Why is variance invariant with regards to translation?

1 Answer
Jul 16, 2016

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#