Warning: this started as ELI5, but I got excited.

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Essentially, the main metric mathematicians would _really_ like to use is variance – standard deviation squared. It has a lot of very good features; notably, for a lot of common probability distributions, it is additive in some way. To give a couple examples, binomial distribution (flip a coin of chance of p to get heads n times) has mean np and variance np(1-p). Two independent normally (the bell curve) distributed variables will sum to a normally distributed variable – with mean and variance being sums of their parts. In fact! Summing _any_ two independent variables will sum their variance.

Variance also very easily can be related to covariance (if variance of X is expectation of (X-EX)(E-EX), covariance of X and Y is expectation of (X-EX)(Y-EY)), which is one of the core metrics to measure relations between random variables (more commonly quoted correlation is just covariance divided by both standard deviations).

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As to why it’s so… _quieting mumbling dissipating into white noise._

In seriousness, one intuitive reason is that once you work with random variables, you often want to imagine them as a vector space – and independent variables correspond to perpendicular vectors. If you need to add them up, the length squared, thanks to uncle Pythagoras, is sum of squares of the constituent parts. I won’t go into detail why we swap random variables for vectors, or standard deviation for length, but suffice it to say – the underlying concepts in both cases are surprisingly similar.

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So it’s not so much that standard deviation is a better metric than your suggested average deviation; it’s that variance is a far better metric than both, but we like deviation to be comparable to the mean (instead of squared), and standard deviation is far easier to find from variance.

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