How does the gradient×covariance method combine uncertainty?

We compute partial derivatives ∂f/∂x_i with a five-point central difference around the nominal means, assemble the covariance matrix as cov(x_i,x_j)=ρ_ij u_i u_j, and evaluate g^T C g. The square root gives the standard uncertainty u_y.

What does the Monte Carlo validation check?

It draws 20,000 correlated normal samples using a Cholesky factor with a fixed seed. The tool confirms the sample standard deviation stays within ±2% of the linearized result and the mean stays within ±0.2% of the theoretical value.

Error propagation calculator — sums, products, powers & general functions (with steps)

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.

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

Formula y = f(...)

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 (optional)

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

FAQ

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