Measurement uncertainty calculator: GUM budget with Type A, Type B and expanded U

Updated: September 30, 2026 · ISKRIT

Measurement uncertainty is the parameter that describes the spread of values that can reasonably be attributed to the measured quantity. Under the GUM (ISO/IEC Guide 98-3) it is estimated in four steps: standard uncertainties from Type A and Type B evaluations, their root-sum-square combination into uc, and multiplication by a coverage factor k, usually 2, to give the expanded uncertainty U at about 95 % coverage.

Calculator

Type A and Type B

Type A is evaluated statistically from repeated readings: the standard uncertainty of the mean is uA = s/√n with n - 1 degrees of freedom.

Type B covers everything else: the instrument specification, the calibration certificate of the reference, the resolution, handbook data. A limit is turned into a standard uncertainty through the assumed distribution:

Combined and expanded uncertainty

Combined. For uncorrelated contributions with unit sensitivity coefficients, uc = √(uA² + uB1² + uB2² + ...). The largest contribution usually dominates, so that is the one worth reducing.

Expanded. U = k · uc. With k = 2 the interval result ± U covers about 95 % of the values. When Type A dominates and there are few readings, k is taken from the Student distribution at the effective degrees of freedom.

Effective degrees of freedom. The Welch-Satterthwaite formula gives νeff = uc⁴ / Σ(ui⁴ / νi). Type B contributions are usually taken with infinite degrees of freedom, which leaves νeff = uc⁴ / (uA⁴ / (n - 1)).

Worked example

Five readings: 10.2, 10.5, 10.1, 10.4, 10.3. The mean is 10.30, s = 0.158 and uA = 0.158 / √5 = 0.0707. An instrument with a maximum permissible error of ±0.2 gives 0.2 / √3 = 0.1155, a reference with U = 0.1 at k = 2 gives 0.05, and a resolution of 0.1 gives 0.1 / (2√3) = 0.0289.

uc = √(0.0707² + 0.1155² + 0.05² + 0.0289²) = 0.147. The effective degrees of freedom are 0.147⁴ / (0.0707⁴ / 4) = 75, so the Student factor is 1.99, practically 2. U = 2 · 0.147 = 0.29, and the result is 10.30 ± 0.29 at k = 2.

Questions

What is the difference between error and uncertainty? Error is the difference from the true value, which is unknown. Uncertainty describes the spread of values that can be attributed to the quantity on the available evidence, and it can be evaluated from readings and documents.

How is a maximum permissible error converted to a standard uncertainty? With no other information the distribution is taken as rectangular and the standard uncertainty is a / √3.

When can k = 2 be used? When the effective degrees of freedom are large, in practice above about 30. With few readings and a dominant Type A contribution the Student factor is used instead and is larger than 2.

Is an uncertainty budget required for accreditation? Laboratories accredited to ISO/IEC 17025 have to evaluate measurement uncertainty for their results, and the budget is the usual way to show it.

Repeatability limits, control charts and precision checks on your own series are computed by the laboratory QC calculator.