Blue line = binomial PMF · Red curve = Gaussian approximation · Bars = actual counts
The Galton board (or bean machine), invented by Sir Francis Galton in 1873, is a physical device that demonstrates the emergence of the normal (Gaussian) distribution from many random binary decisions. Balls are dropped through a triangular array of pegs; at each peg, a ball deflects randomly to the left or right with equal probability. After passing through N rows of pegs, each ball has made N independent binary choices and lands in one of N+1 bins at the bottom. The distribution of balls across bins approximates a binomial distribution that converges to the normal distribution as N increases.
The mathematical foundation is the binomial distribution: after N rows, the probability of a ball landing in bin k (counting from the left) is C(N,k)·(0.5)^N. For large N, by the central limit theorem (CLT), this approaches a Gaussian with mean N/2 and variance N/4, giving standard deviation √(N)/2. The CLT states more generally that the sum of many independent, identically distributed random variables—regardless of their individual distribution—converges to the normal distribution. This is why so many natural measurements (height, test scores, measurement errors) follow a bell curve.
This simulator drops balls through the Galton board and shows the growing histogram alongside the theoretical binomial/Gaussian. You can control the number of rows, initial bias (making left/right deflection unequal), and ball release rate to explore how quickly the distribution converges to Gaussian, how non-equal probabilities shift the distribution, and how sample size affects the match between empirical histogram and theoretical curve.
Why does the Galton board produce a normal distribution?
Each ball makes N independent binary decisions (left or right at each peg). The total displacement from center equals the number of right-deflections minus left-deflections, which is a sum of N independent ±1 random variables. By the Central Limit Theorem, the sum of many independent random variables with any distribution converges to a Gaussian. For the Galton board specifically, with N rows and equal probabilities, the result is exactly binomial, which approaches Gaussian as N grows large due to Stirling's approximation of factorials.
What is the Central Limit Theorem?
The Central Limit Theorem (CLT) states that if you sum (or average) a large number of independent, identically distributed random variables with finite mean μ and variance σ², the resulting distribution approaches a normal distribution with mean Nμ and variance Nσ², regardless of the original distribution's shape. This is why Gaussian distributions are ubiquitous in nature: any measured quantity that results from many independent additive contributions (thermal noise, measurement errors, biological traits influenced by many genes) will be approximately normally distributed.
What happens if the pegs are biased (unequal left/right probability)?
If the probability of deflecting right at each peg is p ≠ 0.5, the resulting distribution is a binomial with mean Np and variance Np(1−p). The distribution shifts from center: with p > 0.5, balls accumulate to the right; p < 0.5 shifts the peak left. The shape remains bell-like (still converges to Gaussian by CLT) but is skewed for small N and becomes symmetric only in the limit. Biased Galton boards model processes where outcomes are not equally likely—like a weighted coin or a genetic trait with dominance.
The number of paths through a Galton board leading to bin k (from 0 on the left) after N rows equals the binomial coefficient C(N,k)—precisely the entries of Pascal's triangle. Row N of Pascal's triangle gives the unnormalized bin counts: row 4 is 1,4,6,4,1, meaning with 4 pegs the center bin is 6× more likely than the edge bins. The binomial coefficients grow according to Pascal's rule C(N,k) = C(N-1,k-1) + C(N-1,k), reflecting that each bin receives balls from the two bins directly above it in the previous row.
The normal distribution arising from the Galton board principle underpins statistics and quality control throughout science and engineering. Measurement uncertainties in physics labs follow Gaussian distributions; the z-test and t-test for statistical significance assume normality. Quality control in manufacturing uses control charts based on normal distribution to detect process deviations. Standardized test scores are designed to be normally distributed. In finance, daily asset returns are approximately (though not exactly) Gaussian, forming the basis of the Black-Scholes options pricing model and Value-at-Risk calculations.