How does the gradient×covariance method combine uncertainty?

We evaluate the gradient with a five-point central difference at the nominal values, form the covariance matrix using the entered standard deviations and correlations, and compute g T Cg. The square root gives the combined standard uncertainty u y .

What does the Monte Carlo validation check?

It draws correlated Gaussian samples with a fixed seed via Cholesky decomposition and confirms that the simulated mean and standard deviation match the linearized prediction within the specified tolerances.

Error propagation calculator – sums, products & powers

Overview

Enter the analytic function y = f(x) and combine standard uncertainties through a first-order approximation. Templates cover sums, differences, products, quotients, and powers, while the general mode accepts safe expressions with constants, trig, hyperbolic, and logarithmic functions.

See a live math preview with highlighted variables, a contribution bar chart that shows which input dominates the uncertainty, and a recommended Monte Carlo cross-check to confirm the linearized result.

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

How to use (3 steps)

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

Formula y = f(...)

Preview

Templates:

Variables with means and standard uncertainties
Name	Mean μ	Std. dev. u	Unit / note	Remove row

Correlation matrix (optional)

Keep the diagonal at 1.0 and enter correlation coefficients ρ_ij between −1 and 1. Upper and lower triangles stay synchronized automatically.

Monte Carlo validation

Run Monte Carlo after calculation

Adds a seeded simulation to confirm the linearized estimate. Turn off if you need the fastest run.

Seed Samples N

FAQ

How does the gradient×covariance method combine uncertainty?: We evaluate the gradient with a five-point central difference at the nominal values, form the covariance matrix using the entered standard deviations and correlations, and compute g^TCg. The square root gives the combined standard uncertainty u_y.
What does the Monte Carlo validation check?: It draws correlated Gaussian samples with a fixed seed via Cholesky decomposition and confirms that the simulated mean and standard deviation match the linearized prediction within the specified tolerances.

Error propagation calculator (with steps)

Overview

How to use (3 steps)

FAQ

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