Measurement Uncertainty
Build a measurement uncertainty budget per the GUM: list the contributions (type A or B), pick a distribution and a sensitivity coefficient for each, and read the combined uncertainty u_c, the expanded one U = k·u_c and the percentage weight of every contribution. Demonstrative tool.
GUM uncertainty calculator tool
1 · Uncertainty components
| Component | Value (a / σ) | Type | Distribution | Sensitivity c | Divisor | u_i | % |
|---|
For bounded distributions the value is the half-width a (half the interval); for the normal it is already a standard uncertainty σ, unless the divisor is a certificate k (e.g. 2). Input quantities are assumed independent (no correlation).
2 · Result
Uncertainty budget
How it works
What is the GUM?
Every measurement carries an uncertainty: no value is exact. The GUM (JCGM 100:2008) is the international guide that standardises how to estimate and state it. The idea is to list every source of uncertainty, convert each into a standard uncertainty u_i in the measurand’s unit, combine them, and finally multiply by a coverage factor to get an interval with a stated confidence level.
Every measurement carries an uncertainty: no value is exact. The GUM (JCGM 100:2008) is the international guide that standardises how to estimate and state it. The idea is to list every source of uncertainty, convert each into a standard uncertainty u_i in the measurand's unit, combine them, and finally multiply by a coverage factor to get an interval with a stated confidence level.
What is the difference between type A and type B?
The GUM distinguishes two ways of evaluating a contribution, not two different natures. Type A is statistical evaluation from a series of repeated observations; type B derives from any other information (calibration certificate, resolution, specs, experience), assuming a distribution. Once converted into u_i the two types are treated identically: the distinction only documents where the contribution comes from.
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 standard deviation of the mean, for instance). Estimated from data.
- Type B — evaluation from any other information: a calibration certificate, instrument resolution, manufacturer specs, experience. A distribution is assumed and the standard uncertainty derived from it.
Once converted into u_i, the two types are treated identically: the A/B distinction does not change the combination, it only documents where the contribution comes from.
Distributions and divisors
For type B contributions you start from an interval or a half-width a and assume a probability distribution. The standard uncertainty is the value divided by a divisor characteristic of the shape:
- Rectangular (uniform) — every value in the interval is equally likely: divisor √3. Typical of resolution and spec limits.
- Triangular — central values are more likely than the extremes: divisor √6. When there is a concentration around the centre of the interval.
- Normal (Gaussian) — the value is already a standard uncertainty (divisor 1); if instead it comes from a certificate with a stated k, the divisor is that k (editable here via the normal source + certificate k).
Sensitivity coefficient
When an input quantity is not the measurand but enters it through a model y = f(x₁, …, xₙ), its contribution is weighted by the sensitivity coefficient cᵢ = ∂f/∂xᵢ, which converts the input uncertainty into uncertainty on the measurand. For a quantity that is the measurand itself, c = 1.
How are uncertainties combined?
With independent input quantities, the standard uncertainties combine as the root of the sum of squares (GUM propagation law, without the covariance terms): u_c = √(Σ u_i²). Because contributions add in quadrature, the smaller terms weigh very little; the percentage budget shows each one’s weight (u_i²/Σ u_i²) and tells you where it pays to act.
With independent input quantities, the standard uncertainties combine as the root of the sum of squares (GUM propagation law, without the covariance terms). The result is the combined standard uncertainty u_c:
Because contributions add in quadrature, the smaller terms weigh very little: a source one third of the largest contributes only ~10 % of the variance. The percentage budget above shows exactly this weight (uᵢ²/Σuᵢ²) and tells you where it pays to act to reduce the uncertainty.
What are expanded uncertainty and the coverage factor?
The combined uncertainty u_c is a standard uncertainty (≈ 68 % confidence for a normal distribution). To state an interval at higher confidence you multiply by a coverage factor k, giving the expanded uncertainty U = k·u_c. The reference value is k = 2 (≈ 95 % for an approximately normal result); k = 3 ≈ 99.7 %. The result is stated as y ± U.
The combined uncertainty u_c is a standard uncertainty (≈ 68 % confidence for a normal distribution). To state an interval at higher confidence you multiply by a coverage factor k, giving the expanded uncertainty U. The reference value is k = 2, which for an approximately normal result corresponds to about 95 % confidence; k = 3 ≈ 99.7 %.
The measurement result is then stated as y ± U, always declaring the coverage factor k used.
Reference: JCGM 100:2008 (GUM) — cited by name, text not reproduced.