Distributions

Where does the bell curve come from?

Watch it emerge live — from something as dumb as a ball bouncing off pegs.

Below is a Galton board: a triangle of pegs with bins along the bottom. Every ball you drop makes n tiny coin flips as it falls — left or right at each row. Its final bin equals the number of "rights" it took. Drop a hundred balls and watch what shape the pile settles into.

The Galton board · a bell curve building itself

Rows of pegs · n12

Balls landed

In middle bins

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the mode of the bell

🔵 blue = balls piling up in bins · red curve = the normal N(n/2, √(n/4)) the CLT predicts. The pile should hug the curve after a few hundred drops.

Try n = 4. Only 4 coin flips per ball — so only 5 possible bins. The pile is clearly discrete but already vaguely bell-shaped. Push n up to 18. Now each ball is a tiny statistical process, and the pile smooths into a proper bell. That transition, from "wobbly" to "bell", is the CLT.

It works for anything, not just coins

The magic isn't the coin flip. Whenever you sum many small independent random things — whatever they individually look like — the total is approximately normal. Pick any source distribution below (even a skewed one) and average n draws together. Bell curve, every time.

Any source · bell curve out

Source distribution n · draws per sample mean30

Empirical mean

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Empirical std

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theory σ/√n

🔵 blue = the raw source distribution · 🟠 orange = the sample-mean distribution · red = the normal the CLT predicts

Try n = 1. The orange histogram is the source. Now push n up. Watch how quickly (or slowly, for lognormal) the shape symmetrises into a bell.

The √n rule

The bell doesn't just appear — it also tightens. Doubling n only shrinks its width by √2. That's why quadrupling the sample size halves the standard error, not doubling.

Empirical std vs sample size · log-log

Slope on log-log should be −½ if the √n rule holds. Every step to the right = 4× the sample size = ½× the error.

The small-schools puzzle. A famous US education initiative funded splitting big schools into small ones because small schools kept showing up among the top test performers. But small schools also showed up among the bottom performers — for exactly the same statistical reason. σ/√n is bigger when n is small. Wider extremes on both sides. The signal wasn't school quality; it was sample size.

Related models

📈 Power laws — the opposite tail shape. Thin tail vs heavy tail, bell curve vs straight log-log line, same data generators flip the result.
🚶 Random walks — walker positions become normal by the CLT. The √t rule in Ch 13 is CLT in disguise.
🎲 Entropy — the normal distribution is max-entropy given a mean and variance, so this one is literally the max-ignorance answer.

← Entropy Power laws →