The central limit theorem
The central limit theorem says that for independent identically distributed observations with finite variance, the sample mean is approximately normal for large n — WHATEVER the population distribution. That universality is what makes normal-based inference apply to skewed data, and the usual rule of thumb is n ≥ 30, though heavily skewed populations need more.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Check independence and finite varianceThe CLT needs both. Heavy-tailed distributions without finite variance are genuine exceptions.
- Identify n and decide adequacyn ≥ 30 is the standard rule; near-symmetric populations work with much less, strongly skewed ones need more.
- Write the approximating distributionx̄ ≈ N(μ, σ²/n), or for a sum, ΣX ≈ N(nμ, nσ²).
- Standardise and use normal tablesz = (x̄ − μ)/(σ/√n), then proceed exactly as for any normal probability.
Worked example
A population has μ = 20 and σ = 6. For n = 36, find P(x̄ > 22).
- By the CLT, x̄ ≈ N(20, 6²/36) = N(20, 1), so the standard error is 1.
- Standardise: z = (22 − 20)/1 = 2.
- P(Z > 2) = 1 − Φ(2) = 1 − 0.9772.
- = 0.0228.
Answer. About 2.3% — even though the population shape is unspecified, the CLT makes the sample mean approximately normal.
Where marks get dropped
These are the specific errors that cost credit on the central limit theorem questions — QED's rubric penalises each of them separately.
- Claiming the CLT makes the DATA normal. It applies to the sampling distribution of the mean, not to individual observations.
- Using σ/√n as though it were σ when standardising a single observation. The two questions need different denominators.
- Applying the CLT to dependent observations, such as a time series with strong autocorrelation, where the effective sample size is much smaller.
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The central limit theorem — frequently asked questions
How large must n be?
n ≥ 30 is the conventional rule. A symmetric population may need only n = 5, while a strongly skewed one may need several hundred.
Does the CLT apply to proportions?
Yes — a proportion is a mean of 0/1 indicators, which is why p̂ ≈ N(p, p(1−p)/n) when np and n(1−p) both exceed about 10.
Are there distributions where it fails?
Yes, those with infinite variance such as the Cauchy. The sample mean of Cauchy observations is Cauchy again, no matter how large n gets.
The rest of Statistics
Describing data, distributions, estimation and hypothesis tests. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Mean, median & mode
- 2Variance, standard deviation & spread
- 3Shape, skew & outliers
- 4Boxplots, histograms & quartiles
- 5The normal distribution & z-scores
- 6Sampling, bias & the sampling distribution
- 7The central limit theorem
- 8Confidence intervals for a mean
- 9Hypothesis testing & p-values
- 10t-tests & comparing two means
- 11Chi-square tests for independence
- 12Correlation & least-squares regression
- 13Type I / Type II errors & power
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