Central Limit Theorem

TL;DR

For large enough samples, the distribution of sample means tends toward a normal distribution even if the population distribution is not normal.
This property lets you estimate population parameters from samples and justify statistical tests that assume normality.
The theorem requires a sufficiently large sample size; without it, normality-based tests may not be valid.

Definition

The Central Limit Theorem is a fundamental statistical principle that states that, given a sufficiently large sample size, the distribution of sample means will approach a normal distribution.

Explanation

The theorem applies to the distribution of sample means: when you repeatedly draw samples from a population and compute each sample’s mean, the distribution of those means will approximate a normal distribution as the sample size increases.
This approximation holds even when the underlying population distribution is not normal.
Because the distribution of sample means becomes approximately normal, we can use that normal approximation to estimate population means and standard deviations from sample data and to apply statistical tests that assume normality.

Examples

High school grades (sample of 10 students)

The grades of students in a high school are not normally distributed. If a sample of 10 students is taken and the mean of their grades is calculated, the distribution of these sample means will approximate a normal distribution as the sample size increases.

Finance — stock returns (sample of 100 stocks)

Stock prices are not normally distributed, but if a sample of 100 stocks is taken and the mean of their daily returns is calculated, the distribution of these sample means will approximate a normal distribution as the sample size increases.

Estimating a population mean (sample of 100 students)

If a sample of 100 students is taken and the mean of their grades is calculated, the Central Limit Theorem allows using that sample mean to estimate the mean of the entire population of students.

Use cases

Justifying the use of t-tests and ANOVA tests, which assume normally distributed sampling distributions.
Making inferences about population parameters (for example, estimating a population mean) based on sample statistics.

Notes or pitfalls

The theorem requires a sufficiently large sample size; if the sample size is not large enough, the distribution of sample means may not approximate a normal distribution.
Without the Central Limit Theorem, statistical tests that assume normality (such as t-tests and ANOVA) would not be valid because the distribution of sample means would not approximate a normal distribution.

Normal distribution
t-test
ANOVA
Sample mean
Population