Effective degrees of freedom and when k = 2 is not enough
When Type A components rest on few observations, the combined uncertainty has limited degrees of freedom and k must come from the t-distribution. Welch–Satterthwaite gives the number.
The coverage factor 2 assumes a normal output distribution. A Type A component estimated from n readings has ν = n − 1 degrees of freedom and, if it dominates, the output follows a t-distribution instead. The GUM's Welch–Satterthwaite formula combines the degrees of freedom of all components into an effective value, from which k is read from the t-table at 95 %.
Example
Repeatability from 4 readings (ν = 3) is the dominant term. Effective degrees of freedom come out near 4; k for 95 % is about 2.8, not 2. Taking 10 readings instead raises ν to 9 and brings k to roughly 2.3 — the cheapest uncertainty reduction available.
Where distributions are far from normal, or the model is strongly non-linear, GUM Supplement 1's Monte Carlo method avoids the assumption altogether.
Frequently asked questions
- Do I need this for every budget?
- Check it whenever a Type A component from fewer than about 10 readings contributes more than a quarter of the combined variance. Otherwise k = 2 is adequate.
References
- [1]JCGM 100:2008 — Guide to the expression of uncertainty in measurement (GUM)
- [2]JCGM 101:2008 — Supplement 1 to the GUM: Propagation of distributions using a Monte Carlo method
General technical guidance written against the cited sources. It is not regulatory or legal advice and does not replace the applicable standard, guideline or a qualified reviewer's judgement.
Have a question on this topic?
Ask ValiTrac AI and see the evidence and calculation behind the answer.
Ask ValiTrac AIFree calculator: Uncertainty budget
Combine Type A and Type B components per the GUM: divisors by distribution, sensitivity coefficients, u_c, effective degrees of freedom and U at k = 2.
Open the calculatorMetrology on ValiTrac
Measurement science, traceability, uncertainty and technical evidence.
Ask a metrology questionRelated articles
What the coverage factor k means
The coverage factor multiplies the combined standard uncertainty to give an expanded uncertainty with a stated level of confidence. k = 2 is roughly 95 % for a normal distribution.
Monte Carlo evaluation of uncertainty (GUM Supplement 1)
Instead of propagating standard uncertainties analytically, sample every input from its distribution, compute the result many times, and read the interval off the output histogram.