Measurement uncertainty
Measurement models, Type A/B evaluation, combined and expanded uncertainty, worked budgets and conformity decisions.
Tags: inspection, metrology, uncertainty, conformity
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What uncertainty describes
Measurement uncertainty characterizes the dispersion of values attributed to a measurand. Include relevant systematic effects, reference standards and repeated-observation variation. JCGM VIM3, 2.26
The length example applies a −0.002 mm correction to a 25.012 mm mean. Four distinct, uncorrelated contributions yield combined standard uncertainty 0.004010 mm. With justified k = 2, report (25.010 ± 0.008) mm after rounding. Include a separate rounding term only if repeatability does not already account for it.
Error is the difference from the relevant reference value. Correcting a systematic effect leaves uncertainty in the correction. NPL discusses this in Good Practice Guide 11, Issue 2 with March 2001 amendments. NPL, uncertainty guide
Build the uncertainty budget for the production measurement and its conditions. A certificate supplies only its own contribution.
Start with the measurement model
State the measurand precisely: for example, a defined diameter at specified locations and temperature conditions, rather than an unspecified “part size.” Identify how the reported result is obtained from observations, reference values and corrections. A general model is Y = f(X₁, X₂, …), with input estimates and their associated uncertainties. Sensitivity coefficients describe how each input affects the output. NIST, combining uncertainty components
Include the effects relevant to the actual procedure: reference calibration, indication, resolution, repeated positioning, environmental correction, alignment, contact deformation, sample preparation or other influences as applicable. Do not append every conceivable effect without assessing whether it is present, significant and already represented elsewhere. Keep the model connected to physical measurement behaviour.
NPL emphasizes that uncertainty sources can arise from the instrument, measured item, operator and environment. Representative observations and traceable references support the evaluation; a spreadsheet cannot compensate for an undefined measurand or inappropriate measurement method. NPL, sources and evaluation
Type A and Type B are evaluation methods
Type A uses statistical analysis of observations. For independent repeated observations used to estimate a mean, a common standard-uncertainty contribution is s/√n, where s is sample standard deviation and n the number of observations. That expression applies to the mean under the stated model, not automatically to a single future observation. NIST also identifies regression and analysis of variance as Type A methods. NIST, Type A evaluation
Type B uses other information, such as calibration certificates, specifications, prior data and knowledge of instrument behaviour. Convert inputs to standard uncertainties, expressed as standard deviations, before combining them. For a certificate giving expanded uncertainty U with stated coverage factor k, the corresponding standard uncertainty is U/k. For an effect bounded by ±a with a justified rectangular distribution, it is a/√3. NIST, Type B evaluation
These labels do not mean “random versus systematic” or “measured versus guessed.” Both require judgement about relevance, distributions and dependencies. State the source, bounds, conversion and assumptions for each entry rather than assigning a divisor merely because it gives a convenient result.
Combine contributions and select coverage
For uncorrelated inputs in an appropriate first-order model:
uᶜ² = Σ[cᵢ² uᵢ²],
where cᵢ is a sensitivity coefficient and uᵢ an input standard uncertainty. Correlated inputs require covariance terms. Shared standards or corrections can create dependencies, so root-sum-of-squares is not universally valid. NIST describes this as a first-order propagation method; check its suitability for the model and distributions involved. NIST, propagation and covariance
Expanded uncertainty U = k × uᶜ. State k and the basis for any coverage-probability claim. NIST notes that k = 2 gives approximately 95% coverage when a normal distribution applies and uᶜ reliably estimates the standard deviation. It is not an unconditional 95% rule for every small sample or distribution. NIST, expanded uncertainty
Original reconciled length example
Assume a simplified additive model for a corrected length result. All contributions below are in millimetres, have sensitivity coefficient +1, are uncorrelated, and represent distinct effects. The example deliberately omits application-specific influences; it is not a ready-made budget for calipers or micrometers.
| Contribution | Assumed evidence/model | Standard uncertainty |
|---|---|---|
| Calibration correction | Certificate U = 0.004 mm, k = 2 | 0.002000 mm |
| Mean of four repetitions | s = 0.006 mm; independent observations | 0.003000 mm |
| Unresolved indication rounding | Increment 0.001 mm; rectangular ±0.0005 mm | 0.000289 mm |
| Residual environmental correction | Rectangular bounds ±0.003 mm | 0.001732 mm |
Here, assume the rounding effect is not already adequately represented in the estimated repeatability contribution. A real evaluation must justify that treatment to avoid double counting.
uᶜ = √[0.002² + 0.003² + (0.001/√12)² + (0.003/√3)²]
uᶜ = 0.004010 mm.
If k = 2 is justified, U = 0.008021 mm. Suppose the raw mean is 25.012 mm and the applied calibration correction is −0.002 mm. The corrected result is 25.010 mm. A suitably rounded statement is:
Length = (25.010 ± 0.008) mm, expanded uncertainty, k = 2, with the assumed coverage basis stated.
The correction changes the result; its uncertainty remains in the budget. Increasing repetitions reduces the assumed mean-repeatability term, but it does not divide the calibration contribution or every systematic effect by √n. Retain guard digits in intermediate calculations and round the final result consistently with its reported uncertainty.
Uncertainty and conformity are separate decisions
A measurement interval does not by itself define pass or fail. JCGM 106:2012, October 2012, discusses uncertainty in conformity assessment, including acceptance intervals, guard bands and the risks of false acceptance or rejection. Specify the applicable decision rule before interpreting a marginal result. Do not assume one rule is mandatory for every product, customer or jurisdiction. JCGM 106:2012
Original decision illustration: suppose a fictional specification has an upper limit of 25.015 mm. The example result, 25.010 mm, is below that limit. Its interval using U = 0.008 mm extends to 25.018 mm. A simple rule comparing only the reported value would accept; an agreed rule requiring y + U ≤ USL would not accept it. The second outcome does not prove the item is nonconforming—it means conformity has not been demonstrated under that rule. This illustration prescribes neither rule and does not quantify decision risk.
Reporting and maintaining the evaluation
NIST TN 1297's reporting guidance calls for the combined or expanded uncertainty, the coverage factor where used, component information and evaluation methods, and the basis of any probability interpretation. A useful record includes the measurand, model, corrected result, units, input sources, distributions, sensitivities, correlations and relevant degrees of freedom or coverage method. NIST TN 1297, section 7
Review the budget when the procedure, instrument, reference, range, specimen or operating conditions change. Reduce uncertainty by addressing dominant physical contributions rather than merely changing numerical formatting. Report any conformity statement with its specification and decision rule; the uncertainty budget supports that decision but does not replace engineering acceptance requirements.