded uncertainty (U + bias). In 

this case, the bias is simply added to the expanded uncertainty and 

is reported as such.

bias ( ) bias U uk + = ×+ c Eqn. (3)

3.2.5.4 Option 2. 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) the equation below may be used.

A rectangular distribution is considered where the value might 

possibly be anywhere within the range shown for each laboratory 

data point. This approach may also be used in the case where a 

standard is predictably drifting with use over time. However, 

because it is known that the limits of the potential error are nearer 

the center of the distribution and not near the extremes, a mid-range 
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value is chosen and ud (uncertainty for differences) is calculated as 

follows, where d is the bias:

bias 1 0.3 bias

2 3

u

d = ≈ Eqn. (4)

3.2.5.5 Option 3. When uncertainties for the laboratory data are considered 

secondary to a reference value (e.g., the difference between the 

laboratory data and data from a higher-level calibration with smaller 

uncertainties) the equation below may be used. 

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 bias is treated as an 

uncorrected systematic error and the following equation may be 

used:

bias 0.6 bias

3

u

d = ≈

Eqn. (5)

Step 3. Quantify uncertainty estimates.

All uncertainty estimates identified in the previous step must be quantified in, or 

converted to, units that represent the final measurement results. 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 or Pareto chart. 

Step 4. Convert all factors to standard uncertainties.

The impact of the uncertainty estimates quantified in 3.3 on the final measurement 

result, in the correct units for the measurement result, can be calculated using the 

measurement equation identified in 3.1. One of several methods can be employed,

e.g., partial derivatives, baseline, or Kragten methodology that use the 

measurement equation to convert the uncertainty estimates into the units of the 

measurement result.

In 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 “one-standard-deviation” values or a k = 1 coverage level. 

The appropriate distribution factor must be used when converting estimated 

uncertainty values to standard uncertainties. Each SOP provides specific guidance 
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if the component is not treated as a rectangular distribution (e.g., a meniscus may 

be treated as a triangular distribution and some of the mass procedures will use onehalf of the value treated as a rectangular distribution). According to the Guide to 

the Expression of Uncertainty in Measurement, a rectangular distribution is 

generally used when detailed information about the distribution is unknown. 

value

3 n u = Eqn. (6)

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:

2222 2

c ps oud u = s u u u u ++++ Eqn. (7)

Table 2. Symbol descriptions.

Symbol Description

U Expanded uncertainty (this is represented with an upper-case U)

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

Step 6. Calculate the expanded uncertainty.

The combined standard uncertainty is multiplied by a coverage factor, k, based on 

the degrees of freedom, or the effective 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:

Uuk = ×c Eqn. (8)

where k = a value from the table in Appendix A, taken from the column 95.45, or 

calculated based on statistical functions to give comparable k values. When there 

are a small number of degrees of freedom, the coverage factor must be determined 

by calculating the coverage factor or be taken from a statistical table such as 
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provided in the Guide to the Expression of Uncertainty in Measurement (See 

Appendix A). 

A situation where the degrees of freedom associated with the value for a standard 

in the traceability chain requires special care to ensure that an appropriate coverage 

factor is used for calculating the uncertainty of subsequent reported values is when

there are a 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. In this case, the effective degrees of freedom

must be used to determine an appropriate coverage factor. Use the WelchSatterthwaite equation provided in NISTIR 6969, Section 8 or provided in the 

Guide to the Expression of Uncertainty in Measurement. 

Step 7. Evaluate the expanded uncertainty.

The expanded uncertainty must be evaluated. The first level of evaluation should 

be to determine if the calculated results make sense and are reasonable, to ensure 

that obvious blunders have not been made in the calculations. Evaluation may be 

against established criteria such as documentary standard or SOP decision limits, 

tolerance limits, customer requirements, and/or calibration and measurement 

capabilities listed on the laboratory scope. Laboratories that are accredited 

generally may not report uncertainties smaller than what has been published on 

their scope by their accreditation bodies. 

For conformity assessments, the uncertainty and risk decisions must be documented 

if not contained in the documentary standard and need to be communicated to the 

customer. For example, the specifications for mass standards clearly state that the 

expanded uncertainty must be less than 1/3 of the maximum permissible errors 

(tolerances).

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:

“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. 
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Appendix A

Value of tp(ν) from the t-distribution for degrees of freedom ν that defines an interval tp(ν) to +tp(ν) that 

encompasses the fraction p of the distribution. Note: This table is taken from NIST Technical Note 1297

and matches the comparable table in the Guide to the Expression of Uncertainty in Measurement. The 

column marked 95.45(a) should be used to give an “approximate 95 % confidence interval.”

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.
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