Galton board · Central Limit Theorem simulation

Why is the bell curve everywhere? If you sum many independent random ±1 steps, does the result always become Gaussian — and how fast?

▶ Run the simulationSee the measured result

Units: dimensionless (log–log convergence exponent of the symmetric board's pointwise gap; secondary knowns: −1 biased, −1/2 CDF, amplitudes φ(0)/2 = 0.1994711 and φ(0) = 0.3989423, Edgeworth prefactor M|1−2p|/(6pq) with M = 0.54763, Lévy exponent 1/α = 2/3)

How the lab tests it

Drop thousands of balls through n rows of pegs (each a ±1 Bernoulli step with probability p); the count of rights is a sum of n independent steps, binned into a histogram with the exact Gaussian N(np, np(1−p)) overlaid. A separate exact experiment plots the maximum pointwise gap Δ(n) between the binomial and its Gaussian limit, log-log, across n = 4…256.

What it checks

the CLT: the histogram fills the bell with measured mean → np and standard deviation → √(np(1−p)); and the convergence RATE — Δ ∝ n^(−3/2) for a symmetric board (the leading skewness correction vanishes at p=½), slowing to Δ ∝ 1/n once the board is biased (p≠½)