A simple example
Roll one die and every face from 1 to 6 is equally likely—nothing like a bell curve. But average 30 dice and repeat thousands of times, and those averages form a bell-shaped normal distribution centered on 3.5. Their spread, the standard error, equals the original standard deviation σ divided by √n, so quadrupling the sample halves the error.
Why it matters
Because sample means behave normally even when the population doesn't, the CLT underpins confidence intervals, t-tests, and most p-value calculations, including a poll's "margin of error ±3 points." The n ≥ 30 rule is only a rule of thumb: heavily skewed or heavy-tailed data such as income need far larger samples, and the theorem fails for infinite-variance distributions like the Cauchy.
FAQ
- Does the data itself become normal?
- No—the distribution of the sample mean does, not individual observations.
- Is a bigger sample always better?
- Error shrinks only with √n, and a biased sample stays biased no matter how large.
Sources reviewed: 2026-10-26
References
- Fischer, H. (2011). A History of the Central Limit Theorem. Springer.
- Laplace, P.-S. (1812). Théorie analytique des probabilités.
- Wasserman, L. (2004). All of Statistics. Springer.