Scientific Measurement Uncertainty Propagation Calculator

Method, example, and limits

Formula: For independent inputs, uᵣ(y) = √[(p·uᵣ(x))² + (q·uᵣ(z))²]; absolute standard uncertainty is |y|uᵣ(y).

Inputs and units: x and z are positive measured values; ux% and uz% are their relative standard uncertainties; p and q are powers. Output units follow the chosen input units and powers.

Worked example: For y = 10 × 5, with independent 1% standard uncertainty on each input and p = q = 1, y = 50, relative standard uncertainty ≈ 1.4142%, and absolute standard uncertainty ≈ 0.707107.

Assumptions: Uses first-order uncertainty propagation for y = xᵖzᑫ, with independent input errors and standard uncertainties. Correlations/covariances are set to zero, and percentages are not expanded uncertainties or guaranteed bounds.

Method reference: JCGM 100:2008, Evaluation of measurement data—Guide to the expression of uncertainty in measurement.

Limits: Linearized propagation may be inaccurate for large uncertainties or strongly nonlinear behavior. Correlated measurements require covariance terms; uncertainty coverage factors and distribution assumptions are not inferred.