The Law of Small Numbers

The Law of Large Numbers says that as a sample size grows, its mean gets closer to the average of the whole population.

The Law of Small Numbers is a sarcastic term coined by psychologists Tversky and Kahneman. It refers to the mistaken belief that small samples represent the population.

They do not.

The Kidney Cancer Map

A study looked at kidney cancer rates in US counties. The counties with the lowest rates were mostly rural, sparsely populated, and Republican.

Why? Is it the clean air? The hard work?

Then they looked at the counties with the highest rates. They were also rural, sparsely populated, and Republican.

The Explanation

It is not politics. It is math.

In a small county with 1,000 people, one cancer case changes the rate dramatically.

If you have 0 cases, your rate is 0%. If you have 1 case, your rate is 0.1%.

In a city of 1 million, one case changes nothing.

Small samples have high variance. They swing wildly from the mean. We see these swings and invent stories to explain them.

The Diagnosis

When you see an extreme statistic—"The safest city," "The worst school"—check the size.

If the N is small, the result is likely noise. Ignore it.