Why is variance squared rather than cubed (or any other exponent)?

Question

Variance is

$$\dfrac{\sum_{i=1}^{n}(x_i-\bar x)^2}{n-1}$$

But why square the difference? Why not cube it, or any other exponent?

Related question.

Answer 1

You can also use other exponents, but then it is not called variance anymore. See this wikipedia page, for example.

Variance comes up quite naturally in various contexts, and carries a lot of information about the distribution.

Answer 2

Still, I am not convinced by this answer. The naming convention can be separate topics to discuss. Here what is the issue if we take cubes or quad etc?