# A measurement without uncertainty is not a measurement > Uncertainty budget per the GUM guide: type A vs B, /√3 and /√6 divisors, quadrature combination and expanded uncertainty U=k·u_c for a defensible calibration. Published: 2026-06-24 Category: software Tag: metrology, uncertainty, gum, jcgm, calibration, uncertainty-budget, measurement Page: --- 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 `a` is 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](/en/tools/measurement-uncertainty/) online; the method and the formulas are in the [dedicated wiki](/en/wiki/measurement-uncertainty/). Need to state the uncertainty of a calibration or a measurement chain and want a defensible budget? [Let's talk](/en/contact/).