Law of Small Numbers, explained.
Belief in the law of small numbers is an overexpectation that small samples will closely represent the population from which they are drawn.
Why it happens
Twenty groups each perform five fair flips; another twenty each perform 100. Predict which set is likelier to contain heads fractions at or below 20% or at or above 80%, then inspect realised proportions and extreme counts.
Belief in the law of small numbers is an overexpectation that small samples will closely represent the population from which they are drawn.
Read the result
Compare proportions rather than absolute heads counts. An extreme rate is more plausible in a tiny sample even when every group has the same generating probability.
A worked example
Two sets of teams with identical odds
Predict whether five-flip or hundred-flip groups will more often show heads fractions outside the twenty-to-eighty percent middle range.
Reveal both sets of twenty groups. Count the extremes without assuming the small groups have different coin probabilities.
Use a new round to see batch variation. A surprising small-sample result is not by itself evidence of a special underlying process.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Twenty groups each perform five fair flips; another twenty each perform 100. Predict which set is likelier to contain heads fractions at or below 20% or at or above 80%, then inspect realised proportions and extreme counts.
Where this idea is useful
A practical use
A tiny team's apparently exceptional success rate may partly reflect sampling variability rather than an unusually strong process.
A common misconception
“Every small sample should look approximately like its population.”
Sampling variability is greater in proportions based on fewer independent observations. Representativeness is an expectation to examine, not a rule each sample obeys.
What this explanation leaves out
- All flips are independent and fair. Particular batches can depart from the expected pattern. This exercise does not diagnose anyone's bias or explain every difference between small and large organisations.
Does a hundred-flip group never look extreme?
It can. The tail event is much less likely under fair independent flips, but the model assigns it a positive probability.
How much of an impressive rate could reflect the denominator rather than a different mechanism?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.