Definition
The law of large numbers (LLN) states that as the number of independent observations of a random variable grows, their sample average converges to the expected value. Flip a fair coin a million times and the proportion of heads will be very close to 0.5. The result was first proved by Jacob Bernoulli in 1713.
There are two main versions: the weak law asserts convergence in probability, and the strong law asserts almost-sure convergence. Both pin down the precise sense in which long-run frequencies stabilise.
Why it matters
How it works
For independent and identically distributed random variables X1, X2, ..., with finite mean mu, the sample average (X1 + ... + Xn) / n approaches mu as n grows. The weak law uses Chebyshev's inequality and the fact that the variance of the sample mean shrinks as 1/n; the strong law requires more delicate arguments about almost-sure convergence.
The practical consequence is that with enough data, statistical estimates become reliable. The companion result, the central limit theorem, then characterises the rate of convergence and the distribution of the deviations.