The Law of Small Numbers

Key Insight

Studies on phenomena like kidney cancer incidence in US counties reveal a common cognitive error: the mind's tendency to seek immediate causal explanations. For instance, extremely low or high cancer rates were observed in mostly rural, sparsely populated counties. People quickly attributed this to lifestyle factors, but a rural lifestyle cannot logically explain both very low and very high incidences.

The underlying key factor is not rurality or political affiliation, but small population size. Statistically, extreme outcomes are far more likely in small samples than in large ones. This principle can be illustrated by drawing marbles from an urn: samples of 4 marbles yield extreme results (all red or all white) 8 times more often than samples of 7 marbles, a purely mathematical fact without a causal explanation.

Therefore, the widely varying cancer rates in small counties are often statistical 'artifacts' resulting from sampling accidents, not true underlying differences. While people generally 'know' the law of large numbers, they frequently fail to apply its corollary: small samples inherently produce more extreme results. This fundamental misunderstanding reflects the human mind's difficulty with 'merely statistical' facts and its strong bias toward finding causal narratives.