Coverage factor k explained: why k = 2 means about 95 %

A combined standard uncertainty u_c corresponds to about 68 % coverage for a normal distribution — too small an interval for most practical statements. The expanded uncertainty U = k·u_c widens it: k = 2 gives roughly 95 %, k = 3 roughly 99.7 %.

When k = 2 is not 95 %

If the combined uncertainty is dominated by a single rectangular component, or the Type A evaluation is based on very few observations, the output distribution is not normal and k = 2 may not deliver 95 %. The GUM's Welch–Satterthwaite formula for effective degrees of freedom, or a Monte Carlo evaluation, addresses this.

Reading certificates

An uncertainty quoted without k cannot be compared with one quoted at k = 2. Ask for it; accredited certificates are required to state it.

The numbers behind k

For a normal output distribution, k is simply how many standard uncertainties wide the interval is, and the coverage follows from the normal distribution.

The convention of k = 2 rather than 1.96 is deliberate. It is a round number, it is marginally conservative, and it is what ILAC and the accreditation bodies expect, so certificates across the world are comparable without anyone doing arithmetic.

Two cases where k = 2 is not 95 %

The first is a budget dominated by a single rectangular component — a coarse display resolution, for example. A rectangular distribution has no tails, and its entire range lies within about 1.73 standard uncertainties of the centre. If one such term dominates, multiplying u_c by 2 produces an interval covering close to 100 % rather than 95 %: over-conservative, and not what the certificate claims.

The second is a Type A component from very few observations. A standard deviation from four readings is itself highly uncertain, and the output distribution has heavier tails than normal. The GUM's answer is to compute effective degrees of freedom by the Welch–Satterthwaite formula and take k from the t-distribution: with ν_eff of 6, k for 95 % is 2.45, not 2 — a 23 % wider interval.

Reading and writing it

• An uncertainty quoted without k cannot be compared with one quoted at k = 2. Ask for it; accredited certificates are required to state it.
• Write approximately 95 %, not 95 %, unless effective degrees of freedom were actually evaluated. The approximation is in the coverage, and the word is doing real work.
• Dividing by k to recover u_c from someone else's certificate is correct and routine — that is exactly what you do when their result becomes an input to your budget.
• Where k is not 2, say why on the certificate. A k of 2.45 with no explanation looks like an error rather than a Welch–Satterthwaite result.