Welch-Satterthwaite calculator
Enter each uncertainty component as a standard uncertainty with its degrees of freedom and get the combined standard uncertainty, the effective degrees of freedom, the coverage factor k from Student’s t, and the expanded uncertainty U. Uses the GUM (JCGM 100:2008) Welch-Satterthwaite method. Everything runs in your browser.
Uncertainty components
Enter each component as a standard uncertainty u (not an expanded value) and its degrees of freedom. For a well known Type B limit, enter inf for degrees of freedom.
| Component | Standard uncertainty u | Degrees of freedom v (n−1, or inf) | Remove |
|---|
Type A components evaluated from n repeated readings have v = n − 1. Type B components with a well characterized limit are commonly entered as infinite degrees of freedom (inf), so they do not pull the effective value down.
All components must share the same units.
k converges to 1.960 as the effective degrees of freedom grow.
Result
Method: Welch-Satterthwaite, GUM (JCGM 100:2008) clause G.4. u_c = √(Σ u_i²) for independent components; v_eff = u_c⁴ / Σ(u_i⁴ / v_i); k = t at 95% two-sided for v_eff (converging to 1.960); U = k · u_c. Degrees of freedom are yours to assign; correlations between components are not modeled.
Effective degrees of freedom, in plain terms
Why a fixed k = 2 is not always enough
The expanded uncertainty is U = k times the combined standard uncertainty. With k = 2 that interval covers about 95% of the outcomes, but only when the combined distribution is close to normal, which holds when several comparable components combine and each is well determined. When the budget leans on a handful of repeated readings, the combined uncertainty is itself uncertain, and the honest coverage factor is a little larger than 2. Effective degrees of freedom are how the GUM turns that idea into a number.
The Welch-Satterthwaite formula
Each component carries its own degrees of freedom v_i: a Type A term evaluated from n readings has v = n − 1, and a well known Type B term is usually treated as having infinite degrees of freedom. The combined standard uncertainty is u_c = √(Σ u_i²), and the effective degrees of freedom arev_eff = u_c⁴ / Σ(u_i⁴ / v_i). Because each term is raised to the fourth power, the component that is both large and thin on data dominates the denominator, and that is the one that holds the effective value down. A component with infinite degrees of freedom adds nothing to the denominator, so references and resolution steps do not penalize you.
Reading the coverage factor from Student’s t
Once you have the effective degrees of freedom, the coverage factor for 95% coverage is the two-sided Student t value, t0.975, at that number of degrees of freedom. It is about 2.23 near ten degrees of freedom, falls through 2.09 near twenty and 2.04 near thirty, and settles toward the normal value 1.960 as the degrees of freedom grow large. This tool reads k from a two-sided 95% t table with interpolation, which is why a thin budget returns a factor above 2 and a well determined one returns a factor close to it.
What this tool leaves to you
Two things are deliberately yours. The degrees of freedom behind each component are a judgement: v = n − 1 for a Type A term is clear, but the value for a Type B term rests on how well you know its own uncertainty (the GUM gives a method at clause G.4.2), and a wrong assignment moves v_eff. And this calculator combines the components as independent standard uncertainties, so it does not add covariance terms for sources that move together; confirm your components are uncorrelated before you rely on the combined figure. It reports a 95% two-sided factor and states its method rather than choosing your policy. References: JCGM 100:2008 (GUM), Annex G; ISO/IEC 17025:2017.
Common questions
What is the Welch-Satterthwaite formula?
The Welch-Satterthwaite formula estimates the effective degrees of freedom of a combined standard uncertainty. It is v_eff = u_c^4 divided by the sum of u_i^4 / v_i, where u_c is the combined standard uncertainty, each u_i is a component standard uncertainty, and each v_i is that component degrees of freedom. The point is to summarize how much data stands behind the combined uncertainty in a single number, so a defensible coverage factor can be read from the Student t distribution. It is given in the GUM (JCGM 100:2008) at clause G.4.1.
What are effective degrees of freedom, and why do they matter?
Effective degrees of freedom describe how well determined the combined standard uncertainty is. A combined uncertainty built mostly from a few repeated readings is less certain than one built from large samples or well characterized references, and the Welch-Satterthwaite value captures that. It matters because the coverage factor for a given level of confidence depends on it: fewer effective degrees of freedom means a larger k for the same 95% coverage. When one or two Type A components with limited data dominate the budget, v_eff is small and k rises above 2.
How do I choose the degrees of freedom for each component?
A Type A component evaluated from n repeated readings has v = n - 1. A Type B component whose limit is well known (a resolution step, a certificate value at a stated k) is commonly taken as having infinite degrees of freedom, so it does not pull the effective value down. The GUM at clause G.4.2 also gives a way to assign a finite value to a Type B component from how well you know its own uncertainty, roughly v_i = 1 / (2 times the relative uncertainty of u_i squared). Assigning these values is a judgement your laboratory owns; this tool applies the numbers you enter.
Why is the coverage factor k not always 2?
A coverage factor of k = 2 gives about 95% coverage only when the combined distribution is approximately normal, which is reasonable when many comparable components combine and the effective degrees of freedom are large. When the effective degrees of freedom are small, the combined result follows a Student t distribution with heavier tails, so a larger factor is needed to keep 95% coverage. As v_eff grows the Student t factor falls toward the normal value 1.960, which is why k is close to 2 for well determined budgets and higher for thin ones.
Does this calculator use 95% or 95.45% confidence?
It uses a two-sided 95.00% level of confidence, the Student t value t_0.975 by degrees of freedom, which converges to 1.960 as the effective degrees of freedom grow. Some GUM examples instead use a 95.45% level, for which the normal factor is exactly 2.000, so the factors come out slightly larger. This tool states the 95% convention plainly so you can compare it against your own policy. If your quality system fixes a different level, treat the k reported here as the 95% two-sided figure rather than a replacement for your stated rule.
How does this relate to the uncertainty budget calculator?
The uncertainty budget calculator combines your components by root-sum-square into a combined standard uncertainty and expands it at a coverage factor you choose, and it deliberately leaves the degrees of freedom out. This calculator is that missing step: bring each component standard uncertainty and its degrees of freedom here to get the effective degrees of freedom and a coverage factor read from Student t, rather than assuming k = 2. Used together they give a combined uncertainty and an expanded uncertainty with a defensible k.
This calculator applies the Welch-Satterthwaite formula to the components you enter and reports the combined standard uncertainty, the effective degrees of freedom, a two-sided 95% coverage factor, and the expanded uncertainty. It does not choose your degrees of freedom, does not model correlations between components, and does not replace JCGM 100:2008 or ISO/IEC 17025. Treat the result as a working estimate and validate it against your documented uncertainty procedure before you use it.
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