What Is Central Limit Theorem?
The central limit theorem is one of the most important results in statistics. It states that if you take many random samples from a population and compute the mean of each, the distribution of those sample means will be approximately normal — even when the underlying population is not normal — provided the samples are large enough.
Because of this theorem, the sampling distribution of the sample mean has a predictable center and spread: its mean equals the population mean μ, and its standard deviation, called the standard error, equals σ divided by the square root of the sample size n. This is what allows confidence intervals and hypothesis tests about means to work in practice.
Formula
Mean of sample means: μₓ̄ = μ
Standard error of the mean: σₓ̄ = σ / √n
where:
μ = population mean
σ = population standard deviation
n = sample size
Applications
- Estimating the average height of a population from many small samples
- Building confidence intervals for the mean of a production process
- Designing sampling plans for quality control inspections
- Justifying normal-based methods even when raw data are skewed
Sources
- Moore, D. S., McCabe, G. P., & Craig, B. A. (2021). Introduction to the Practice of Statistics (10th ed.). Macmillan.
- Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers (7th ed.). Wiley.