---
title: "The Law of Large Numbers and the Central Limit Theorem"
description: "Averages settle on the population mean as samples grow, and their distribution approaches a normal one whatever the population looked like. The two results are what makes inference possible at all."
canonical: https://lightmysky.com/learn/mathematics/the-law-of-large-numbers-and-the-central-limit-theorem-mt_NknG3L5rQp
source: https://lightmysky.com/learn/mathematics/the-law-of-large-numbers-and-the-central-limit-theorem-mt_NknG3L5rQp.md
retrieved: 2026-09-12
---

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# The Law of Large Numbers and the Central Limit Theorem

Averages settle on the population mean as samples grow, and their distribution approaches a normal one whatever the population looked like. The two results are what makes inference possible at all.

Subject: Mathematics · Area: Data & Statistics · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/the-law-of-large-numbers-and-the-central-limit-theorem-mt_NknG3L5rQp

## Ready when they can

- State both results with their conditions and say what each claims
- Apply the theorem to a skewed population and check the sample size
- Separate the claim about the average from the claim about a single observation

## Lesson: Averages settle, shapes turn normal

Single observations swing wildly, but averages settle down. The law of large numbers says the sample mean converges to the true mean mu as n grows. The central limit theorem adds shape: for large n the sample mean looks approximately normal, whatever the population looked like. Its centre is mu and its spread is sigma over the square root of n. Both claims are about the average, never about one single observation.

**Example.** Watch the spread shrink. A population with sigma 12 and samples of 36 gives spread 12 over 6 = 2. With sigma 15 and samples of 25 you get 15 over 5 = 3. Skewed town incomes need bigger samples: means from n = 100 look far more normal than means from 2, 5, or 10. A working rule calls 30 large enough, though heavy skew wants more.

To use the theorem you standardise: subtract mu and divide by sigma over root n, then read normal tables. Nora flips 100 coins, sees mostly tails, and expects heads on flip 101. The theorem covers averages over many trials, so flip 101 stays 50-50 and past tails change nothing. Never confuse the average with the next single trial.

**Tip.** Check conditions before trusting the bell curve: independent trials, the same distribution, and n large for the skew. Proportions are averages in disguise, since each trial writes 1 or 0. For those, both expected successes and failures should be comfortably large, often at least 10 each.

**Recap.** Averages converge to the mean and, for large samples, vary normally with spread sigma over root n.

## Practice

13 questions on this page, each with its working shown.

## Needs first

- [Sampling Distributions and the Central Limit Theorem](https://lightmysky.com/learn/mathematics/sampling-distributions-and-the-central-limit-theorem-mt_nugaEeUSii)
- [Joint Distributions, Covariance and Independence](https://lightmysky.com/learn/mathematics/joint-distributions-covariance-and-independence-mt_YyTFTAqhSF)

## Opens up

- [Point Estimation: Bias, Variance and Consistency](https://lightmysky.com/learn/mathematics/point-estimation-bias-variance-and-consistency-mt_I374r2UU_i)
- [The Microcanonical and Canonical Ensembles](https://lightmysky.com/learn/science/the-microcanonical-and-canonical-ensembles-mt_NSCYLK-brt)
- [Brownian Motion and Its Defining Properties](https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z)
- [Overfitting and the Bias-Variance Trade-off](https://lightmysky.com/learn/computing/overfitting-and-the-bias-variance-trade-off-mt_xznyNxeNNw)