Quincunx

Francis Galton built one of these out of wood and glass in 1873 to argue a point about heredity. Balls fall through a triangular lattice of pins. At every pin a ball goes left or right. At the bottom they pile up in bins.

The gold line is not fitted to the bars. It is the exact distribution the machine is sampling, drawn on the same scale — so what you are watching is the bars converging onto a curve that was already there.

Why the middle fills up

A ball that bounces right $k$ times out of $n$ rows lands in bin $k$. Nothing else about its journey matters — not the order of the bounces, not the path.

So the question “which bin?” is really “how many rights, out of $n$?”, and the number of routes into bin $k$ is the number of ways to choose which $k$ of the $n$ bounces went right:

There is exactly one route into the far-left bin: go left every single time. There are $n$ routes into the next one along. And there are a great many routes into the middle, which is the entire reason the middle fills up. The pile is not telling you the balls prefer the centre — it is telling you the centre is easier to reach.

Set the rows to 4 and drop a few hundred. The bins go 1 : 4 : 6 : 4 : 1, and at that size you can count the routes by hand and check.

Tilt it

Turn the bias off a half and the pile slides. The peak moves to $np$ — with twenty rows and a one-in-four chance of going right, it settles over bin 5 — and the shape stays recognisably the same bell, just narrower.

Narrower because the spread is

which is largest at $p = \tfrac{1}{2}$ and shrinks toward zero at either extreme. Push the bias all the way to 0 or 1 and the machine stops being a machine: every ball takes the same route into the same bin, and the distribution collapses to a spike.

That is worth sitting with, because the bell is often described as what you get “when things are random”. It isn’t. It is what you get when things are random and roughly balanced. A lopsided coin is exactly as random as a fair one and produces a much tighter pile.

Where the bell comes from

The binomial is not the normal distribution, but it converges to it as the rows increase — de Moivre worked this out in 1733, for coin tosses, nearly a century before Gauss. The quincunx is that theorem you can watch: each extra row of pins is one more coin toss added to the sum, and the outline of the pile is the central limit theorem arriving one row at a time.

Where it came from

A p5.js sketch of mine, which drew the pins and dropped the balls but kept the left/right chance nailed to a half. It had a bias variable ready and waiting — with the comparison written backwards, so raising the chance of going right would have sent fewer balls right. Harmless while the number never moved; the first thing to fix once it became a control.