Measurement Uncertainty Calculator
Combine Type A (statistical) and Type B (other) uncertainty contributions into a combined standard uncertainty uc and an expanded uncertainty U at a chosen coverage factor. Aligned with JCGM 100 (GUM), UKAS M3003, and ISO/IEC 17025:2017 §7.6.
When to use this calculator
This calculator is built for the everyday case a quality manager or metrologist faces: you have a measurement, an instrument with a manufacturer accuracy spec, a reference standard with a stated uncertainty on its calibration certificate, and you need to put a defensible number on the dispersion of the result. The calculator combines those contributions in root-sum-square (RSS) per the GUM, applies the appropriate divisor for each probability distribution (rectangular, triangular, U-shape, normal), and reports both the combined standard uncertainty and the expanded uncertainty at a coverage factor you control.
Use it for instrument verifications, in-house calibrations of working standards, tolerance decisions under ISO 9001 Clause 7.1.5.2, and as a working step before formalising a budget for an ISO/IEC 17025 certificate. It is also useful when you need to sense-check an upstream provider’s certificate — if your own contributions alone exceed the expanded uncertainty quoted on the certificate, the certificate’s budget is probably under-stated.
Do not use it as a substitute for an accredited calibration laboratory’s uncertainty budget. UKAS-accredited results carry a documented best measurement capability (CMC) and are subject to peer review; this tool is a working aid, not an accreditation deliverable. If your measurement is feeding a regulatory submission, a forensic process, or a customer-facing accuracy claim, anchor it to a UKAS-accredited certificate from your reference standard supplier.
The formula in one line
For a measurand y dependent on inputs xi with standard uncertainties u(xi) and sensitivity coefficients ci = ∂y/∂xi:
u_c(y) = sqrt( Σ ( c_i · u(x_i) )² ) and U = k · u_c(y)
For each contribution row you choose what you’re entering: a half-width / limit (the tool divides by the appropriate distribution divisor — √3 for rectangular, √6 for triangular, √2 for U-shape, and 1 for normal at one standard deviation), a standard uncertainty already at one standard deviation (the tool uses it as-is), or an expanded uncertainty taken from an upstream calibration certificate at k=2 (the tool divides by 2 to recover the standard uncertainty). The combined standard uncertainty is the RSS of those scaled contributions; the expanded uncertainty is the combined standard uncertainty multiplied by the chosen coverage factor (default k=2 for a 95% level of confidence per UKAS M3003).
| Contribution | Type | Input is… | Distribution | Value | ci | Conversion | ui · ci | |
|---|---|---|---|---|---|---|---|---|
| u = value (already 1σ) | — | |||||||
| u = value ÷ 2 (k=2 → k=1) | — | |||||||
| u = limit ÷ √3 | — |
Combined uc
0
Coverage k
k=2 ≈ 95% confidence (default)
Expanded U (k=2)
0
Assumptions & caveats — read before relying on the result
- Educational screening tool, not a validated budget. It combines contributions in root-sum-square (the GUM combined standard uncertainty) and applies the coverage factor you choose. It is not a validated uncertainty budget for a UKAS-accredited calibration certificate, and it supports §7.6 evaluation — not §7.8.4.1 certificate reporting.
- Uncorrelated inputs. Contributions are assumed independent (no covariance). If any inputs are correlated, root-sum-square understates the combined uncertainty — add the covariance term yourself.
- Consistent units. Enter every contribution in the same unit as the measurand, or use the sensitivity coefficient ci to convert each one. The tool does not check dimensions.
- Type A — enter the standard uncertainty of the mean. For repeatability from n measurements that is s/√n (the sample standard deviation divided by √n), not the raw sample standard deviation s.
- A “normal” half-width is treated as already 1σ (divisor 1). If your limit represents, say, a 95% bound, enter it as an “expanded U at k=2” instead so it is divided by 2.
- k is the coverage factor you choose. The tool does not compute effective degrees of freedom (Welch–Satterthwaite); with few Type-A measurements the effective degrees of freedom may justify k > 2 for ~95% coverage.
Have a competent metrology practitioner review any budget before it informs a certificate or a conformance decision.
Why does uncertainty matter?
