Why do bell curves show up everywhere?

It is tempting to think the bell curve is how nature likes things to be: that most people are “average” because averageness is somehow built in. That gets the cause backwards. The bell is not a property of heights or of test-takers. It is a property of addition. Take almost anything that is the total of many small pushes, each a bit random and none of them in charge, and the totals will pile up in a bell. Change how the pushes combine and the bell goes away.

The cleanest way to see this is a machine built for the purpose.

A board full of coin flips

In the 1870s the English scientist Francis Galton built a board he called a quincunx, after the five-dot pattern of its pegs. Lead shot poured in at the top and fell through rows of pegs. At each peg a ball goes left or right, and at the bottom the balls collect in bins. A ball’s bin is simply the number of times it went right. Galton described the device in detail in his 1889 book Natural Inheritance, where he called the falling balls “shot”.

Below is a model of that board in which every bounce really is an independent coin flip. Press Release balls to watch them fall, or Drop 1,000 to add a thousand at once. Then change the number of rows, or tilt the odds with the bias slider, and watch where the pile ends up.

With 12 rows there are 2¹² = 4,096 equally likely left–right paths. Only one of them ends in the far-left bin (left every time), so that bin catches about 1 ball in 4,096, or 0.02%. But 924 different paths end in the middle bin (six lefts and six rights, in any order), so it catches 924 in 4,096, about 22.6%. The middle is crowded not because balls are drawn to it but because there are vastly more ways to get there.

Counting paths like this gives the binomial distribution. With n rows and a chance p of going right at each peg, the chance of landing in bin k is

P(k) = C(n, k) · p^k · (1 − p)^{n − k}

where C(n, k), “n choose k”, is the number of paths with exactly k rights. The average bin is np, and the typical distance from it — the standard deviation, a measure of spread — is √(np(1 − p)). The short level lines in the figure are this formula; the bars are what actually happened, and they wobble around the ticks less and less as more balls land.

The green curve is something else: the normal distribution, the smooth bell defined by nothing but a centre and a spread. In 1733 the mathematician Abraham de Moivre showed that for a fair coin and many rows, the binomial counts hug this curve ever more closely. Try 3 rows, where the fit is rough, then 20, where it is hard to tell apart. Then push the bias to 10% with only a few rows: the pile leans against the left wall and the bell fits badly. Add rows and, even with the bias, the pile pulls away from the wall and becomes a bell again — it just takes longer.

It doesn’t matter what you add

A coin flip is a very simple effect: 0 or 1. The surprise is that the bell does not depend on that. Around 1810 Pierre-Simon Laplace showed that sums of many independent errors of almost any shape come out bell-shaped. This is the central limit theorem: add up many independent random effects, none of which dominates and each of which has a finite spread, and the total is close to a normal distribution — whatever the shape of the individual effects.

In the next figure, pick the shape of a single effect: a fair die, a lopsided die that lands on 1 half the time, a two-humped die that mostly lands on 1 or 6, or a coin that pays 1 only one time in ten. Then slide n, the number of them you add. The bars are the exact probabilities of each total, worked out by combining every possible outcome, not by simulation.

Start with the lopsided die at n = 1: the bars are just the die itself, a tall spike at 1 and a flat tail. Its skewness — a number that is 0 for a symmetric shape and grows as one tail gets longer — is 1.00. At n = 4 the sum already looks like a lopsided hill; by n = 30 it is a bell with skewness 0.18. The skewness of a sum of n identical effects falls like 1/√n: four times as many effects, half the lopsidedness.

The two-humped die, which looks least like a bell, turns into one surprisingly quickly: it is symmetric, so it has no skew to lose, and by about 10 effects its sum is closer to the bell than the fair die’s. The 1-in-10 coin is slowest, because almost everything about it is in its rare tail. With 60 of them the total still leans visibly — the count of rare events behaves like this, which is why counts of rare things are skewed unless there are very many chances.

The next figure tracks that same largest gap for all four effects as n grows to 200. Both axes are logarithmic, so a gap shrinking like 1/√n is a straight line sloping down. Drag across the chart, or focus it and press the arrow keys, to read the gaps at any n; pick a shape to see its Berry–Esseen ceiling as a dashed line.

All four lines end up parallel, falling a factor of ten for every hundredfold increase in n. That is the 1/√n rate. What differs is the height of the line. At n = 200 the 1-in-10 coin is still 5.9% off; the fair die was that close after only 4 throws, so the coin needs about 50 times as many terms. Even the symmetric two-humped die keeps the 1/√n rate, because a sum of dice can only be a whole number: its cumulative chance rises in small steps while the bell rises smoothly, and the steps shrink only like 1/√n.

When the bell fails

The theorem has conditions, and the bell appears only when they hold. The two that fail most often in real life are adding (the effects must combine by addition) and no giant effects (none may be capable of swamping the rest).

Many things grow by multiplying. A colony of cells, a population, or money in an investment grows by some percentage each period, and a good year multiplies whatever is already there. The next figure starts at 100 and applies n random effects, each anywhere between −40% and +40% of the current value, or (in Add mode) of the starting 100. Each setting runs the experiment 20,000 times.

In Multiply mode the pile is crammed against zero with a long tail to the right. The mean stays near 100, but the median — the middle run — ends near 57: most runs lose money, while a few lucky ones gain enormously and drag the average up. Tick Log scale and the bell comes back. Multiplying numbers is the same as adding their logarithms, so the logarithm of the result is a sum of many small independent effects, and the central limit theorem applies to it. A quantity whose logarithm is normal has a log-normal distribution; incomes (apart from the very richest), the sizes of particles from grinding and the time a disease takes to show symptoms are often close to log-normal.

Switch to Add and the bell returns on an ordinary scale. Make the effects small (±5%) in Multiply mode and the result is nearly a bell too: when every change is small, multiplying is almost the same as adding.

Now try Add, but heavy-tailed. The effects are drawn from a Cauchy distribution: most are about as small as before, but now and then one is enormous. Its tails are so heavy that it has no finite standard deviation, so the theorem’s condition fails. The sum never settles into a bell. Instead of growing like √n, the spread grows in proportion to n, and a single huge effect often decides the total on its own. With 20 effects, about half of all runs land more than four standard deviations (of the ordinary adding version) from 100; a true bell puts only 0.006% of runs there. Press Run again a few times and watch the sample mean in the Mean readout (and the skewness) leap about: a Cauchy distribution has no average at all, so the average of a sample never settles down. Its median does, which is why the median is still a fair summary.

So the rule of thumb is this. Expect a bell when a quantity is the sum of many small, independent nudges, none in charge. Expect a skewed, log-normal shape when the nudges multiply. And expect fat tails, with occasional extreme outliers, when one effect can dwarf the rest — as in earthquake sizes, insurance losses or the price swings of a market crash.