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 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