Without an uncertainty statement, a measurement is just a number — you cannot say whether a part is in tolerance, whether two laboratories agree, or whether a drift signal is real. ISO/IEC 17025:2017 §7.6 requires every calibration result to report measurement uncertainty, and UKAS M3003 gives the practical method. Read the underlying clauses:
- Calibration certificate guide — what the uncertainty statement must look like.
- Certificate completeness checker — verify a supplier certificate reports uncertainty correctly.
References
- BIPM JCGM 100:2008 — Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM)
- UKAS M3003 — The Expression of Uncertainty and Confidence in Measurement (current edition)
- ISO/IEC 17025:2017 — General requirements for the competence of testing and calibration laboratories (§7.6 covers measurement uncertainty)
Frequently asked questions
› What is measurement uncertainty?
Measurement uncertainty is a quantitative parameter that describes the dispersion of values reasonably attributable to a measured quantity. It is not the same as error: error is the difference between a single measurement and the true value, while uncertainty characterises how confident you are in the reported value. The Guide to the Expression of Uncertainty in Measurement (JCGM 100, commonly called the GUM) is the international reference. Every calibration result issued under ISO/IEC 17025 must report measurement uncertainty alongside the measured value, because without it the result is unverifiable. In practice, uncertainty is expressed as a value with a coverage factor (typically k=2 for a 95% level of confidence) — for example, 25.000 mm ± 0.004 mm at k=2.
› What's the difference between Type A and Type B uncertainty?
Type A uncertainty is evaluated by statistical analysis of repeated observations — typically the standard deviation of a series of measurements, divided by the square root of the number of readings. Type B uncertainty is evaluated by all other means: manufacturer specifications, calibration certificates from upstream standards, instrument resolution, drift estimates, environmental influence, or expert judgement. Both types contribute equally to the combined standard uncertainty; the GUM is explicit that there is no qualitative difference between them once they have been converted into standard uncertainties (one standard deviation). Most calibration uncertainty budgets are dominated by Type B contributions, because in routine work the instrument is read once or twice rather than dozens of times.
› Why is the coverage factor k=2 used?
The coverage factor k expands the combined standard uncertainty u_c into an interval that is expected to contain a large fraction of the values that could reasonably be attributed to the measurand. For an approximately normal distribution with sufficient effective degrees of freedom, k=2 corresponds to a coverage probability of approximately 95%. UKAS M3003 (current edition published by UKAS — see the PDF link below) recommends k=2 as the default for accredited UK calibration certificates, and ISO/IEC 17025 requires the coverage factor to be stated explicitly. If the underlying distribution is not approximately normal, or if the effective degrees of freedom are small, a larger k (calculated via the Welch–Satterthwaite formula and a t-distribution) may be more appropriate.
› Do I need to estimate uncertainty for every measurement?
Under ISO/IEC 17025 §7.6, calibration laboratories must estimate measurement uncertainty for every calibration result they report. For ISO 9001-only quality systems (Clause 7.1.5.2), uncertainty must be considered when monitoring and measurement results are used to demonstrate conformity to specified requirements — but a full GUM-style budget is not always required. As a working rule: if the measurement is being used in a tolerance decision (pass/fail against a spec), you need an uncertainty estimate to apply a guard band or decision rule. If the measurement is purely informational (trend monitoring, indicative checks), an approximate estimate is usually sufficient. Document your decision rule either way — auditors check the reasoning, not just the number.
› How does CalProof help with uncertainty for calibration certificates?
Each calibration record in CalProof has an optional uncertainty field where you record the uncertainty stated on the certificate (for example the expanded uncertainty and coverage factor). CalProof keeps that figure with the calibration and its certificate PDF and audit trail — it stores what you enter, it does not compute or propagate an uncertainty budget. Use this calculator to work out the budget in line with JCGM 100 / UKAS M3003, then record the result against the calibration. Your validated budget and accredited procedures stay in your own quality system.
How CalProof helps with uncertainty
CalProof records the measurement uncertainty figure you enter from each calibration certificate against the calibration record, alongside the certificate PDF and the audit log. Pro-tier ISO 9001 + ISO/IEC 17025 audit reports include the recorded uncertainty values.
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Related tools and guides
Free tool — no signup, no data leaves your browser. Outputs are guidance only; your accredited calibration provider remains responsible for the certified uncertainty budget on a UKAS-issued certificate.