Measurement uncertainty per the GUM

No measurement is exact: every value must be accompanied by an estimate of its uncertainty. The GUM — Guide to the Expression of Uncertainty in Measurement (JCGM 100:2008) — is the international guide that standardises how to estimate and state it. This page summarises the procedure the tool puts into practice: list the contributions, convert them to standard uncertainty, combine them and expand them with a coverage factor.

The GUM in a nutshell

The uncertainty of a measurement quantifies the doubt about the value. The GUM procedure is always the same, whatever the measurement:

1. identify the measurand and, where needed, a model y = f(x₁, …, xₙ) linking the input quantities to the result;
2. list every source of uncertainty and evaluate each as a standard uncertainty uᵢ, in the measurand’s unit;
3. combine the uᵢ into the combined standard uncertainty u_c;
4. multiply by a coverage factor k to obtain the expanded uncertainty U, which defines an interval at a stated confidence level.

The result is stated as y ± U, always declaring the k used.

Type A and type B uncertainty

The GUM distinguishes two ways of evaluating a contribution, not two different natures of uncertainty:

• Type A — statistical evaluation from a series of repeated observations. The canonical example is the standard deviation of the mean of n readings, σ/√n. Estimated from the collected data.
• Type B — evaluation from any other information available: a calibration certificate, the instrument resolution, manufacturer specifications, experience or a manual. A probability distribution is assumed and the standard uncertainty derived from it.

Key point: once both types have been converted into a standard uncertainty uᵢ, they are treated identically. The A/B distinction does not change the combination, it documents where each contribution comes from.

Distributions and divisors

For type B contributions you typically start from a half-width a (half the interval within which the value is assumed to lie) and assume a distribution shape. The standard uncertainty is the half-width divided by a divisor characteristic of the shape:

$$u=\frac{a}{\text{divisor}},\begin{array}{ll}\text{rectangular:}& \text{divisor}=\sqrt{3}\\ \text{triangular:}& \text{divisor}=\sqrt{6}\\ \text{normal:}& \text{divisor}=1(\text{or certificate }k)\end{array}$$

• Rectangular (uniform) — every value in the interval is equally likely, with no concentration around the centre. Typical of a display’s resolution, spec limits, quantisation. Divisor √3.
• Triangular — central values are more likely than the extremes (for instance from the sum of two uniform contributions). Divisor √6.
• Normal (Gaussian) — the supplied value is already a standard uncertainty, so divisor 1. If instead it comes from a calibration certificate stating an expanded uncertainty U with its k, the standard uncertainty is recovered by dividing by that k (divisor = certificate k).

Sensitivity coefficient

When an input quantity is not directly the measurand but enters it through the model y = f(x₁, …, xₙ), its contribution must be weighted by the sensitivity coefficient:

$$c_i=\frac{\partial f}{\partial x_i},u_i=\frac{a_i}{\text{divisor}}\cdot c_i$$

The coefficient cᵢ converts the uncertainty of input xᵢ into the uncertainty it produces on the measurand y (it also changes units, if different). For a quantity that is the measurand itself, c = 1.

Combination in quadrature

With independent input quantities (zero covariances), the standard uncertainties combine as the root of the sum of squares — the GUM propagation law without the correlation terms. The result is the combined standard uncertainty:

$$u_c=\sqrt{\sum _i{u}_i^2}$$

An important practical consequence: because contributions add in quadrature, the smaller terms weigh very little. How little is worth seeing, because it decides where the time goes:

The right-hand column is the instructive one, because it answers the question you actually have: if I removed this contribution entirely, how much would I gain? A source one third of the largest carries 10 % of the variance, and zeroing it — not reducing it, zeroing it — brings u_c down by 5 %. One at a tenth buys half a percentage point.

Hence the operational rule: attack the largest source, and stop when the first two contributions are comparable. Refining the third, when the first dominates it threefold, is work that does not show up in the result.

Uncertainty budget

The uncertainty budget is the table that lines up every contribution with its standard uncertainty uᵢ and its percentage weight on the variance:

$${\text{weight}}_i=\frac{u_i^2}{{\sum }_j{u}_j^2}\times 100\%$$

It is the metrologist’s diagnostic tool: it shows at a glance which sources dominate and where it pays to act to reduce the uncertainty (improve the calibration, increase the repetitions, choose a finer-resolution instrument). The web tool draws this budget as horizontal bars ordered by decreasing contribution.

A worked example

Measuring a resistance with a multimeter, three contributions:

• repeatability (type A, from 10 readings): u₁ = 0.020 Ω
• display resolution (type B, rectangular, half-width 0.005 Ω): u₂ = 0.005/√3 = 0.0029 Ω
• accuracy from the certificate (type B, normal, U = 0.030 Ω with k = 2): u₃ = 0.015 Ω

In quadrature: u_c = √(0.020² + 0.0029² + 0.015²) ≈ 0.025 Ω, and expanded U = 2·u_c ≈ 0.050 Ω (k = 2). The percentage budget: repeatability 63 %, accuracy 36 %, resolution 1 %. The immediate diagnosis is that repeatability dominates, so it pays to increase the repetitions before buying a more accurate multimeter.

