Error propagation calculator (with steps)

Overview

Enter y = f(x), then combine standard uncertainties with a first-order approximation. Templates cover sums, differences, products, quotients, and powers. General mode accepts safe expressions with constants and common functions.

Use the live formula preview, variance contribution chart, and optional Monte Carlo check to confirm the linearized result.

Keyboard tips: Ctrl/⌘+S exports CSV, Ctrl/⌘+L copies a shareable URL.

How to use (3 steps)

1. Type the formula y = f(...) and add each variable with its mean and standard uncertainty.
2. Set correlations if needed and choose whether to run the Monte Carlo validation.
3. Review the combined uncertainty, expanded uncertainty, and contribution ranking.

FAQ

How does the gradient×covariance method combine uncertainty?

It computes sensitivity coefficients at nominal values, builds a covariance matrix from standard deviations and correlations, and evaluates g^TCg. The square root is the combined standard uncertainty u_y.

What does the Monte Carlo validation check?

It samples correlated Gaussian inputs with a fixed seed and checks whether simulated mean and standard deviation agree with the linearized estimate.

When should I enter correlations instead of leaving them at zero?

Enter correlations when two inputs share the same instrument, calibration, or physical constraint. If the inputs are independently measured, leaving the matrix at zero is the usual starting point.

What changes when correlation is positive or negative?

Positive correlation usually increases the combined uncertainty when sensitivities move in the same direction. Negative correlation can partly cancel contributions, so it is worth documenting the sign instead of guessing.

Does this report standard uncertainty or expanded uncertainty?

The main output is the combined standard uncertainty. If you need an expanded uncertainty for reporting, apply your chosen coverage factor after checking the assumptions behind it.