Foundations · Sampling distributions
Central Limit Theorem Calculator
Why averages are so well-behaved. Give a population's mean, standard deviation and a sample size, and see the sampling distribution of the sample mean — centred on the same mean but narrower by a factor of √n — with its standard error and probabilities like “how likely is a sample mean below this value?”. The population can be any shape; the mean's distribution is still a bell.
The sample mean averages n draws. The central limit theorem says that average is approximately normal — N(μ, σ∕√n) — almost whatever the population looks like, once n is reasonably large.
Result
In plain English
Individual values can be distributed any way at all — incomes are skewed, dice are flat, defects are lumpy. The central limit theorem is the near-magical result that the average of a sample doesn't inherit that mess: as the sample grows, the distribution of the sample mean settles into a normal bell curve, centred on the true mean and narrowing in proportion to √n. It is why so much of statistics can lean on the normal distribution even when the raw data are nothing like it.
- sampling distribution of the mean
- If you took many samples of size n and plotted each one's average, this is the distribution you'd see — not the spread of raw values, but the spread of the averages.
- standard error (SE)
- The standard deviation of that sampling distribution: σ ∕ √n. It measures how much a sample mean typically misses the true mean by, and it shrinks as n grows.
- the √n law
- Precision improves with the square root of sample size: to halve the standard error you need four times the data, not twice.
- what it does not say
- The population does not become normal, and individual values do not. Only the distribution of the mean (and of sums) is pulled toward normal.
- how large is large?
- n ≥ 30 is the usual rule of thumb, but heavily skewed or heavy-tailed populations need more; near-symmetric ones need far less.
Frequently asked
What does the central limit theorem actually say?
That the sampling distribution of the sample mean (or sum) of independent draws from a population with finite variance approaches a normal distribution as the sample size n grows — regardless of the population's own shape. The mean of that distribution is the population mean μ, and its standard deviation is the standard error σ ∕ √n. It is the reason the normal distribution shows up everywhere in inference.
What sample size is needed for the central limit theorem?
The common rule of thumb is n ≥ 30, but it is only a rough guide. A nearly symmetric population is close to normal in the mean for n as small as 5–10; a strongly skewed or heavy-tailed one (incomes, insurance claims) may need hundreds before the sample mean looks normal. When in doubt, the more skew, the more data.
What is the difference between the standard deviation and the standard error?
The standard deviation σ measures the spread of individual values and does not change with sample size. The standard error σ ∕ √n measures the spread of the sample mean, and shrinks as you collect more data. The central limit theorem is what connects them: with n observations, the mean's standard deviation is √n times smaller than a single value's.
Does the central limit theorem apply to statistics other than the mean?
A version does. The theorem is usually stated for the sample mean and for sums, but many statistics — a proportion, a difference between two means, a regression coefficient, even the median under the right conditions — have sampling distributions that approach normal as the sample grows. The mean is simply the cleanest, most-quoted case. What the theorem never does is make the raw data normal; only the distribution of the summary statistic is pulled toward the bell.