Measurement uncertainty explained, from Type A and B to the coverage factor k

Measurement uncertainty is a non-negative number that describes how widely the values you could reasonably attribute to a measured quantity are spread, based on the information you have (VIM 2.26). In practice it is the “± something” after a result: 10.000 025 kg ± 58 mg (k = 2) says that, with a coverage probability of about 95 %, the mass lies within 58 mg of the stated value. A result without its uncertainty is incomplete (GUM 3.1.2; EA-4/02 §2.1). Uncertainty is not the same as error, accuracy, tolerance or resolution, and k = 2 does not always mean 95 %. This guide explains each idea with one real calibration, from the list of contributions to the pass/fail decision.

What measurement uncertainty is (and what it isn’t)

The international vocabulary defines uncertainty as a parameter that characterizes the dispersion of the values attributed to the measurand (the quantity you intend to measure), and it includes components from systematic effects, such as the uncertainty of corrections and of the reference standard’s value (VIM 2.3, 2.26 and Note 1). The parameter can be a standard deviation, a multiple of one, or the half-width of an interval with a stated coverage probability (VIM 2.26 Note 2). The newest introduction to the GUM describes it more simply, as the doubt you are still left with about the true value once the measurement is done (GUM-1:2023, §3.4).

Most confusion comes from five neighboring terms. They answer different questions:

The distinction that matters most is error versus uncertainty. If a calibration shows an error, you know about it and can correct it. Uncertainty is what is left after correction: even a perfectly corrected result remains an estimate, because the random effects and the correction itself are not known exactly (GUM 3.3.1). The GUM also warns against the reverse habit: don’t enlarge the uncertainty to avoid applying a known correction (GUM 6.3.1 Note). UKAS adds that safety or comfort factors don’t belong in a budget either (UKAS M3003, §3.48).

Where uncertainty comes from

The GUM lists the usual sources (GUM 3.3.2; EA-4/02 Appendix C):

• An incomplete definition of the measurand, or an imperfect realization of it.
• A sample that does not represent the measurand.
• Environmental conditions that are poorly known or poorly controlled (temperature, humidity, air buoyancy).
• Personal bias in reading analog instruments.
• The finite resolution of the instrument.
• Inexact values of reference standards and reference materials, and of constants taken from external sources.
• Approximations and assumptions in the measurement method.
• Variation in repeated observations made under apparently identical conditions.

Two cautions. These sources are not always independent: some of them already show up as scatter in your repeated readings, so don’t count them twice (EA-4/02 Appendix C2; GUM 4.3.10). And mistakes are not uncertainty. A misread scale or a wrong unit is a blunder to be found and removed, not a contribution to the budget (GUM 3.4.7; NPL GPG 11, §5.3). On the other side, leaving out a significant source is not harmless either: the stated uncertainty becomes too small, which can undermine traceability and lead to wrong conformity decisions (GUM-1:2023, §4.7).

Type A and Type B evaluation

Every contribution has to become a standard uncertainty u: a value at the level of one standard deviation. There are two ways to get it (GUM 2.3.2–2.3.3; VIM 2.28–2.29):

• Type A: by statistical analysis of repeated observations. You take n readings, compute their experimental standard deviation s, and, if you report the mean, the standard uncertainty of that mean is s/√n (GUM 4.2.2–4.2.3; EA-4/02 §3.2.2). With fewer than about 10 readings the estimate of s is itself shaky; a pooled standard deviation from earlier work on the same process is often better (EA-4/02 §3.2.2).
• Type B: by any other means. Calibration certificates, manufacturer specifications, handbook data, earlier measurements and experience all count (EA-4/02 §3.3.1–3.3.2). A well-founded Type B evaluation can be just as dependable as a Type A one, especially when the Type A rests on few readings (EA-4/02 §3.3.2).

The labels describe how you evaluated a contribution, not what kind of effect it is. Both types are expressed as standard deviations and are combined the same way (GUM 3.3.4). It is a common mistake to read A as “random” and B as “systematic”: the GUM says the A/B categories are not substitutes for those words, and NIST warns that there is no simple correspondence between the two classifications (GUM 3.3.3; NIST TN 1297, §2.3).

For Type B, you usually start from limits or from an expanded uncertainty, and you convert them with a divisor that depends on what you know about the distribution:

The uncertainty budget: a worked example

An uncertainty budget is the statement of an uncertainty, its components and how they were calculated and combined (VIM 2.33). The usual format is a table with, for each input quantity, its estimate, standard uncertainty, distribution, sensitivity coefficient and contribution, with the combined result at the bottom (EA-4/02 §4.8–4.9).

Here is a complete worked example published by European Accreditation in its guide for calibration laboratories. A 10 kg weight of OIML class M1 is calibrated against a class F2 reference weight of the same nominal value on a mass comparator, using three ABBA substitution cycles (EA-4/02, S2.1–S2.8). The model is simple: the mass of the weight equals the mass of the standard, plus the drift of the standard since its calibration, plus the observed difference, plus corrections for the comparator and for air buoyancy. Every sensitivity coefficient is 1.

