Combined Uncertainty (GUM)
Root-sum-square of up to five uncertainty components, with coverage factor for an expanded uncertainty.
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The engineering
The GUM (Guide to the Expression of Uncertainty in Measurement) treats each independent uncertainty source as a standard uncertainty and combines them in quadrature — root-sum-square — to get the combined standard uncertainty uₙ. Multiplying by a coverage factor k gives the expanded uncertainty U that defines your reported interval. Use this whenever you build an uncertainty budget for a calibration or test report.
Because components add as squares, the budget is dominated by the largest one: a component half the size of another contributes only a quarter of its variance. The 'largest share of variance' row tells you where to spend your effort — chasing a 10% contributor is usually wasted work. This card assumes uncorrelated inputs; correlated terms need the full covariance treatment.
k = 2 gives roughly 95% coverage for an approximately normal result with adequate degrees of freedom — the default for most accredited reports. Drop to k = 1 for a standard-uncertainty statement, or use a Student-t value when your effective degrees of freedom (Welch-Satterthwaite) are low.
Where this math comes from
Before the GUM, every lab expressed measurement uncertainty its own way — some added errors linearly, some used worst-case bounds, some quoted only repeatability. Comparing results across borders was a mess, so in 1977 the BIPM asked its Working Group on Uncertainty to recommend a unified approach. The result, championed by figures like Klaus Weise and Wolfgang Wöger, was the distinction between Type A (statistical) and Type B (everything else) evaluations, both expressed as standard deviations so they could be combined by the same rule.
The first Guide to the Expression of Uncertainty in Measurement appeared in 1993 under seven international bodies (BIPM, IEC, ISO, and others), and was reissued essentially unchanged as JCGM 100:2008 by the Joint Committee for Guides in Metrology. The quadrature-combination law of propagation on this card is its central equation.
- 1980BIPM Working GroupIssues Recommendation INC-1, establishing Type A / Type B uncertainty evaluation.
- 1993ISO/IEC/BIPM et al.Publish the first Guide to the Expression of Uncertainty in Measurement (GUM).
- 1994Welch & Satterthwaite (applied)Effective degrees of freedom formula adopted for choosing coverage factors.
- 2008JCGMReissues the GUM as JCGM 100:2008 — the current standard reference.
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