The Central Limit Theorem (CLT)

In the real world, we rarely know the true distribution of a population. Are stock returns normally distributed? Is trade volume exponentially distributed? We often don't know, and the data is usually messy.

The Central Limit Theorem provides a powerful, almost magical solution. It guarantees that if we take a large enough number of samples from any population and calculate the mean of each sample, the distribution of those sample means will be approximately normal (a bell curve).

This is the bridge from messy, unknown real-world data to the predictable world of statistical inference. It allows us to use the properties of the normal distribution to perform hypothesis tests and construct confidence intervals for a population's mean, even when we know nothing about the population itself.