Question

Are measures of central tendency invariant to translation?

Answer 1

No, measures of central tendency get translated the same amount that the data values themselves do, so they are not invariant. Measures of spread are invariant to translation. They don't change when all the values are translated by the same amount.

Explanation:

For example, if your data set consists of the numbers `4, 5, 7, 10, 11`, the mean is 7.4 and the median is 7. If these data values all get translated along a number line, for instance, to the right by 10 units to become `14, 15, 17, 20, 21` the mean and median (both measures of central tendency) are also translated up by 10 units to 17.4 and 17, respectively.

On the other hand, the (sample) standard deviation for both data sets is about 3.05 and the range for both data sets is 7 (both measures of spread).