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If I'm not mistaking standard deviation is defined as $\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2}$ .
You could rewrite this as $\sigma = (\frac{1}{N} \sum_{i=1}^N |x_i - \mu|^c)^\frac{1}{c}$ where $c=2$.

This makes me wonder: What is special about 2 ?
Why don't we ever use any other value for $c$ ?

I'm probably mistaking but to me it seems that the only effect of another $c$ would be that the average distance of the mean would be either more or less visible (depending on whether you choose a larger or smaller $c$).

Or is $c=2$ just a convention because $\sqrt{x}$ is easier to calculate then (just a random example): $x^\frac{1}{8.15}$

Garo
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    You might be interested in the "$p$-norm" of a finite list of numbers (what people often call a vector), and the fact that only for $p = 2$ does this notion of length arise from an "inner product" and give rise to Euclidean geometry. – Andrew D. Hwang Jan 27 '22 at 13:40
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    Standard deviation is the square root of the variance of a random variable, and this is, by definition, the second moment of said random variable (shifted by the mean). It is completely natural to consider other moments (e.g. numbers other than 2). However, the normal distribution (one of the most commonly occurring distributions), is uniquely determined by its (co)variance and mean. – o0BlueBeast0o Jan 27 '22 at 13:43
  • If $c=1$ the statistic is called the mean absolute deviation from the mean. Also see this answer. – David K Jan 27 '22 at 18:09

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