The central limit theorem
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Flip to reveal answersWhat does the Central Limit Theorem say about the sample mean X̄?
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Question
What does the Central Limit Theorem say about the sample mean X̄?
Answer
For a large sample size n, X̄ is approximately Normal — even if the population itself is not Normal.
Question
What is the mean of the sample mean X̄?
Answer
It equals the population mean μ (averaging does not shift the centre).
Question
What is the standard deviation of the sample mean (the standard error)?
Answer
σ/√n — the population standard deviation divided by the square root of the sample size.
Question
As a formula, what is the approximate distribution of X̄ for large n?
Answer
X̄ ≈ N(μ, σ²/n), i.e. Normal with mean μ and standard error σ/√n.
Question
Why does the standard error shrink as n grows?
Answer
Averaging cancels out highs and lows; dividing σ by √n means larger samples give steadier (less variable) means.
Question
To halve the standard error, how much more data do you need?
Answer
Four times as much — because √n must double, and √4 = 2.
Question
When is X̄ EXACTLY Normal for any n?
Answer
When the population is already Normal — then no large-sample approximation is needed.
Question
Single value vs sample mean — which standard deviation do you use?
Answer
A single value uses σ; the mean (or total) of n values uses the standard error σ/√n.
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Topic 4.15 hub
Central limit theorem (HL only)
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