Propagate measurement uncertainties through addition, multiplication, powers, logarithms, and exponentials with error budget analysis and step-by-step derivations.
Error propagation determines how measurement uncertainties combine when you compute a derived quantity. If you measure mass with uncertainty ±2g and volume with uncertainty ±0.5mL, what's the uncertainty in the calculated density? This calculator answers that question for all common mathematical operations.
The calculator supports six operation types: addition/subtraction (errors add in quadrature), multiplication/division (relative errors add in quadrature), powers/roots, natural logarithm, exponential, and multi-variable mean with standard error. For each calculation, it shows the propagation formula, step-by-step derivation, and an error budget breaking down which measurement contributes most to the total uncertainty.
Understanding error propagation is essential for laboratory scientists, engineers, and any field where measurements are combined into derived results. The GUM (Guide to the Expression of Uncertainty in Measurement) standardizes these methods internationally, and this calculator implements its core formulas. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
Correct error propagation is required for every laboratory report, engineering tolerance analysis, and scientific publication. Doing it by hand is error-prone, especially for complex formulas. This calculator automates the process, shows the step-by-step derivation for learning, and identifies which measurement dominates the error budget.
The error budget feature is particularly valuable for experimental design: if you know which measurement contributes most to the final uncertainty, you can invest in improving that specific measurement rather than wasting effort on one that barely matters.
Addition/Subtraction: δ(A±B) = √(δA² + δB²) Multiplication/Division: δ(A×B)/(A×B) = √((δA/A)² + (δB/B)²) Power: δ(Aⁿ)/Aⁿ = |n| × δA/A Logarithm: δ(ln A) = δA/|A| Exponential: δ(eᴬ) = |eᴬ| × δA General: δf = √(Σ(∂f/∂xᵢ × δxᵢ)²)
Result: 7500 ± 80.78 (1.077% relative uncertainty)
Multiplying A=150±2 by B=50±0.5: relative errors are 1.333% and 1.000%. Combined in quadrature: √(0.01333² + 0.01000²) = 1.077%. Applied to 7500 gives absolute uncertainty of 80.78. The error budget shows A contributes 64% and B contributes 36% of the total variance.
The "Guide to the Expression of Uncertainty in Measurement" (GUM), published by the Joint Committee for Guides in Metrology, is the international standard for evaluating and reporting measurement uncertainty. It classifies uncertainty evaluations into Type A (statistical analysis of repeated observations) and Type B (other methods: manufacturer specs, experience, calibration certificates). Both types feed into the same propagation formulas used by this calculator.
In chemistry, error propagation determines uncertainty in molarity from mass and volume uncertainties. In physics, it combines timing and distance errors into velocity uncertainty. In engineering, it converts component tolerances into assembly tolerances. Medical diagnostics use it to establish confidence intervals for clinical measurements. Each field has domain-specific conventions, but the underlying mathematics is identical.
For simple operations, the analytical formulas (used by this calculator) are exact for linear approximations. For highly nonlinear functions, Monte Carlo simulation — generating thousands of random samples from each input's distribution and computing the output distribution — can be more accurate. However, analytical propagation is sufficient for most practical applications and provides immediate insight through the error budget that Monte Carlo obscures.
Linear addition (δA + δB) gives the worst case if both errors are maximally correlated. Quadrature addition √(δA² + δB²) is appropriate when errors are independent and randomly distributed, which is the usual assumption for separate measurements. Quadrature gives a smaller, more realistic combined uncertainty.
Never — uncertainties are always positive and always increase the total. Even for A − B, the propagated uncertainty is √(δA² + δB²), not δA − δB. Subtracting two values with uncertainties gives a result with more uncertainty, not less.
Absolute uncertainty (δA) is in the same units as the measurement (e.g., ±2 grams). Relative uncertainty (δA/A) is dimensionless (e.g., ±1.3%). Multiplication/division rules use relative errors; addition/subtraction rules use absolute errors.
The error budget shows which measurement contributes most to the total uncertainty. If 90% of the propagated error comes from one measurement, improving the other one won't help much. Focus your effort on reducing the dominant source of uncertainty.
This calculator assumes independent measurements. For correlated quantities, the propagation formula includes a covariance term: δf² = Σ(∂f/∂xᵢ)²δxᵢ² + 2ΣΣ(∂f/∂xᵢ)(∂f/∂xⱼ)cov(xᵢ,xⱼ). Positive correlation increases uncertainty; negative correlation can decrease it.
It computes the standard error of the mean (SEM) from repeated measurements of the same quantity. This is useful when your uncertainty comes from measurement variability rather than instrument precision. The SEM decreases as √n, so more measurements improve the estimate.