Why do the analytic and Monte Carlo σ_z values sometimes disagree?

The analytical formula is a first-order Taylor approximation valid only when the function is nearly linear across the range spanned by the input uncertainties; for genuinely nonlinear formulas like z=x² or z=√(x²+y²), pushing σ_x or σ_y larger relative to the mean makes the true distribution skew away from Gaussian, which the Relative difference stat quantifies directly.

Why is z = x + y exact but z = x² only approximate?

Addition is exactly linear, so its partial derivatives are constant everywhere and the first-order formula is exact at any σ; squaring has a derivative 2x that changes with x itself, so the propagated uncertainty is only a local linear approximation that breaks down as σ_x grows.

What does raising the MC samples slider actually improve?

More samples reduce statistical noise in the Monte Carlo estimate itself, producing a smoother histogram — it does not change whether the underlying function is linear, so it will not fix a disagreement caused by genuine nonlinearity.

Home▸Statistics▸Error Propagation & Uncertainty (2D)

Error Propagation & Uncertainty (2D)

2D Canvas histogram lab: draws Monte Carlo samples of z = f(x,y) and overlays the analytical first-order error-propagation Gaussian so you can see exactly when they agree and when nonlinearity makes them diverge.

Statistics2DModerate60 FPS📱 Mobile-adapted⇄ 3D version

This 2D companion draws the same Monte Carlo samples and the same first-order analytical formula as the 3D version, but renders the histogram with plain Canvas 2D drawing instead of a WebGL shader: pick a formula z = f(x,y) from the dropdown, set the mean and σ for x and y, and watch the red analytical Gaussian curve and the pink sample histogram either sit on top of each other (linear formulas) or visibly pull apart (nonlinear ones) as the dashed blue line tracks the Monte Carlo mean and the stats panel reports the relative gap between the two methods.

⚙ Under the hood

error propagationuncertaintyMonte Carlostandard deviationpartial derivatives

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install