# Law of large numbers

As a sample gets bigger, the sample mean gets closer to the true mean. Early results can wobble a lot, but the average settles down.

Statistics (university) · Part 05: Sampling and estimation · Card 2 of 8 · Last updated 2026-10-19

## Analogy

A basketball player's shooting percentage after 5 shots is unreliable. After 500 shots it tells you the truth.

## Example

**Flipping coins**

10 flips: 7 heads = 70%
100 flips: 56 heads = 56%
10,000 flips: 5,030 heads = 50.3%
Share drifts toward 50%

## In practice

Casinos rely on it: single bets are random, but millions of bets follow the odds closely.

## Practice

### Flashcards

- **What does the law of large numbers say?** The sample mean approaches the true mean as n grows.

## Questions people ask

### Does the law of large numbers mean results must balance out?

No. It says the proportion settles, not that early gaps get repaired. After 10 heads in a row, the next flip is still 50:50. The early streak simply becomes a smaller and smaller share of the total as flips pile up.

### How is the law of large numbers different from the central limit theorem?

The law of large numbers says the sample mean approaches the true mean. The central limit theorem describes the shape of the leftover random error around it, showing that it is approximately normal. One gives the target, the other the pattern of the miss.

## Related concepts

- [Population and sample](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-5/statistics-university-part-5-concept-1)
- [Sampling distribution and standard error](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-5/statistics-university-part-5-concept-3)
- [Central limit theorem](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-5/statistics-university-part-5-concept-4)

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