Question

One can think of the mean as a balance point. How does this relate to the necessity of squaring the deviations in order to determine the variance?

Answer 1

The squaring of deviations becomes important because the way the mean is defined the sum of all the deviations from the mean for any data set would always come out to be 0. This is derived as followed:

where n is the total number of data points

Now as sample mean is a constant, we can bring it out to get:

Now, by definition:

Therefore,

Putting this in the equation of sum of deviations, we get:

Hence we see that the sum of deviation from the mean would always come out to be 0 and therefore it acts like a balance point. This is the reason that we take square or modulus of the deviations to get a meaningful term.