What the example shows:

1. Each row is converted first. The certificate’s 45 mg at k = 2 becomes 22.5 mg; the ±15 mg drift limit and the ±10 mg comparator and buoyancy limits become a/√3; the pooled 25 mg standard deviation becomes 25/√3 = 14.4 mg because the result is the mean of three cycles (EA-4/02, S2.3–S2.8).
2. Contributions combine in quadrature. For independent inputs, the combined standard uncertainty is the square root of the sum of the squared contributions, each multiplied by its sensitivity coefficient: the law of propagation of uncertainty (GUM 5.1.2, eq. 10; EA-4/02 §4.1). Here that gives 29.2 mg. Adding the five numbers would give 57.1 mg, almost twice as much. Correlated inputs need extra covariance terms (GUM 5.2.2; EA-4/02 §4.6); EA states that none are significant here (S2.7).
3. One contribution dominates. The reference standard alone is about 59 % of the variance and repeatability about 24 %. If you wanted a smaller uncertainty, a better-calibrated standard is the lever; tightening the ±10 mg buoyancy estimate would barely change anything.
4. The standard’s certificate is only one row. The 45 mg on the reference weight’s certificate is not the uncertainty of this calibration: four other contributions add to it.

Expanded uncertainty and the coverage factor k

A standard uncertainty covers a narrow interval, so results are normally reported with an expanded uncertainty U = k · u_c, written as y ± U (GUM 6.2.1; VIM 2.35). The coverage factor k is a number larger than 1, typically between 2 and 3 (VIM 2.38; GUM 6.3.1).

k = 2 means about 95 % only under conditions. For an exactly normal distribution, ±2u covers 95.45 %, and exactly 95 % corresponds to k = 1.96 (UKAS M3003, §3.44 and §3.46). k = 2 gives approximately 95 % when the distribution of the result is approximately normal and the effective degrees of freedom are large enough (GUM 6.3.3 and G.6.6). EA-4/02 uses k = 2 when normality can be assumed and the standard uncertainty is reliable enough; otherwise k must come from the effective degrees of freedom (for example, a Type A contribution with few readings that dominates the budget) or from the actual distribution (EA-4/02 §5.1–5.6). Normality is a fair assumption when three or more independent contributions of similar size, with well-behaved distributions, combine (EA-4/02 §5.2).

The official counterexample is a handheld multimeter calibrated at 100 V with a resolution of 0.1 V. The resolution contribution (0.029 V of a combined 0.030 V) dominates the budget, so the result is essentially rectangular and EA uses k = 1.65 for 95 %, reporting an error of indication of (0.10 ± 0.05) V (EA-4/02, S9.10–S9.14). The reverse case is a budget dominated by a Type A evaluation with few readings: the GUM’s own reporting example uses k = 2.26 for 95 % with 9 degrees of freedom (GUM 7.2.4). NIST, for its part, uses k = 2 by convention and departs from it only for documented reasons (NIST TN 1297, §6.5).

What the interval does and doesn’t promise. The VIM defines a coverage interval as one that contains the true value of the measurand with a stated probability, based on the information available (VIM 2.36). The probability is about what you know, not a guarantee. If a significant source was left out of the budget, the interval can simply miss (GUM-1:2023, §4.7). That is why reviewers look at the budget, not only at the final U.

How to calculate measurement uncertainty: the steps in brief

The GUM summarizes the method in eight steps (GUM clause 8), and EA-4/02 adapts them to calibration (EA-4/02 §6.1). Combining both, in short:

1. 1

   Define the measurand and write the model

   Y = f(X₁, X₂, …), including every correction you apply (temperature, drift, buoyancy).

2. 2

   List the input quantities and their sources

   Reference standard, instrument, method, environment, operator, item. Avoid counting a source twice.

3. 3

   Estimate each input

   Its best value: a reading, a certificate value, or zero for a correction you do not apply.

4. 4

   Evaluate each standard uncertainty

   Type A from repeated readings (s/√n), or Type B from certificates, specs and resolution with the right divisor.

5. 5

   Find the sensitivity coefficients

   How much the result changes per unit change in each input (∂f/∂xᵢ). For a simple sum they are all 1.

6. 6

   Combine

   Root sum of squares of the contributions, plus covariance terms if inputs are correlated.

7. 7

   Expand

   Choose k for about 95 % coverage: k = 2 if the result is close to normal and well known; otherwise use the effective degrees of freedom or the actual distribution.

8. 8

   Report

   y ± U with units, k and the coverage probability, and keep the budget that supports it.

A spreadsheet is often enough; complex cases may need specialist software or a statistician (GUM-1:2023, §4.6). The GUM offers two ways to propagate uncertainty, the law of propagation and a Monte Carlo propagation of the full distributions, and the choice depends on the model and the inputs (GUM-1:2023, §4.10). When the model is strongly nonlinear or the output distribution is far from normal, the law of propagation may no longer be valid, and Monte Carlo is the alternative (JCGM 101, §5.8). The 2026 amendment to the GUM adds notes on exactly this point: when the nonlinearity of the model is significant, use Monte Carlo or higher-order terms (JCGM 100:2008/Amd.1:2026).

