Error Propagation Calculator — Uncertainty Propagation

Formula

How this is calculated

When a calculated result f depends on measured quantities a and b, any uncertainty in a and b propagates into f. For independent, uncorrelated measurements the standard rule — sometimes called quadrature addition — combines the partial derivatives: σ_f² = (∂f/∂a)² σ_a² + (∂f/∂b)² σ_b². This calculator applies that general rule to the five most common operations.

For addition or subtraction, the absolute uncertainties add in quadrature: σ_f = √(σ_a² + σ_b²). For multiplication or division, the relative uncertainties add in quadrature: σ_f/|f| = √((σ_a/|a|)² + (σ_b/|b|)²). For a power law f = aⁿ, the propagation gives σ_f = |n| × |a^(n−1)| × σ_a (equivalently, the relative uncertainty scales by |n|). All formulas assume the uncertainties are small relative to the measured values (linear approximation) and that the variables are statistically independent.

The result is expressed as f ± σ_f, representing the central value and one standard deviation of the output distribution (68% confidence interval if the inputs are normally distributed). The bell-curve plot visualises this: the shaded region from f − σ_f to f + σ_f covers roughly 68% of the probable values of the result. For systematic uncertainties or correlated inputs the formulas are more complex — see the GUM (Guide to the Expression of Uncertainty in Measurement) for the full treatment.