Central Limit Theorem

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Edexcel A Level Further Maths — Further Statistics 1

The Central Limit Theorem

If you take a big enough sample from literally any population and average it, that average behaves like a normal distribution — even if the original population doesn't look normal at all.

Quick Summary

A random sample is a set of independent, identically distributed copies of a population variable X.
The sample mean, X̄, is not a fixed number — it changes depending on which sample you happen to draw.
If the population itself is normal, then X̄ ~ N(μ, σ²/n) exactly, for any sample size n.
The Central Limit Theorem (CLT) says that for any population distribution, as long as n is large, X̄ is approximately N(μ, σ²/n).
You always need to find μ = E(X) and σ² = Var(X) of the population first, then divide the variance by n.
Bigger samples → sample means cluster tighter around μ → smaller spread.

1. Samples and the Sample Mean

1.1 What is a "random sample" of a random variable?

Imagine you flip a fair coin 10 times and count the heads. That's one trial of X ~ B(10, 1/2), and here E(X) = np = 5, Var(X) = np(1-p) = 2.5.

Now suppose you repeat this whole experiment (10 flips → count heads) once a day, for 7 days. Each day gives you a new random variable:

X₁ ~ B(10, ½) X₂ ~ B(10, ½) … X₇ ~ B(10, ½)

Think of it like this: X is the "recipe" (the distribution). X₁, X₂, ..., X₇ are seven separate meals cooked from the exact same recipe. They're all made the same way (identical distributions), but that doesn't mean each meal tastes exactly alike (they're still independent, random outcomes). One day you might get 4 heads, the next day 7, the next day 5 — even though every day used the same underlying B(10, ½) rule.

X here is called the population distribution, with population mean μ and population variance σ². The collection X₁, X₂, ..., Xₙ is called a random sample of size n — each Xᵢ is an independent copy of X.

1.2 The sample mean, X̄

Definition

X̄ = (X₁ + X₂ + ... + Xₙ) / n

Plain English: add up all n observations in your sample, then divide by how many there are — exactly like an ordinary average, except here the whole thing is treated as a random variable in its own right.

Crucial idea: X̄ is not a fixed number. Every time you draw a fresh sample of size n, you get a slightly different X̄. Do this 7-day coin experiment once and you might get a sample like 4, 6, 5, 5, 3, 5, 6 → mean 4.857. Do it again next week and you'll get a different set of 7 numbers, and a different mean. If you did this hundreds of times, all those different sample means would themselves form a distribution — and that distribution is what the rest of this chapter is about.

Why this matters Statisticians almost never know the true population mean μ. What they usually have is one sample, and they use its mean X̄ as an estimate of μ. To judge how trustworthy that estimate is, we need to know how X̄ itself is distributed — how spread out sample means tend to be around the true μ.

1.3 The distribution of X̄ when the population IS normal

If your random sample X₁, X₂, ..., Xₙ comes from a normal population, X ~ N(μ, σ²), then the sample mean has an exact distribution too:

Distribution of the sample mean (normal population)

X̄ ~ N( μ, σ²/n )

Mean of X̄

The catch only

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Also in the full note

2. The Central Limit Theorem (CLT)
What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes
2.1 What does the CLT actually say?
2.2 Worked mini-example: Poisson population
2.3 Finding μ and σ² from the population
2.4 Full worked example (exactly how exam questions are phrased)

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