eli5: In statistics, why do we square the deviation before dividing by n to get the variance instead of just using absolute values?

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I’m assuming there’s a good reason this doesn’t work but I’ve never had it explained to me why.

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Anonymous 0 Comments

Both are used in statistics depending on what you’re doing. There isn’t much real reason other than that squaring makes the math easier. Another equally valid question is why do we say x^2 instead of |x^1.9999 | or |x^2.0001 |.

In fact, the use of other exponents is ubiquitous in math. Just look up the L-p norms. The 2 seems somewhat arbitrary, except that euclidean distances use the 2 as an exponent, so maybe there’s some physical intuition there. But for most statistical questions, the 2 is pretty arbitrary.

Also look up moments of a distribution. The first moment is the mean, which is the average of the raw values. the second moment is the variance, which is the average of squares. The third moment is skewness, which is the average of cubes, the fourth moment is kurtosis, which is the average of fourth powers, etc. Again, why not a 2.5th moment? There’s no real reason why not.

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