Everybody knows about the Central Limit Theorem, but have you ever seen a visual demonstration?

The Central Limit Theorem states that, given certain conditions, the mean of a large number of iterates of independent random variables will be approximately normally distributed, regardless of the underlying distribution.

Formally,

Let {X₁, … , X_n} be a sequence of independent and identically distributed random variables drawn from distributions of expected values given by µ and finite variances given by σ², then

        has a distribution like that of a standard normal distribution N(0,1) for large values of n.

Formulas are nice (if you can understand them!), but it’s always easier to learn things when given a visual demonstration. So let’s try!

As the theorem states, the underlying distribution is not a problem. Therefore, let’s choose an exponential distribution with labmda equal to two for our example.

We draw one thousand random sample of size two from this exponential distribution, take the mean of each pair of two, and plot the histogram of the results.

In this case, the n of theorem would be two, and as we can observe the distribution doesn’t look like a normal distribution:

If we take samples of size ten (n is now ten) and repeat the previous process, the distribution is a little bit more normal:

And as n gets larger, it’s easy to see how the distributions of the sample mean looks more like a normal distribution.

So that’s it! Here you have a nice, easy way to understand what the Central Limit Theorem says!

If you want a bit more fun understanding this theorem, go and visit this video from The New York Times.