Central Limit Theorem

The Central Limit Theorem (CLT) is a key concept in statistics that tells us when we take a large number of samples from any population, the distribution of the sample means (averages) will be approximately normal (shaped like a bell curve). This happens regardless of the original distribution of the population. It’s a powerful idea because it allows us to use normal distribution techniques for statistical inference even when the population is not normally distributed.

According to the CLT, as you increase the size of your samples ($n$), the distribution of these sample averages will look more and more like a normal distribution. The mean of this distribution of averages will be the same as the population mean ($\mu$), but the spread (standard deviation) will get smaller as you increase your sample size. This smaller standard deviation in the distribution of sample averages is calculated as $\sigma / \sqrt{n}$.

This concept is very useful in statistics because it means that by using larger samples, we can get more accurate estimates of population parameters.

Random Distribution Left Skewed Right Skewed Pick 30 Samples from Data Pick 30 Samples 1,000 Times