Why does the normal distribution describe data collected in real life so well?

Question

$$ P(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp \left( - \frac{(x-\mu)^2}{2\sigma^2} \right) $$

Is there any intuition behind choosing $e^{-x^2}$ instead of some other function?

Answer 1

The question, at least in the way you asked it, really doesn't make much sense, because that funny exponential formula is the definition of "normal".

Your comments sound like you sort of don't get what I'm getting at here. An analogy: Suppose someone asked this:

Q: How do you prove that a triangle has three sides?

The answer would be this:

A: Huh? Having three sides is what it means to be a triangle!

Now, it may well be that the question you meant to ask makes more sense. But we can't tell, all we can see is what you actually wrote...

Answer 2

The derivation of the Normal Distribution (Gaussian Distribution) should answer your question; it also explains why there is a pi and so forth:

https://www.sonoma.edu/users/w/wilsonst/papers/Normal/default.html