Definition
The central limit theorem (CLT) states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the original population distribution. This is true as long as the samples are independent and the sample size is sufficiently large.
How It Works
No matter what the original data looks like - skewed, uniform, bimodal - the averages of repeated samples will form a bell curve.
Rolling a single die gives a flat (uniform) distribution - each number from 1 to 6 is equally likely.
But if you roll 30 dice and record the average, then repeat this 1,000 times, the distribution of those averages will be bell-shaped, centered around 3.5.
The more dice per roll, the closer the distribution of averages gets to a perfect normal curve.
Why It Matters
The central limit theorem is arguably the most important theorem in statistics. It justifies the use of confidence intervals, hypothesis tests, and many other methods that assume normality. Without the CLT, these tools would only work on data that is already normally distributed, which is rare in the real world.
The CLT also explains why averages are more reliable than individual measurements. The variability of sample means decreases as sample size increases (by a factor of 1/square root of n), which is why larger studies produce more precise estimates.
The central limit theorem guarantees that sample averages are approximately normal for large enough samples. This is why most statistical methods work regardless of the shape of the original data.