Why is the Variance defined the way it is? Instead of $\operatorname{Var}(X)=E[(X-E[X])^2]$ I don't see why we could not define it as for example $\sigma=E[|X-E[X]|]$ or something different.
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2People study the [Mean Absolute Deviation](https://en.wikipedia.org/wiki/Average_absolute_deviation) all the time. Nothing unusual about it. The usual variance has some nice analytic properties...in particular, it is sometimes inconvenient that the absolute value can't be differentiated at $0$. – lulu May 18 '21 at 12:47
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1Read https://math.stackexchange.com/questions/4787/motivation-behind-standard-deviation and https://math.stackexchange.com/questions/3071367/whats-so-special-about-standard-deviation and https://math.stackexchange.com/questions/717339/why-is-variance-squared – Henry May 18 '21 at 12:50