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Data Analysis › Statistics

Law of Large Numbers

Averages settle toward the true value as you gather more observations.

Data Analyst · Mid-level

Also known as: LLN, law of averages, law of large numbers

The law of large numbers says that as the number of independent observations grows, the sample mean converges on the expected value of the underlying distribution. Flip a coin a thousand times and the proportion of heads is usually close to a half; flip it a million times and it is usually closer still.

It is a statement about long-run behaviour. It does not say that a run of tails is owed a head, and it does not say that any particular sample will be close. Small samples stay noisy, which is exactly why the standard error shrinks as 1/√n rather than in proportion to n.

Where it changes an analysis:

  • A short window is a weak estimate of a rate. Tuesday’s conversion rate is a much worse number than the same rate for the whole quarter, and comparing two short windows invites regression to the mean to be read as a real change.
  • More rows is not the same as more information. The law applies to the mean of independent observations, not to the sum of correlated ones. A metric that counts the same active user across 200 sessions is not 200 times as informative.
  • Convergence is slow. Halving the error takes four times the data. That arithmetic is what drives sample size planning.

The mistake is to treat “we have a lot of rows” as an answer to “is this estimate trustworthy?”. Volume fixes noise, not bias: a million rows drawn from a biased sample converge confidently on the wrong number.

For the related claim about how the spread of averages behaves, see the central limit theorem; for what a range around an estimate means, see confidence intervals.