Central limit theorem

Math

Definition

A fundamental theorem stating that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's original distribution. Generally, n ≥ 30 is considered sufficient for the approximation.

Examples

  • Even if individual die rolls are uniformly distributed, the average of 50 rolls will be approximately normally distributed
  • Quality control: the mean weight of samples of 40 cereal boxes will follow a normal distribution even if individual box weights are skewed
Key Fact

For large n: x̄ ~ N(μ, σ/√n) regardless of population shape

Related Terms

Normal distribution Sampling distribution of a mean Standard deviation Confidence intervals for proportions

Study This Concept

Practice central limit theorem with free review games in these units:

AP Statistics Unit 5: Sampling Distributions