Picturing the central limit theorem

The central limit theorem tells us that if we add data that contains random errors then we tend to get a sum that's normally distributed, independent of the distribution of the original data. I was wondering how many errors you need to add together to get something that looked close to a normal distribution, so I created some random data from a uniform distribution in Mathematica that represents such errors. I then and plotted one of these errors, the distribution of the sum of two of these errors and the distribution of the sum of three of these errors. Here's what I got.

One random error.

The distribution of the sum of two random errors.

The distribution of the sum of three random errors.

It certainly looks like you're fairly close to a normal distribution after adding together only three errors. I was surprised to see how quickly this happened.