Central limit theorem, simulated
Pick an ugly population. Take samples, record each sample's mean, and watch those means pile up into a bell curve anyway.
The population you are sampling from
Distribution of 0 sample means (n = 5)
What you just watched
The top chart is a population that is nothing like a bell. The bottom chart collects the mean of each random sample. Try this sequence: choose "Right-skewed", set n = 1, and draw 2,000 samples. The means just copy the population's skew, because a sample of one is only a single value. Now set n = 5 and draw again: the pile is already more symmetric. At n = 30 it sits almost exactly on the normal curve.
The theorem, in words
Take independent samples of size n from any population with mean μ and finite standard deviation σ. As n grows, the distribution of sample means gets closer to a normal distribution with:
- centre μ, the same as the population's mean;
- spread σ/√n, the standard error, which shrinks as samples get bigger;
- shape approaching normal, whatever the population looked like.
Compare the "Theory: σ/√n" and "Observed SD of means" boxes after a few thousand samples: they agree closely. Quadruple n and the spread halves.
Why it matters
The CLT is why a poll of 1,000 people can describe a country, why quality-control charts work, and why confidence intervals and z and t tests are valid even when the raw data are not normal. It is also what the Galton board and the dice sums show: add up enough independent random pieces and a bell curve appears.
The fine print
"n ≥ 30" is only a rule of thumb. Very skewed populations or ones with extreme outliers need larger samples, and populations without a finite variance (like the Cauchy distribution) never settle down at all. The samples must also be independent: 30 measurements of the same person do not count as 30 people.
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Questions students ask
What does the central limit theorem say?
For independent samples of size n from any population with mean μ and finite standard deviation σ, the distribution of the sample mean gets closer to normal as n grows, with mean μ and standard deviation σ/√n.
How large does n need to be?
The common rule of thumb is n ≥ 30, but it depends on the population. Symmetric populations need very few; heavily skewed ones need more. Try the skewed population here with n = 5 and n = 50.
Does the CLT mean my data become normal?
No. The raw data keep their shape. It is the distribution of sample means (or sums) across many samples that becomes normal.
Why does the central limit theorem matter?
It is why z and t procedures, confidence intervals and many tests work even when the underlying data are not normal.