Binomial PMF
Gaussian
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.
Rows (5 to 16) sets how many peg rows a ball bounces through, which equals the number of independent binary left/right decisions N in the underlying binomial model. More rows means more possible landing bins (N+1 total) and finer discretisation of the bell curve, so the histogram increasingly resembles a smooth Gaussian rather than a blocky staircase — this is the Central Limit Theorem visibly kicking in as the number of summed random steps grows.
The variance of the binomial distribution produced by the board is Np(1-p), which for the standard p = 0.5 case is N/4 — so the spread of possible outcomes grows proportionally with N even though the standard deviation only grows as the square root of N. With more rows the ball can land in more distinct bins spread further from the center, so the visible width of the histogram increases even as its shape becomes more precisely bell-curved.
Ball Radius (3 to 8 pixels) is purely a rendering setting that changes how large each dropped ball appears on screen. The underlying peg-bounce logic that decides whether each ball deflects left or right at every row is unaffected by this slider, so the statistical distribution across bins stays the same regardless of whether the balls are drawn as tiny dots or larger circles.
A single ball's final bin is just one random sample from the binomial distribution, which tells you almost nothing about the underlying shape. Dropping hundreds or thousands of balls builds up a histogram of many independent samples, and by the law of large numbers this empirical histogram converges toward the true binomial (approximately Gaussian) probability distribution, which is why the "Drop many" style buttons reveal the bell curve far more clearly than dropping balls one at a time.
The default simulation uses an equal probability of deflecting left or right at each peg, matching the classic symmetric Galton board and producing a distribution centred on the middle bin. Even small departures from exactly 50/50 (as covered in the biased-peg FAQ above) would shift the mean of the binomial distribution toward one side, but the overall bell shape survives because the Central Limit Theorem applies to any fixed per-peg probability, not just p = 0.5.