Roll one die, then two, then ten. The distribution of the average converges to a bell curve.
From the chapter: Chapter 4: Probability
Glossary: central limit theorem, gaussian distribution
Transcript
A single die has a flat distribution. Six values, each equally likely.
Roll two dice and add. The sum runs from two to twelve, but it is no longer flat. Seven is the most likely outcome. The distribution forms a triangle.
Roll three. The shape rounds. The peak is at ten and a half, the tails grow shorter.
Roll five. Roll ten. Roll thirty. With each new die added to the sum, the distribution looks more and more like a Gaussian. A bell curve.
This is the central limit theorem. The sum, or equivalently the average, of many independent random variables converges to a Gaussian, no matter what shape each individual variable had to begin with.
The same thing happens when we roll loaded dice. When we draw from an exponential. From a uniform. The shape of each one washes out.
What survives is just the mean and the variance. Two numbers describe the limit.
This is why the Gaussian shows up everywhere in statistics, in measurement error, in noise models. It is the universal attractor of summation. Every time many small independent effects add up, you get a bell curve, no matter what the small effects looked like.