Up to a point, though — and the point can be computed. Type A uncertainty goes as s/√n, so it can be reduced at will; the other two contributions cannot. Their combination in quadrature is a floor:

The floor is 0.015275 Ω, that is 61 % of the value at ten readings. Going from 10 to 100 readings gains 34 %; from 100 to 1000, another 7 %; beyond that, nothing. So the “increase the repetitions” diagnosis is right but bounded: if you need an uncertainty better than 0.015 Ω, no number of readings will do and the only route is a tighter calibration certificate. It is the same reading of the percentage budget, carried through to its consequence.

Expanded uncertainty and the coverage factor

The combined uncertainty u_c is a standard uncertainty: for a normal distribution it corresponds to about 68 % confidence. To state an interval at higher confidence you multiply by a coverage factor k, giving the expanded uncertainty:

$$U=k\cdot u_c$$

The reference value is k = 2. The two equivalences always quoted — k = 2 ≈ 95 %, k = 3 ≈ 99.7 % — hold only if the resulting distribution is well approximated by a normal one and the effective degrees of freedom are adequate: they are properties of the Gaussian, not of k. The final result is written y ± U (k = 2).

“Adequate degrees of freedom”: how many, in numbers

That “adequate” can be computed, so the vagueness can go. The budget’s effective degrees of freedom are estimated with the Welch-Satterthwaite formula:

$${\nu }_{\text{eff}}=\frac{u_c^4}{\sum _i\frac{u_i^4}{{\nu }_i}}$$

where νᵢ are each contribution’s degrees of freedom: n − 1 for a type A from n readings, and conventionally infinite for a type B coming from a specification or a certificate — so only the statistical contributions limit ν_eff. With ν_eff in hand, the correct coverage factor is Student’s t at that number of degrees of freedom, not 2.

Applied to the previous section’s example, changing only the number of readings:

The picture is clear. With 10 readings — what a lab would consider a normal sample — ν_eff is 22.6, the correct factor is 2.117 and using k = 2 understates the expanded uncertainty by 6 %: little, but not zero. With 3 readings ν_eff drops to 2.8, the correct factor is 3.48 and k = 2 understates by 74 % — an interval stated at 95 % that in fact covers much less. With two readings only the number is absurd (t = 8.7), which is mathematics’ way of saying you cannot estimate a standard deviation from two measurements.

The practical rule that follows: k = 2 is defensible from about twenty readings up, where the correction falls below a couple of percentage points. Below ten, stating U (k = 2) and calling it 95 % is a claim the numbers do not support. This tool applies whatever k you set and does not compute ν_eff: the table above is there to decide which k to set.

Getting the divisor wrong: what it costs

The most common mistake of all is not in the formula but in the choice of distribution, and it has an asymmetric price. On the same ±0.030 Ω specification of the example:

Treating a certificate that states k = 2 as k = 1 inflates u_c by 43.7 %: you declare an uncertainty half again as large as the true one and fail an acceptance test that would have passed. Reading the same specification as rectangular instead of normal k = 2 costs 5.8 %, and as triangular it lowers u_c by 6.1 % — in that case you declare less than you should, which is the worse of the two errors.

The rule that keeps you right: a calibration certificate states an expanded uncertainty with its k, and the divisor is that k. A catalogue specification limit, on the other hand, is not an uncertainty: it is an interval within which the manufacturer guarantees the value falls, and it is treated as rectangular (divisor √3) unless the manufacturer says otherwise. They are two different documents to be read in two different ways.

Limitations

• Independence is assumed between input quantities: no covariance terms. With correlated quantities the combination needs the cross terms, not covered here.
• A linear model around the operating point (first-order expansion): the sensitivity coefficients are constant, strong non-linearities are not handled.
• The coverage factor k is applied directly; effective degrees of freedom (Welch-Satterthwaite) are not computed, nor is the Monte Carlo method of GUM Supplement 1 (JCGM 101) used.
• Demonstrative tool to build and read an uncertainty budget, not a substitute for the calculation and data-management system used in an accredited laboratory. “Accredited metrology software” is not an existing qualification: accreditation applies to the laboratory and the activities it is accredited for, not to the program.

References

• JCGM 100:2008 — GUM, Guide to the Expression of Uncertainty in Measurement — the reference document for the evaluation and expression of uncertainty (cited by name/acronym; the text is not reproduced).
• Related tool — the GUM uncertainty calculator puts this page into practice: list the type A/B components, choose distributions and sensitivities and read u_c, U and the percentage budget.