Question

Why is variance invariant with regards to translation?

Answer 1

Because the variance is calculated in terms of the deviations from the mean, which stays the same under a translation .

Explanation:

The variance is defined as the expectation value
`E[(x-mu)^2]` where `mu` is the mean value.

When the data set is translated, then all the data-points are shifted by the same amount

`x_i -> x_i +a`

The mean also shifts by the same amount

`mu -> mu +a`

so that the deviations from the mean stay the same :

`x_i -mu -> (x_i +a) -(mu +a) = x_i -mu`