What is the Central Limit Theorem?

The Central Limit Theorem (CLT) is a fundamental concept in probability theory and statistics. It states that, under certain conditions, the sum of a large number of independent and identically distributed (iid) random variables will be approximately normally distributed, regardless of the underlying distribution of the individual variables.

More specifically, the Central Limit Theorem states that if we take a sample of size n from any population, with a finite mean and variance, and calculate the sample mean of that sample, then as n approaches infinity, the distribution of the sample mean will approach a normal distribution with mean equal to the population mean and variance equal to the population variance divided by n.

This result is important because it allows us to use the normal distribution to make inferences about the population mean, even if we don’t know anything about the underlying distribution of the population. The Central Limit Theorem is widely used in statistical inference, hypothesis testing, and confidence interval estimation, among other applications.