Law of large numbers
The law of large numbers states that the sample mean converges in probability to the true population mean as sample size increases without bound, providing the theoretical basis for using sample statistics to approximate population.
See it move
The card shows P(|x̄ₙ − μ| > ε) → 0 as n → ∞: the probability that the sample mean x̄ₙ strays from the true mean μ by more than any tolerance ε shrinks to zero as sample size n grows. Three points follow: convergence needs independent, identically distributed draws with an existing mean; it says nothing about how large n must be; and it underpins every sampling-based estimate.
The formula
Variables
- sample mean of n independent, identically distributed observations (units of X)
- population mean (units of X)
- any fixed positive tolerance (units of X)
- sample size (dimensionless)
- probability (dimensionless)
Weak law of large numbers: the sample mean converges in probability to the population mean as n grows without bound.
Check yourself
A statistician repeatedly rolls a fair six-sided die, recording the running average after each roll. The population mean of a single fair die roll is 3.5. Which statement best describes what the law of large numbers guarantees about the running average as the number of rolls increases?