The "normal distribution" is often described as if it were one of the great wonders of the world, or something like that. I am unable to see why. I definitely agree things in the universe, like the length of human beings, tend to be centered around an average value, and that an average deviation from that center can be calculated. But, the mean absolute deviation does that pretty well, without involving the squaring of the distance of each variable's distance from the average, (x-u)^2. The fact that squaring is superior to just using the absolute value, is not initially apparent to me.
Diving into how curves that have a "bell shape" can be created mathematically, taking some base like e to the power of the absolute value of x, e^-|x|, creates a pretty ugly and spiky curve. Adding an exponent, like e^-|x|^2 or e^-|x|^1.5 or e^-|x|^3, flattens the spike. With e^-|x|^2, the absolute value operation can be skipped, leaving e^-x^2. This is what the "normal distribution" uses, and it also subtracts the average to center the curve around that value, as e^-(x-u)^2.
Given that a curve e^-(x-u)^2 is chosen to be used, also calculating the root mean square deviation, instead of mean absolute deviation, seems like it could be beneficial, since it is proportional to (x-u)^2 (the variance is proportional, the mean square deviation. )
So assuming the curve e^-(x-u)^2, or e^-(x-u)^2/(2s^2), I can see why the "standard deviation" format is chosen instead of mean absolute deviation.
But, I do not see why e^-x^2 is somehow the "perfect match" for phenomena in the universe. It seems a bit arbitrary. It seems like there might be many other ways to create that "bell" shape.
Is there some easy-to-understand rationale, that is easy and intuitive but not too dumb to be meaningless (so easy in a smart sense, a clever explanation) for why exactly the "normal distribution" equation is uniquely suited to fit how things tend to be distributed and for probability and frequency distributions? An example of an "easy" explanation is 3Blue1Brown's explanation of the Fourier Transform.