How it Works
Linear error propagation (first-order Taylor expansion) gives the output uncertainty as a quadrature sum of partial derivatives times input uncertainties. For uncorrelated inputs x and y with standard deviations σ_x and σ_y:
Monte Carlo simulation draws N samples from the Gaussian input distributions, evaluates z for each sample, and computes the empirical mean and standard deviation. The histogram should match the analytical Gaussian for linear functions and reveal skew/asymmetry for nonlinear ones.
Frequently Asked Questions
What is error propagation?
Error propagation (uncertainty propagation) is the effect of input uncertainties on the uncertainty of a computed output. If z = f(x, y), the output uncertainty σ_z depends on the input uncertainties σ_x, σ_y and the partial derivatives of f.
What is the linear (first-order) error propagation formula?
For uncorrelated inputs: σ_z² = (∂f/∂x)²σ_x² + (∂f/∂y)²σ_y². This is derived from a first-order Taylor expansion of f around the mean values. It is exact for linear functions and approximate for nonlinear ones.
When does linear error propagation fail?
Linear propagation fails when the function is strongly nonlinear over the range of input uncertainty, or when input uncertainties are large relative to the function's scale. In these cases, Monte Carlo simulation gives more accurate results.
How does Monte Carlo uncertainty work?
Monte Carlo uncertainty analysis draws many random samples from the input distributions (usually Gaussian with mean ± standard deviation), evaluates the function for each sample, and computes the standard deviation of the output samples.
What is the uncertainty of a sum z = x + y?
For z = x + y with uncorrelated x, y: σ_z = √(σ_x² + σ_y²). The absolute uncertainties add in quadrature.
What is the uncertainty of a product z = x·y?
For z = x·y: (σ_z/z)² = (σ_x/x)² + (σ_y/y)². The relative uncertainties add in quadrature.
What is the uncertainty of z = x^n?
For z = x^n: σ_z/z = |n| · σ_x/x. Powers amplify relative uncertainty by factor |n|.
What is a coverage factor and confidence interval?
A coverage factor k multiplies the standard uncertainty to give expanded uncertainty at a desired confidence level. k=1 gives 68%, k=2 gives 95%, k=3 gives 99.7% for normal distributions.
What is the difference between accuracy and precision?
Accuracy describes closeness to the true value (systematic error / bias). Precision describes reproducibility — closeness of repeated measurements (random error / standard deviation).
What is the GUM and why is it important?
The Guide to the Expression of Uncertainty in Measurement (GUM) by BIPM provides internationally standardized methods for evaluating and expressing measurement uncertainty. It defines Type A (statistical) and Type B evaluations.