Why is variance invariant with regards to translation?

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Answer 1 · 2016-07-16T18:11:07

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

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 :

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