The Galton Board

In the 1870s Francis Galton built a board to make a point he could not make with words alone. Lead shot poured from a funnel at the top, bounced through rows of pins, and collected in slots at the bottom. He called it a quincunx, after the five-spot pattern of the pins, and in Natural Inheritance (1889) he used it to show something that still feels slightly wrong when you watch it: a crowd of completely unpredictable events producing a completely predictable shape.

Watch a single bead and there is nothing to learn. At every pin it goes left or right, and no amount of staring tells you which. Watch a thousand and the pile has the same outline every time: tall in the middle, falling away symmetrically, thin at the edges. That outline is the binomial distribution, and as the rows increase it becomes the normal curve, the bell. The curve on this board is drawn from a formula before any bead has fallen. The beads have no way to know it is there, and they fill it anyway.

The tiles below the board keep score. The centre of the pile settles on the number of rows times the chance of going right. Its spread settles on the square root of the rows times both chances. And the share of beads sitting in the wrong bin, compared with the exact prediction, keeps falling as you pour, at a rate you can also predict: a hundred times more beads brings the pile ten times closer. Chance does not average out by being cancelled. It averages out by being outnumbered.

Switch to Sum of dice and every row now rolls a die instead of meeting a pin, moving the bead anywhere from zero to five slots. Load the dice so they favour sixes. The step is now lopsided and nothing like a coin, and after a dozen rows the pile is a bell again. This is the central limit theorem: add up enough small, independent random nudges of almost any kind and their total is normally distributed. It is why heights, measurement errors and exam marks so often fall on the same curve, and why the bell turns up in places nobody put it.