Effective degrees of freedom and when k = 2 is not enough

The coverage factor converts a combined standard uncertainty into an interval with a stated level of confidence. k = 2 for about 95 % is so routine that it gets applied automatically, but it rests on an assumption: that the combined standard uncertainty is itself well known. When a Type A component evaluated from a few observations carries most of the budget, that assumption fails, and the interval you report is narrower than the confidence you claim.

What the table actually says

GUM Table G.2 gives the factor for a 95.45 % interval as a function of effective degrees of freedom. The tail is steep.

Note ν = 1: a factor of 13.97. Two observations give you one degree of freedom and almost no information about the spread, and the table is honest about that even though nobody wants to report it.

What it costs in a real budget

Take a budget of three components: repeatability at 0.040 °C as a Type A standard deviation, a reference certificate at U = 0.030 °C with k = 2, and a resolution term of ±0.005 °C. Repeatability dominates. Now vary only how many readings it came from.

Three readings give an expanded uncertainty 126 % larger than the k = 2 shortcut would have reported. Ten readings bring the penalty down to about 12 %, and thirty to 3 %. The curve flattens quickly, which is the practical message: you do not need thirty readings, you need to stop taking three.

Where the coverage factor actually comes from

k = 2 is a shorthand. The figure it approximates is the two-tailed 95.45 % quantile of the Student t-distribution, and which t-distribution depends on how many degrees of freedom the uncertainty rests on. With effectively infinite degrees of freedom the t-distribution has converged on the normal distribution and the quantile is 2.00; with ten it is 2.28, and with four it is 2.87. Effective degrees of freedom exist to tell you which row of that table you are on.

This is why the answer is never to raise k arbitrarily and call it conservative. The t-distribution gives a specific multiplier for a specific number of degrees of freedom, and a budget that reports k = 2 while sitting on four degrees of freedom is not slightly optimistic, it is reporting an interval around 30 % narrower than the confidence it claims.

The Welch-Satterthwaite formula

ν_eff = u_c⁴ divided by the sum over components of (c_i u_i)⁴ / ν_i. Type B components with stated limits are conventionally treated as exactly known, so their ν is infinite and their term in the denominator is zero. That is why a budget made entirely of Type B limits has infinite effective degrees of freedom and k = 2 is simply correct.

It also explains why ν_eff in the table above exceeds the repeatability component's own ν. The Type B components contribute variance without contributing to the denominator, so they dilute the small-sample penalty. A Type A term that carries only 20 % of the variance barely moves k at all.

What to do about it

• Record the number of observations behind every Type A component and carry it into the budget as degrees of freedom. A budget that does not track ν cannot compute ν_eff.
• Take more readings before reaching for a better instrument. Going from 3 to 10 readings costs minutes and removes most of the penalty.
• State the coverage factor and its basis on the certificate. "k = 2" alone does not say whether it was assumed or derived.
• Where a dominant Type A term is unavoidable and small, report the derived k. Reporting k = 2 with ν_eff = 2.6 overstates your confidence by a wide margin.

None of this matters if your Type A term is a minor contributor. Check before you spend effort on it: if repeatability carries 10 % of the variance, the correction to k will be in the third decimal place.