Building an uncertainty budget step by step

The eight steps

• 1. State the measurand precisely: e.g. the correction to the indicated temperature of the unit under test at 5 °C.
• 2. Write the model: correction = t_ref + δt_ref_drift + δt_bath_gradient + δt_bath_stability − t_uut − δt_uut_resolution … each δ a correction with an expected value of zero and an uncertainty.
• 3. List the inputs and gather information: certificate uncertainties, specifications, repeatability data, resolutions.
• 4. Evaluate each as a standard uncertainty (Type A from data; Type B from limits with a distribution).
• 5. Determine sensitivity coefficients from the model.
• 6. Combine: u_c² = Σ (c_i u_i)².
• 7. Check effective degrees of freedom if any Type A component is based on few observations.
• 8. Expand: U = k u_c, usually k = 2, and report with the value rounded consistently.

A minimal temperature example

Reference certificate 0.02 °C (k=2 → 0.010); reference drift ±0.01 (rect → 0.0058); bath gradient ±0.02 (rect → 0.0115); bath stability observed SD 0.008; UUT resolution 0.1 (rect → 0.0289); UUT repeatability 0.012. Combined ≈ 0.035 °C; U ≈ 0.07 °C at k = 2. The resolution term dominates — a common finding for 0.1 °C loggers.

Read the table, not just the answer

The last column is the point of the exercise. Resolution contributes 68 % of the variance — more than everything else combined — because a 0.1 °C display is a ± 0.05 °C rectangular half-interval, and squaring makes large terms dominate. Buying a better reference thermometer here would move the total from 0.0351 to about 0.0339: a 3 % improvement for real money. Reading the logger at a finer resolution, or interpolating across the display step, is worth ten times as much.

That is the general lesson. A budget is a diagnostic, not a certificate decoration. Sort by contribution and work on the top row.

Rounding, and the mistake almost everyone makes

The combined uncertainty above is 0.0351 °C and the expanded is 0.0702 °C. Report it as U = 0.07 °C, not 0.0702 °C. GUM §7.2.6 asks for the uncertainty to be given to at most two significant figures, and the result rounded to the same decimal place. Quoting four figures of uncertainty claims a precision about the uncertainty itself that the estimate cannot support — and an assessor will notice.

The correction is then reported alongside it at the matching resolution: for instance correction +0.03 °C, U = 0.07 °C (k = 2, approximately 95 %). Round at the end, once, never at each intermediate step.

This article is the method. The uncertainty budget calculator is the arithmetic: enter each component with its distribution and sensitivity coefficient, and it applies the divisors, combines in quadrature, computes effective degrees of freedom and reports U — including which component dominates the variance, which is usually the only output that changes what you do next.

Checks before you sign it

• Does every input have a sensitivity coefficient, even where it is 1? Writing c_i = 1 deliberately is different from forgetting it exists.
• Is anything counted twice? Reference drift and reference certificate overlap if the certificate already covers the interval.
• Is anything missing that you know about? Immersion, self-heating and hysteresis are the usual omissions in temperature work.
• Are the divisors right? Rectangular is √3, triangular √6, U-shaped √2, and a certificate at k = 2 divides by 2 — not by √3.
• Does the answer pass a sanity check against the tolerance? A U comparable to the tolerance means the conformity statement needs a decision rule.