Coverage Factor and Expanded Uncertainty Calculator (Welch Satterthwaite)
Combine up to four standard uncertainty components, find the effective degrees of freedom by the Welch Satterthwaite formula and the coverage factor k for about 95 % coverage, then the expanded uncertainty.
- uc = √Σui² = 0.01673
- νeff = uc⁴ ÷ Σ(ui⁴ ÷ νi) = 15.12
- k = Student t value for νeff (rounded down) at 95.45 % = 2.181
Welch Satterthwaite (GUM Annex G)
νeff = uc⁴ ÷ Σ(ui⁴ ÷ νi)
k = tp(νeff), U = k × uc
ui are standard uncertainties in the same unit (already divided by their distribution divisor), νi their degrees of freedom (n − 1 for Type A; infinite for most Type B components), and tp the Student t value for coverage probability p. νeff is rounded down, as GUM recommends.
Worked example
With u₁ = 0.012 (ν = 4), u₂ = 0.010 and u₃ = 0.006 (both infinite): uc = 0.0167, νeff ≈ 15.0, so k ≈ 2.18 at 95.45 % and U ≈ 0.0365.
Frequently Asked Questions
When is k = 2 enough?
When the effective degrees of freedom are large (above about 50), k = 2 gives about 95 % coverage.
What degrees of freedom does a Type B component have?
Usually infinite, when the limits are well known, such as a certificate value or a manufacturer specification.
Why round ν<sub>eff</sub> down?
Rounding down gives a slightly larger, conservative k, as recommended in the GUM.
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