When we deliver a measurement — a value read from an instrument, a quantity computed by an acquisition chain — the first question we ask isn’t “what is it”, but “what is it ± how much”. A number on its own, without its uncertainty, can’t tell you whether two measurements agree, whether a part is within tolerance, whether a result confirms or refutes a hypothesis. That’s why we say it plainly: a measurement without uncertainty is not a measurement, it’s a numerical opinion.
The uncertainty budget
Uncertainty isn’t a single number falling from the sky: it’s built, line by line, in an uncertainty budget. You list every source contributing to the spread of the result — reading noise, display resolution, calibration drift, temperature effect, the uncertainty of the reference standard — and assign each a contribution. The methodological reference framework is the GUM guide (JCGM 100:2008, Guide to the Expression of Uncertainty in Measurement), which fixes the vocabulary and the rules by which these contributions are to be evaluated and combined. The budget’s value isn’t only the final number: it’s that it forces every assumption to be made explicit, and that’s where forgotten contributions surface.
Type A and type B: how a contribution is evaluated
The GUM splits sources into two categories by how the contribution is estimated, not by their physical nature.
Type A uncertainty is evaluated statistically, by repeating the measurement: you take N readings, compute the sample standard deviation, and from it the standard deviation of the mean. It’s the uncertainty you “measure”, the dispersion observed when the experiment is repeated.
Type B uncertainty is estimated from any source other than repetition: the instrument’s calibration certificate, display resolution, limits stated on the datasheet, experience. Here you don’t observe a dispersion, you infer it from what you know a priori.
Divisors: from half-tolerance to standard uncertainty
Type B sources almost always arrive as an interval — “± half a resolution count”, “± the stated 0.1%”, “between these two limits” — not as a standard deviation. To combine them with the others they must be converted into standard uncertainty, and the conversion depends on the probability distribution we assume inside that interval.
- Rectangular distribution — every value in the interval is equally likely. It’s the prudent default assumption for resolution, quantisation, datasheet limits with no further information. The half-width
ais divided by √3. - Triangular distribution — central values are more likely than those at the edges. Used when there’s reason to believe the extremes are improbable. The half-width is divided by √6.
Getting the divisor wrong — for instance treating a resolution as if it were a standard deviation, forgetting the /√3 — is one of the most common errors, and silently inflates or deflates the budget.
Combine, then expand
Once each contribution is expressed as a standard uncertainty u_i, and assuming the sources are independent, they combine in quadrature: the combined standard uncertainty is the square root of the sum of squares, u_c = √(Σ u_i²). The sum in quadrature, not arithmetic, reflects the fact that it’s unlikely all contributions err in the same direction at once; consequently the smaller contributions weigh little — below a third of the dominant one they become nearly irrelevant, which tells you where to invest to reduce uncertainty.
The last step is to communicate the result with a stated confidence level. The u_c corresponds to roughly 68% (one standard deviation); to reach the customary ~95% you compute the expanded uncertainty by multiplying by a coverage factor k: U = k · u_c, with k ≈ 2 for ~95%. The result is then written as value ± U (k=2) — and this is the form that makes a measurement comparable with another, with a tolerance limit, with a reference value.
What makes a pass/fail defensible
In calibration and instrumentation the uncertainty budget isn’t certificate bureaucracy: it’s what makes a “pass/fail” defensible. Building it forces you to know what enters the measurement chain — resolution, drift, reference standard, environment — and to quantify it instead of hoping. To assemble a concrete budget, with type A/B, divisors, quadrature and coverage factor, we put a measurement uncertainty (GUM) calculator online; the method and the formulas are in the dedicated wiki.
Need to state the uncertainty of a calibration or a measurement chain and want a defensible budget? Let’s talk.