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New formula/modification for standard deviation/variance?

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The reason we use the regular equation for variance with the square is because (As I was told) the absolute value function is difficult to deal with. It is not only defined piecewise but is also non-differentiable.
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Couple of things: mu is used for the population mean, and it is unlikely that you would have that when calculating your measure of variability.  You likely would use x-bar, the sample mean.

Second, you may want to look at the MAD, which has many properties that we would prefer when looking at measures of variability.  The absolute deviation (the top of your fraction) is more robust against the effects of outliers than the standard deviation.
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What you wrote down is essentially the centered L^1 norm, the variance is the centered L^2 norm. They both measure deviation from the mean, but they bias different parts of the distribution. Because of the square in the variance, large values are weighted more and small values are weighted less. More generally you can define L^p norms for any p>0. You can see a graphical comparison of the different norms on Wikipedia.

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