Coverage factor k and the Welch–Satterthwaite formula
The expanded uncertainty is U = k · uc. Almost every certificate says k = 2, and in most calibrations that is correct. But k = 2 gives a coverage probability of about 95 % only if the combined uncertainty is known well, that is, if it has many degrees of freedom. When a Type A component with few readings dominates the budget, k = 2 understates U.
Why the number of readings matters
A standard deviation calculated from 4 readings is itself an uncertain estimate. To keep the same coverage probability, the interval must be wider. Statistically, the coverage factor then comes from Student's t-distribution instead of the normal distribution, with the degrees of freedom ν of the estimate.
The Welch–Satterthwaite formula
A budget combines several components, each with its own degrees of freedom νi (n − 1 for Type A, usually ∞ for Type B). The effective degrees of freedom of the combined uncertainty are approximated by:
Components with νi = ∞ add nothing to the sum. The fourth powers mean that only the large components with few degrees of freedom matter. If νeff is not an integer, EA-4/02 says to round it down to the next lower integer. Then k is the t-quantile for that ν and the chosen coverage probability.
Worked example: micrometer with four readings
A 10 mm pin is measured four times with a digital micrometer: 10.012, 10.018, 10.009 and 10.015 mm.
| Source | Evaluation | u(xi), mm | νi |
|---|---|---|---|
| Repeatability, 4 readings | Type A, s = 0.00387, s / √4 | 0.00194 | 3 |
| Micrometer calibration | Normal, U = 0.004, k = 2 | 0.00200 | ∞ |
| Resolution (0.001 mm) | Rectangular, a = 0.0005 | 0.00029 | ∞ |
| Combined uc | 0.00280 |
For ν = 13 and 95.45 %, k = 2.21. So U = 2.21 × 0.00280 = 0.0062 mm. With k = 2 out of habit, U would have been 0.0056 mm, about 10 % too small. Repeatability makes up almost half of the variance here, and three degrees of freedom are not many.
Two ways to bring k back towards 2: take more readings, or use a pooled standard deviation from an earlier repeatability study with many degrees of freedom (see Type A and Type B uncertainty).
Table of coverage factors for 95.45 %
Values of the t-distribution for a coverage probability of 95.45 %, as in GUM table G.2 and EA-4/02 table E.1:
| νeff | 1 | 2 | 3 | 4 | 5 | 6 | 8 | 10 | 20 | 30 | 50 | ∞ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| k | 13.97 | 4.53 | 3.31 | 2.87 | 2.65 | 2.52 | 2.37 | 2.28 | 2.13 | 2.09 | 2.05 | 2.00 |
Beyond about 50 effective degrees of freedom, k is within 2.5 % of 2. Below 10 the difference becomes significant.
A note on EA-4/02 table E.1. While validating Metroloom we found that the table prints k = 2.24 for ν = 11. The exact t-quantile is 2.2549, which rounds to 2.25 and fits between 2.28 (ν = 10) and 2.23 (ν = 12). Metroloom uses the exact value. Details in the validation report.
95 % or 95.45 %?
EA-4/02 and most accredited calibration certificates use 95.45 %, the probability that corresponds exactly to k = 2 for a normal distribution. Some fields use 95 %, which gives k = 1.96 for ν = ∞. The difference is small, but state on the certificate which one you used, together with k.
When Welch–Satterthwaite is not enough
The formula assumes uncorrelated inputs and a combined distribution close to normal. If one rectangular component dominates the budget, the combined distribution is closer to rectangular than normal, and the t-based k can be too large. For such cases, and for strongly non-linear models, the Monte Carlo method of GUM Supplement 1 (JCGM 101:2008) gives a more reliable interval.
See the full budget method in How to make a measurement uncertainty budget.