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Uncertainty budget calculator
Result
U = 0.034
k = 2.04 at 95 %, ν_eff = 32.5 · 0.337 % of the measurand
unrounded 0.0336762 · combined standard uncertainty u_c = 0.0165429
| Contribution | ÷ | u_i | share |
|---|---|---|---|
| u1 Repeatability of the reading (Type A, standard deviation of the mean) · dominant | 1 | 0.012 | 52.6 % |
| u2 Reference standard, from its certificate at k = 2 | 2 | 0.01 | 36.5 % |
| u3 Resolution of the unit under test, half a digit | 1.732 | 0.002887 | 3.0 % |
| u4 Drift of the reference since calibration | 1.732 | 0.004619 | 7.8 % |
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The method
What the numbers mean.
How is each contribution turned into a standard uncertainty?
By its divisor. A value already stated as a standard deviation — a Type A repeatability term — has divisor 1. A value quoted at k = 2 on a calibration certificate is halved. A limit inside which the true value is equally likely anywhere — a resolution, a manufacturer's specification — is rectangular, divisor √3. The divisor is the whole of the difference between a number on a data sheet and a number in a budget.
How are they combined?
As the root sum of squares of the standard uncertainties, each multiplied by its sensitivity coefficient where the input is not already in the units of the measurand. The share each contributes to the combined variance is shown, because the largest share is the only one worth reducing.
Where does the coverage factor come from?
From the effective degrees of freedom by Welch–Satterthwaite, and then from the t-distribution at the chosen confidence. A budget dominated by a Type A term with few readings has few effective degrees of freedom and a k above 2; a budget dominated by well-characterised Type B terms approaches k = 1.96 at 95 %. A laboratory may instead declare a fixed k = 2 and say so; the certificate prints whichever was used.
What does ISO/IEC 17025 require?
ISO/IEC 17025 §7.6 requires a laboratory to identify the contributions and evaluate the uncertainty, and ISO/IEC 17025 §7.8.4 requires the certificate to state it with its coverage. The metrology is the GUM — JCGM 100:2008 — and this page follows it. See the JCGM publications for the guide itself.
The expanded uncertainty is reported rounded up to two significant figures, per GUM 7.2.6, because an uncertainty stated smaller than the budget evaluated is a claim the budget does not support.
About this calculator
- Is this the same calculation a certificate uses?
- Yes. The page runs the CalibraWorks calculation engine, compiled from the same source that evaluates a budget on a certificate: GUM divisors, root-sum-square combination, Welch–Satterthwaite effective degrees of freedom, a coverage factor from the t-distribution, and rounding up to two significant figures. It is not a re-implementation for the web.
- Why is the expanded uncertainty rounded up?
- GUM 7.2.6 has the reported uncertainty rounded to two significant figures, and rounding up is the conservative direction: an uncertainty stated smaller than it was evaluated to be is a claim the budget does not support. The unrounded value is shown beside it.
- What does the effective degrees of freedom do?
- It sets the coverage factor. With few repeated readings, k at 95 % is larger than 2 — for nine degrees of freedom it is about 2.26 — and using k = 2 regardless understates the uncertainty. Welch–Satterthwaite combines each contribution's degrees of freedom weighted by its share of the variance; a Type B contribution with infinite degrees of freedom drops out of the denominator.
- Which contribution should I worry about?
- The one with the largest share of the combined variance, marked as dominant. Reducing any other contribution changes the result very little. If the reference standard dominates, a better master helps; if repeatability dominates, more readings or a better procedure helps.
Related: test uncertainty ratio and guard-band calculator. For a laboratory working towards accreditation, the budget on the certificate is the same evaluation, kept with the raw inputs so it reproduces — how that works.
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