When is a distribution normal?

Question

I see a lot of tutorial videos and papers that teach you how to use the normal distribution to find probabilities or that explain what the terms 'variance' and 'mean' mean, but I can't find one that gives me an explanation on why the function should even be a normal distribution.... I would like to know a real life example and a proof that a distribution function of one variable, say $x$, is actually normal without involving more than one dimension (like the darts example, where they use rotational symmetry).

Answer 1

There are lot of things that are not normally distributed... money in the economy, the brightness of stars, the magnitudes of earthquakes.

But, if you sum (or average) enough non-normally distributed random variables together, they the sum (or average) will increasingly resemble a normal distribution. This is proved in the Central Limit Theorem.

Now there are plenty of processes that are approximately normally distributed -- grades in a class, the heights of the students, etc. But if the height truly followed a normal distribution there would be a non-zero probability of having a student with a negative height.