---
title: "Sampling Distributions and the Central Limit Theorem"
description: "The mean of a sample is itself a random variable. Across repeated samples it centres on the population mean with a spread of σ divided by the square root of n, and its shape approaches a normal one as"
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source: https://lightmysky.com/learn/mathematics/sampling-distributions-and-the-central-limit-theorem-mt_nugaEeUSii.md
retrieved: 2026-09-12
---

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# Sampling Distributions and the Central Limit Theorem

The mean of a sample is itself a random variable. Across repeated samples it centres on the population mean with a spread of σ divided by the square root of n, and its shape approaches a normal one as n grows.

Subject: Mathematics · Area: Data & Statistics · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/sampling-distributions-and-the-central-limit-theorem-mt_nugaEeUSii

## Ready when they can

- Describe how the spread of sample means changes as the sample gets bigger
- Use the σ over root n result for a stated sample size
- Explain why a mean from a large sample is more trustworthy than one from a small sample

## Lesson: Crowds behave better than individuals

The mean of a sample is itself a random variable, centred on the true population mean. Across repeated samples the means cluster tighter as samples grow: averaging lets highs cancel lows. That is why a mean from 100 people beats a mean from 4.

**Example.** The spread of sample means is sigma over root n. With population spread 12 and samples of 36, root 36 = 6 and 12/6 = 2. Quadruple the sample to halve the spread: with spread 20, samples of 25 give 4 while samples of 100 give 2.

For large samples the distribution of sample means turns approximately normal, whatever shape the population started with. About 30 is the usual rule of thumb, sooner if the population already looks symmetric. Bigger samples never shift the expected mean, only tighten the spread.

**Tip.** Odd individuals dominate tiny groups but cancel out in crowds, so big means sit nearer the truth. Sketch the curve of means, mark the target, and convert with the standard error instead of sigma. The only twist versus ordinary normal problems is that extra division by root n.

**Recap.** Sample means centre on the true mean with spread sigma over root n, turning normal as samples grow.

## Practice

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

## Needs first

- [Sampling Methods and Their Trade-offs](https://lightmysky.com/learn/mathematics/sampling-methods-and-their-trade-offs-mt_aSsRQvnKa6)
- [Finding Probabilities and Values from a Normal Model](https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos)

## Opens up

- [Setting Up a Hypothesis Test](https://lightmysky.com/learn/mathematics/setting-up-a-hypothesis-test-mt_JmpLNHK0Nf)
- [The Law of Large Numbers and the Central Limit Theorem](https://lightmysky.com/learn/mathematics/the-law-of-large-numbers-and-the-central-limit-theorem-mt_NknG3L5rQp)
- [Confidence Intervals for Means and Proportions](https://lightmysky.com/learn/mathematics/confidence-intervals-for-means-and-proportions-mt_Pg2fPDW1sn)
- [Overfitting and the Bias-Variance Trade-off](https://lightmysky.com/learn/computing/overfitting-and-the-bias-variance-trade-off-mt_xznyNxeNNw)