CLT — a Galton board

Quincunx

A ball falls through n peg rows, going left/right with probability ½. Its slot counts the rights: k ~ Binomial(n, ½), mean n/2, variance n/4 — a bell as n grows.

Sum of i.i.d.

Pegs only add ±1 steps, so this mode drops them and sums n draws from a source you pick. Each draw uses inverse-transform sampling: U ~ Uniform(0,1), then X = F⁻¹(U) (shown in the inset). The standardized sum z = (S − nμ)/(σ√n) still lands on the same bell.

The curves

• Histogram — empirical density over time.
• KDE — Gaussian kernel density estimate.
• Normal — the N(0,1) the CLT predicts.

CDF & KS test

The CDF view shades the gap between the empirical and normal CDFs. D is the largest gap; the p-value comes from the Kolmogorov distribution at √N·D. We standardize with the known μ, σ, so the test is valid. (The quincunx is discrete, so for small n, D settles above zero — use more rows.)

Controls

Space pause · R reset · V board ⇄ CDF · F full-screen · H help.