What is the Central Limit Theorem and why is it important in statistics?

Example Answer

The Central Limit Theorem explains why averages often have an approximately normal sampling distribution, even when individual observations are not normally distributed. In the standard version, observations are independent and identically distributed with finite, nonzero variance. After centering and scaling, the sample mean approaches a standard normal distribution as the sample size grows. Its standard error is sigma divided by the square root of n.

That gives us a foundation for many approximate confidence intervals and tests. But thirty observations is not a universal cutoff, and the theorem does not make the raw data normal. Strong skew, rare events, dependence, or heavy tails can make a normal approximation poor. For example, a 10% event rate produces only three expected events in thirty observations. I would check the design and distribution rather than treat the sample count as a guarantee.

Why this works

It states the standard theorem’s assumptions and connects the approximation to a practical limitation.

Follow-up to prepare for

What changes with clustered observations, or with a Cauchy distribution that lacks a finite mean and variance?