To build your own budget row by row, use our measurement uncertainty calculator.

Reading the uncertainty on a calibration certificate

On an accredited calibration certificate, uncertainty follows a fixed policy. Global ACI-TECH-1-009, which replaced ILAC P14 on June 26, 2026 with the same content and numbering on these points, requires that (Global ACI-TECH-1-009, §5.2–5.6):

• the result is reported as y ± U with units, and the certificate states k and the coverage probability, with a note that the coverage is approximately 95 %;
• U has at most two significant digits, with rounding done at the end;
• U includes the contributions of your instrument and the short-term effects during calibration, so it is normally larger than the lab’s CMC (its best capability), and never smaller;
• U is in the unit of the measurand or relative to it (for example, %).

ANAB, a US accreditation body, sets the same points for its labs and adds that pass/fail statements must account for uncertainty under an agreement with the customer (ANAB AR 2251, §4.2–4.7 and §4.4).

To use the certificate in your own budget, decode it: divide U by the stated k. A thermometer certificate that says U = 0.06 °C at k = 2 gives your budget a standard uncertainty of 0.03 °C for that reference; if the certificate gives U at 99 % without k, divide by 2.58 (GUM 4.3.3–4.3.4). Then add the rest of the rows for the way you use the instrument. Our guide to reading a calibration certificate covers the other checks.

From uncertainty to pass/fail: TUR and decision rules

Uncertainty earns its keep when you compare a result with a limit. Global ACI’s decision-rule guidance defines the test uncertainty ratio as the tolerance divided by the 95 % expanded uncertainty (Global ACI-TECH-1-002, §2.13, formerly ILAC G8); for a two-sided tolerance that is the tolerance span divided by 2U (A2LA G136). A2LA describes 4:1 or better as the generally accepted industry standard; it is not an ISO/IEC 17025 requirement (A2LA G136).

When a result sits close to a limit, its uncertainty band crosses the limit and the pass/fail call carries risk. That is why ISO/IEC 17025 requires a documented decision rule stating how uncertainty is taken into account whenever a lab issues a statement of conformity; when the customer asks for one, the rule must be agreed with the customer unless the specification already contains it (clauses 3.7, 7.1.3 and 7.8.6, as quoted in Global ACI-TECH-1-002, §3). JCGM 106 gives the theory: the probability of false accept and false reject depends on the uncertainty and on where the result lies (JCGM 106; EA-4/02 Appendix F).

A useful idea for anyone who sets requirements is target uncertainty: the largest uncertainty you can accept, decided from the intended use of the result (VIM 2.34). Write it next to the tolerance in your equipment records, and the question “is 20 % uncertainty bad?” answers itself: compare it with the target, not with a rule of thumb.

Go deeper with decision rules and guard banding, and check your own ratio with the TUR calculator.

What ISO/IEC 17025 and accreditation bodies require

For paid standards we describe the requirement in our own words, from the free documents cited.

The method all of them point to is the GUM, which is now a family of documents rather than one book (GUM-1:2023, §5). JCGM 100:2008 remains the core text, amended in 2026 for nonlinear models. Around it sit GUM-1:2023 (introduction, which replaced JCGM 104), GUM-6:2020 (developing measurement models), GUM-5:2026 (worked examples), Supplement 1 on Monte Carlo (JCGM 101) and JCGM 106 on conformity assessment. All are free from the BIPM.

Accreditation documents have also moved. Since 2026, Global ACI has taken over the work of ILAC and IAF; ILAC P14 and G8 were replaced on June 26, 2026 by Global ACI-TECH-1-009 and TECH-1-002. ILAC P10 (traceability) stays in force until its Global ACI equivalent is adopted, and the replacement for G24 (calibration intervals) is still in process (Global ACI cross-reference table, v14.0). Older lab procedures and ANAB AR 2251 still cite ILAC P14, which is fine: the policy text is the same.

Uncertainty in the classroom vs in the calibration lab

If you came here from a physics or chemistry class, the terms map like this:

• Absolute uncertainty is the uncertainty in the unit of the result (± 58 mg).
• Relative uncertainty is the uncertainty divided by the absolute value of the result (VIM 2.32 defines it for the standard uncertainty; certificates may also give U/|y|, Global ACI-TECH-1-009, §5.2). For the 10 kg weight: 58 mg / 10 kg = 5.8 × 10⁻⁶.
• Percent uncertainty is the relative uncertainty times 100.

The classroom rule of “half the smallest division” for an analog scale is a quick estimate of one contribution, the reading. In a calibration budget, resolution is one row among several; for a digital display it is δx/√12 (GUM F.2.2.1), and it is combined in quadrature with the others, not added.

Common myths

Measurement uncertainty checklist for quality managers

You don’t need to build budgets to manage uncertainty well. You need to ask the right questions of the people and labs who do.

In this section

• Guard banding: methods, formulas and ISO/IEC 17025 decision rules

  What guard banding is, the five common guard band methods with formulas and a worked example, how they trade false accepts for false rejects, and how to state a decision rule under ISO/IEC 17025.