Maths › Further Statistics 1 › The Central Limit Theorem
The Central Limit Theorem
Average enough independent observations from almost any distribution and the average behaves normally, with a spread that shrinks as the square root of the sample size.
Builds on The normal distribution and The Poisson distribution.
IN THIS TOPIC
- State the Central Limit Theorem and the distribution it gives the sample mean.
- Compute probabilities for a sample mean from any parent distribution.
- Recognise that the standard error falls with the square root of n.
WHAT YOU PROBABLY THINK
The sample mean can only be treated as normal when the population it comes from is itself normal.
Averages turn normal
For a population with mean μ and variance σ², the mean of a large independent sample is approximately normal, whatever shape the parent has:
The parent may be skewed, discrete, or bounded; the averaging smooths all of that away, which is precisely what the opening claim denies. The variance of the mean is σ²/n, so the standard error σ/√n falls with the square root of the sample size: quadrupling n halves the spread.
WORKED EXAMPLE
A probability for a sample mean
A population has mean 50 and standard deviation 12, with unknown shape. For a sample of 36, find the probability the sample mean exceeds 53.
By the Central Limit Theorem, the mean is approximately N(50, 144/36), that is N(50, 4) with standard error 2.
z = (53 − 50)/2 = 1.5.
P(Z > 1.5) = 0.0668. The parent's shape never entered the calculation, which is the whole point of the theorem.
Where the square root bites
Because the standard error carries √n rather than n, precision is expensive: halving the spread of the sample mean costs four times the data. That single fact governs how large surveys are designed and why sample sizes climb so steeply as the required margin narrows.
YOUR TURN
Sizing a sample
A population has standard deviation 20. How large must a sample be for the standard error of the mean to be at most 2?
Show the working
Standard error = 20/√n ≤ 2, so √n ≥ 10.
n ≥ 100.
Reducing the standard error to 1 would need n ≥ 400: halving the error quadruples the sample.
THE EXAM BIT
- Quote the theorem's conclusion as a distribution: mean μ, variance σ²/n, approximately normal.
- Divide the variance by n, not the standard deviation; the standard error is σ/√n.
- Say 'approximately' and mention that n is large; the theorem is an approximation, not an identity.
- The parent distribution's shape is irrelevant for the sample mean, but say so rather than ignoring it.
CHECK YOURSELF
A distribution has mean 30 and variance 100. Write down the approximate distribution of the mean of a sample of 25.
Show a hint
Divide the variance by n.
Show the answer
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For large n the sample mean is approximately N(μ, σ²/n), whatever the parent distribution.
The standard error σ/√n falls with the square root: four times the data for half the spread.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- State the Central Limit Theorem and the distribution it gives the sample mean.
- Compute probabilities for a sample mean from any parent distribution.
- Recognise that the standard error falls with the square root of n.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.