Measurement Uncertainty (Propagation) Calculator

Uncertainty propagation (ISO GUM)

For independent random variables, uncertainties combine in quadrature. Sum/difference (z = a ± b): σ_z = √(σ_a² + σ_b²) — sum absolute uncertainties. Product/quotient (z = a·b or a/b): σ_z/|z| = √((σ_a/a)² + (σ_b/b)²) — sum relative uncertainties. Power (z = aⁿ): σ_z/|z| = |n|·σ_a/|a|. The ISO Guide to the Expression of Uncertainty in Measurement (GUM) is the metrological standard. Type A uncertainty comes from statistics (standard deviation of n repeated measures, ≈ σ/√n); type B from non-statistical sources (instrument resolution, manufacturer specs, calibration certificate). Example: a = 10 ± 0.1, b = 5 ± 0.05 → a+b = 15 ± 0.112; a·b = 50 ± 0.707 (relative 1.41%).

Applications

Calibration labs accredited to ABNT NBR ISO/IEC 17025; metrology at INMETRO (RTAC and other technical regulations); pharmaceutical-industry assay validation; physics-undergraduate lab reports; engineering tolerance stack-up; instrument datasheets (resolution, repeatability, accuracy); analytical-method validation (LOD/LOQ); and uncertainty budgets in legal/forensic metrology.

FAQ

Coverage factor k = 2 — what does it mean? Multiplying the combined standard uncertainty u_c by k yields the expanded uncertainty U = k·u_c. For a normal distribution, k = 2 covers ~95% (k = 1 ≈ 68%, k = 3 ≈ 99.7%). Calibration certificates usually report U with k = 2.

When does "sum in quadrature" not apply? When variables are correlated. Then a covariance term appears: σ_z² = σ_a² + σ_b² + 2·cov(a,b). Two readings of the same instrument with the same systematic bias are correlated.

How many significant figures should the uncertainty have? Per GUM, 1–2 significant figures. The measurement itself is rounded so that its last digit matches the uncertainty's last digit (e.g., 9.812 ± 0.045 m/s²).