Rectangular, triangular and normal distributions in uncertainty budgets

A Type B component usually starts life as a half-width a — a specification of ±a, a display resolution of 2a, a drift limit of ±a over the calibration interval. Before it can be combined with anything else it has to become a standard uncertainty, which means dividing it by a factor that depends on how the value is believed to be distributed within those limits. That belief is a modelling choice, and the GUM expects you to make it explicitly.

The four divisors

Each divisor is the standard deviation of the assumed distribution expressed as a fraction of its half-width. They are not conventions to be memorised; they fall out of the distribution's own variance.

A fifth case needs no divisor at all: a component already expressed as a standard uncertainty — a Type A standard deviation of the mean, or a supplier's stated u — is divided by 1 and enters the budget as it stands.

Rectangular is the default, and here is why

When a manufacturer states ±0.2 °C and says nothing else, you know two things: the maker believes the error lies inside those limits, and you have no information at all about where inside. Assuming a peak in the middle would be inventing information you were not given. The rectangular distribution is the one that assumes least, which is why the GUM reaches for it and why an assessor will expect it on any line where you have not argued for something else.

It is also the safer error. Rectangular gives the largest standard uncertainty of the three half-width cases apart from U-shaped, so a wrong rectangular assumption overstates rather than understates — the direction you want to be wrong in when the budget supports a conformity decision.

A worked budget, both ways

Four components for a temperature calibration at a single point: a reference thermometer certificate at U = 0.030 °C with k = 2, a readout resolving to 0.01 °C (half-width 0.005 °C), bath stability observed within ±0.020 °C, and repeatability of 0.012 °C from ten readings. Combined by the law of propagation of uncertainty for uncorrelated inputs, this is what each contributes.

That gives u_c = 0.0226 °C and U = 0.045 °C at k = 2. Now change one assumption: argue that the bath spends more time near its setpoint than at the extremes of its cycle, and call that component triangular instead of rectangular. Its standard uncertainty falls from 0.0115 °C to 0.0082 °C, and the expanded uncertainty falls from 0.045 °C to 0.042 °C — a change of 6.8 %.

6.8 % is worth knowing but it is not where budgets go wrong. A component left out entirely, a sensitivity coefficient of the wrong sign, or a reference used outside the range it was calibrated over will each cost you more than every divisor on this page put together.

Degrees of freedom travel with the distribution

Type B limits are normally treated as exactly known, which is to say infinite degrees of freedom. A Type A component is not: ten readings give ν = 9. Run the same four components with the coverage factor taken from GUM Table G.2 rather than fixed at 2 and the effective degrees of freedom come out at 113, giving k = 2.025 and U = 0.046 °C. The correction is small here because the Type A component carries only 28 % of the variance. Let repeatability dominate a budget built on four readings and the same calculation will hand you a visibly larger k.

Choosing in practice

• Manufacturer's accuracy specification, no confidence level stated: rectangular.
• Manufacturer's specification stated at 95 %: normal, divide by 2 — the maker has told you the shape.
• Display or logger resolution d: rectangular with half-width d/2, so u = d/(2√3).
• Drift between calibrations, bounded by observed history: rectangular over the bound.
• Temperature control cycling about a setpoint: U-shaped is defensible if you have the trace to show it; rectangular if you do not.
• Anything you have actually measured repeatedly: not a distribution question at all — it is Type A, and the standard deviation of the mean goes in directly.

Whatever you choose, write it in the budget next to the value. A reviewer can disagree with a stated assumption and recompute in a minute. An unstated one has to be reverse-engineered from the arithmetic, and that is the point at which a budget stops being evidence.