What does the Central Limit Theorem state?

The Central Limit Theorem asserts that as the sample size increases, the sampling distribution of the sample mean will tend to follow a normal distribution, regardless of the shape of the population distribution from which the samples are drawn. This principle holds true typically when the sample size is 30 or more, though in practice, the normal approximation may become valid even for smaller sample sizes if the population distribution is not heavily skewed.

The theorem is foundational in statistics because it enables statisticians to make inferences about population parameters using sample statistics, which is crucial when the population distribution is unknown. It indicates that the means of sufficiently large samples will resemble a normal distribution, allowing for the use of normal distribution principles in hypothesis testing and confidence interval estimation.

Understanding this concept allows students to effectively apply statistical techniques and make valid conclusions based on sample data, widening the applicability of inferential statistics even when direct observations of the population are challenging.