Sketch · 2026-09-28
Galton board.
A triangular field of pegs. A ball dropped at the top hits a peg, and bounces left or right by chance. Then again, and again, once per row. At the bottom it lands in a bin. Do this a few thousand times and the heap that piles up is a binomial distribution, drawn by hand.
The thin curve draped over the bins is the exact binomial PMF for the current n and p, rescaled to match the total count so far. Watch the histogram lock onto it as samples pile up.
Note
Francis Galton built a version of this device in 1873 and called it the quincunx, after the five-dot pattern on a die. It was his way of showing where a bell curve comes from without invoking calculus. Every ball is an independent sum of n Bernoulli variables, and the central limit theorem says any such sum, for large enough n, looks normal. The device does not prove the theorem, but it demonstrates it, one clatter at a time.
Slide the bias off 0.5 and the pile shifts, but its shape barely changes: a Binomial(n, p) has mean n p and variance n p (1 − p). The variance is the widest at p = 0.5 and shrinks toward the ends. The pile widens or narrows accordingly. Nothing about the peg layout knows about p; only the coins do.
One thing this simulation lies about is the physics. Real Galton boards have wobble: a ball can hit a peg on top and skip a row, or graze one and tunnel sideways. That noise is exactly why Galton liked the machine as a metaphor for heredity. Here every ball obeys the coin exactly, which is why the histogram converges cleanly rather than smeared out.
Sources
- Galton, F. (1889). Natural Inheritance. Macmillan, chapter V (The Charms of Statistics), pages 62-65.
- Stigler, S. M. (1986). The History of Statistics: The Measurement of Uncertainty Before 1900. Harvard University Press, pages 275-277.
- Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. Wiley, chapter VI on the binomial and its normal approximation.