1. Define the measurand and measurement model
The measurand is a defined segment's length under specified reference conditions. Our simplified model adds calibration and resolution corrections to the mean indication. Symmetric rounding gives a zero estimate for the resolution correction, but its uncertainty remains.
This is a teaching model. An actual measurement may require temperature, thermal expansion, contact force, geometry and alignment terms. Include significant influences or justify their exclusion. Sensitivity coefficients follow from the model; assuming that every coefficient is 1 would be incorrect for many other measurement problems.
2. Collect information and convert it into u
The mean indication has Type A standard uncertainty 0.5 µm, evaluated as s/√n for stable, independent observations. The calibration correction has standard uncertainty 0.3 µm, assumed correctly recovered from the certificate's U, k and stated coverage information.
The third contribution, 0.2 µm, is the standard uncertainty assigned to resolution under a justified rounding model. It is not the display increment itself: a uniform error within ±q/2 gives q/√12. For this example, we assume the resolution contribution has not already been included in the repeatability evaluation.
3. Check units, sensitivities and dependencies
Each row identifies the input, estimate, evidence, distribution, standard uncertainty, unit and sensitivity coefficient. Its output contribution is cᵢuᵢ. Here all inputs are directly added lengths: coefficients are dimensionless and equal to 1, with contributions in µm.
Independence is an explicit assumption, not a consequence of separate budget rows. A shared standard, common temperature or correction calculation can introduce correlations requiring terms 2cᵢcⱼcov(xᵢ,xⱼ). In other models, such as differences between quantities, correlation can increase or decrease the uncertainty. The signs of sensitivity coefficients must therefore be retained.
4. Calculate combined standard uncertainty
Adding squared standard uncertainties gives combined variance 0.38 µm² and u꜀ ≈ 0.61644 µm after taking the square root. Note the squared variance units. Simply adding 0.5 + 0.3 + 0.2 would be incorrect under these assumptions.
The 0.5 µm contribution accounts for about 65.8% of the variance, making repeatability or the sampling plan a useful improvement target. More independent observations can reduce uncertainty of the mean, but not a shared calibration uncertainty. Prioritize improvements using squared contributions and physical causes rather than budget labels alone.
5. Calculate U and justify the coverage factor
Using k = 2 for this example gives U ≈ 1.23288 µm. This multiplication does not establish 95% coverage. The three standard uncertainty values alone are insufficient to choose k: distribution shapes, reliability of uncertainty estimates, degrees of freedom and adequacy of the model's linear propagation also matter.
If the output is adequately approximated by a normal distribution and standard uncertainty is reliably estimated, k = 2 usually provides approximately 95% coverage. With limited Type A degrees of freedom, the Welch–Satterthwaite formula and Student t distribution may be appropriate under suitable assumptions. A different k requires recalculating U.
6. Verify and report the final result
Retain intermediate precision and round at the end. For example, report U as 1.2 µm with two significant digits and align the length result's decimal place. State the unit, k and justified coverage probability only after verifying the assumptions.
Finally check model completeness, corrections, duplicate contributions and units. Record conditions and evidence identifiers so the budget can be reproduced. Uncertainty is neither measurement error nor tolerance. Deciding whether the length conforms to a specification requires a separate decision rule addressing the result, requirements and decision risk.
Remember this
Three independent contributions with c = 1 give u꜀ = √0.38 µm and U ≈ 1.23 µm for k = 2. The arithmetic is straightforward; validity depends on the model, evidence and coverage justification.
Sources and further reading
This guide explains fundamental GUM principles. Always adapt your model to the method and the information available about the measurement.
Practise with your own measurement.
Understanding comes with practice. Training or coaching connects theory with your own method and data.