Central limit theorem
The central limit theorem states that the sample mean's distribution approaches normal as sample size grows, regardless of population shape — making normal-based inference valid for large samples.
See it move
However skewed the underlying population, the distribution of sample means becomes increasingly bell-shaped as sample size n grows — the central limit theorem. Its spread, the standard error, equals σ ÷ √n, so larger samples cluster more tightly around the true mean. By around n = 30, the normal approximation is usually reliable enough to justify z-tests and confidence intervals.
The formula
Variables
- Population standard deviation
- Sample size
- Standard error: standard deviation of the sampling distribution of x̄
As n grows, SE falls and the sample mean clusters more tightly around μ.
Variables
- Sample mean
- Population mean
- Population standard deviation
- Sample size
By the CLT, this z-score is approximately standard normal for large n (typically n ≥ 30), regardless of the population's distribution shape.
Check yourself
A food retailer's weekly units sold per product follows a heavily right-skewed distribution (many slow sellers, a few blockbusters). An analyst draws 600 random samples of 45 products each and records the mean units sold for each sample. What does the central limit theorem predict about the distribution of those 600 sample means?