Every ball in the machine below makes the same dumb choice again and again: left or right, a coin flip at each peg. Nobody tells the pile what shape to take. Drop enough balls and it grows into the same smooth hill, every time.
Press Drop 100 and watch where they land. Then try Add 1,000 to see the shape without the wait. The dashed line is the textbook bell curve for these settings. Change the rows or tilt the pegs and the pile starts over.
Slot number = how many times the ball bounced right.
The device is called a Galton board, or a quincunx. Francis Galton built one for a lecture at the Royal Institution in London in 1874. A ball enters at the top and meets one peg per row. At each peg it goes left or right with equal chance, then meets the next peg, and so on until it drops into a slot at the bottom.
A ball's slot is simply the number of times it bounced right. Zero rights puts it in the far-left slot. Every right puts it in the far-right slot. There is exactly one route to each edge, but many routes to the middle.
With 10 rows there are 210 = 1,024 equally likely routes. Only one of them is all lefts, so the edge slot catches about 0.1% of balls. But 252 of those routes contain exactly five rights and five lefts, in some order. That is about 25% of balls in the single middle slot.
Nothing pushes balls toward the centre. There are just far more ways to get a mixed result than an extreme one. Ten rights in a row is possible, only rare, in the same way that ten heads in a row is rare.
The Frenchman Abraham de Moivre noticed in the 1730s that as the number of flips grows, the counts trace a smooth curve. That curve is now called the normal distribution, or the bell curve. Set the board to 20 rows and watch the dashed line hug the bars.
The spread is measured by the standard deviation. For a fair board it equals half the square root of the number of rows. That has a surprising consequence: the pile widens slowly. Four times as many rows only doubles the typical distance from the middle. Slide from 4 rows to 16 and watch the spread readout go from 1 to 2.
Now tilt the pegs so a ball goes right 80% of the time. The whole hill slides toward the right edge, centred near 80% of the rows. With only a few rows the hill is lopsided, jammed against the wall. With 20 rows it is almost symmetric again, only shifted.
A ball's final position is the sum of many tiny independent nudges. That is the heart of the central limit theorem: add up lots of small, independent random effects and the total tends toward a bell curve, almost no matter what each nudge looks like on its own.
Galton was interested in exactly this. Traits like human height come from many small influences stacking together, and measured heights often look bell-shaped. It also explains why a bell curve is such a useful default for measurement error, from a scale reading to a lab instrument.
It is not a law of everything. Things that multiply instead of add, such as the sizes of cities or incomes, usually give lopsided piles with long tails instead. And a few huge shocks can swamp many small ones. The bell curve shows up when the pushes are many, small and independent.
Set the rows to 1 and drop 100 balls: two slots, roughly half each, and the bell curve is a poor fit. Set the rows to 20 and add 1,000: even slots far from the middle stay almost empty, because a route with 20 rights has odds of about one in a million. And at 50% with 12 rows, press Add 1,000 twice and compare: the exact bars shuffle a little, the overall shape does not.