The "law of small numbers" is Kahneman and Tversky's ironic name for a real statistical mistake: treating small samples as if they were as reliable as large ones, which leads to seeing meaningful patterns in what's actually just the higher natural variability of small datasets. Small samples don't just produce results that are randomly off — they systematically produce more extreme results in both directions, and it's tempting to invent a causal story for whichever extreme happens to show up.
The mistake is dangerous specifically because it doesn't feel like guessing — it feels like spotting a real pattern. Recognizing when a striking result might just be small-sample noise, rather than reflexively reaching for a causal explanation, is one of the most consistently useful and consistently ignored statistical instincts to build.
Statisticians noticed that U.S. counties with the very lowest rates of kidney cancer tend to be rural, sparsely populated, and often politically similar to one another — a pattern that invites an obvious causal story about rural living. But the counties with the very highest rates share the exact same profile: rural and sparsely populated. The pattern is a pure statistical artifact — small populations produce more extreme rates in both directions purely by chance, with no real causal link to rural life at all. Kahneman notes the same error, at a much larger scale, drove a real philanthropic decision: an analysis found small schools overrepresented among a state's best-performing schools, leading major foundations to invest heavily in breaking up larger schools — without noticing that small schools were equally overrepresented among the worst-performing schools too.