Bouncing balls always draw the same bell.
A triangle of pegs, a marble dropped from the top, fifty-fifty at every bounce — and somehow, every time, the bins below fill into the same curve. The board is the Central Limit Theorem made tactile: a thousand coin flips becoming the most universal shape in statistics.
Stack N rows of pegs in a triangle. Drop a ball at the top. At every peg it strikes, it goes left or right by a fair coin flip. After N bounces it lands in one of N+1 bins. One ball tells you nothing. A thousand balls draw a bell — always the same bell, no matter how many times you reset.
Take N = 10 rows. The probability of landing in bin k is C(10, k) / 1024. The eleven bin probabilities, multiplied by 1024, read (1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1) — row 10 of Pascal's triangle, already shaped like a bell.
The bell is not a coincidence. It is the Central Limit Theorem on its simplest stage: sum many independent fair coin flips, rescale, and the distribution converges to a Gaussian. Heights, errors, test scores, daily returns — anything that is a sum of many small independent kicks ends up here.
Francis Galton's quincunx, built in 1873 and popularised in his 1889 Natural Inheritance, is a board of N staggered rows of pegs. Release a marble at the apex and watch it strike exactly one peg per row, veering left or right with probability 1/2 at each impact, every collision an independent coin flip. After N rows the marble has dropped into one of N+1 catch bins along the floor, indexed by how many right-veers it took.
Every ball takes its own random path; the histogram is the only thing that converges. Slide N and watch the bell sharpen.
Pegs in cyan, falling balls in amber, the histogram fills in cyan below. The shape sharpens with every drop.
The number of left-right paths leading to bin k is exactly the binomial coefficient C(N, k) — the same numbers as row N of Pascal's triangle. Divide by the total 2ᴺ paths and you have the bin probability. The Galton board is, secretly, a physical lookup table for binomial coefficients.
Total = 1024 = 2¹⁰. The centre fattens; the edges starve. Already a bell, and N is still tiny.
Abraham de Moivre proved in 1733 that as N grows, C(N, k)/2ᴺ converges to the Gaussian density (1/√(2π·N/4)) · exp(−(k − N/2)² / (N/2)). Laplace generalised it to biased coins in 1810. This is the first instance of the Central Limit Theorem — a hundred years before the general statement.
At N = 4 the binomial is a coarse staircase; by N = 40 the Gaussian curve hugs every bar. This is the convergence De Moivre proved in 1733 — the first Central Limit Theorem in disguise.
Cyan bars: the exact binomial. Violet curve: the matching Gaussian. The amber dashed line marks μ = N/2.
The general CLT, rigorously proved by Lyapunov in 1901 and sharpened by Lindeberg in 1922, says far more: take ANY independent random variables with finite variance — skewed, discrete, ugly — sum N of them, rescale. The limit is a Gaussian. The bell is the universal attractor of averaging.
Independent, finite variance, any distribution. The standardised sum always converges to the standard normal: the bell is the universal attractor of averaging.
Adult heights are a sum of thousands of small genetic and environmental contributions — and so are measurement errors, IQ scores, daily logarithmic returns (under tidy assumptions). Each is a sum of many small independent kicks, so each is approximately Gaussian. This is why standard deviation has a name and why bell curves rule statistics.
The CLT needs finite variance. Drop that assumption and stranger attractors take over: the stable distributions, with Cauchy and Lévy at the extremes. Mandelbrot's critique of finance — that returns have «wild» power-law tails — is exactly this point. When a single contribution can dwarf the sum, no bell appears.
Cauchy · Lévy · Pareto · stable distributions with infinite variance. Mandelbrot, 1963: financial returns are «wild», not «mild».
The Explorer lets you push N to 40, bias the coin, watch the histogram fill in real time, and toggle the Gaussian overlay. Drop ten thousand marbles and watch the theorem assemble itself.
→ Open the ExplorerHow and where this technique lives in the world today.
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