# Why is variance invariant with regards to translation?

Jul 16, 2016

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
$E \left[{\left(x - \mu\right)}^{2}\right]$ where $\mu$ is the mean value.

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

${x}_{i} \to {x}_{i} + a$

The mean also shifts by the same amount

$\mu \to \mu + a$

so that the deviations from the mean stay the same :

${x}_{i} - \mu \to \left({x}_{i} + a\right) - \left(\mu + a\right) = {x}_{i} - \mu$