1
$\begingroup$

I know that SD depends on change of scale and not on change of origin. what about mean deviation? Mean deviation from mean = $\frac{\sum|X_{i}- \overline X|}{N}$.

Improve this question
$\endgroup$
5
  • $\begingroup$ How do you define mean deviation? $\endgroup$
    – user2974951
    Commented Jul 9, 2019 at 10:53
  • 2
    $\begingroup$ that is mean absolute deviation to be precise. $\endgroup$
    – gunes
    Commented Jul 9, 2019 at 11:08
  • 1
    $\begingroup$ The mean deviation from the mean is always zero, so that counts as invariant to changes of origin, scale and regime in power. What you mean is widely understood, but as @gunes implies, mentioning absolute at least when you introduce the term is a good idea, $\endgroup$
    – Nick Cox
    Commented Jul 9, 2019 at 11:31
  • 1
    $\begingroup$ A better index is Gini's mean difference: the mean absolute difference between all possible pairs of observations. It doesn't rely on the mean as the centering index. $\endgroup$
    – Frank Harrell
    Commented Jul 9, 2019 at 11:44
  • $\begingroup$ The term "mean deviation" has a long history in statistics and (particularly in older works) is nearly always intended to mean "mean absolute deviation from the mean".. $\endgroup$
    – Glen_b
    Commented Jul 9, 2019 at 15:04

1 Answer 1

Reset to default
5
$\begingroup$

Distance from mean of each observation is clearly invariant to "changes of origin". 5 is as far from 3 as 1005 is from 1003, or, in a more general form: $|(a+c)-(b+c)| = |a-b|$

The scale is different, though. As $|rx-ry| = r|x-y|$, scaling your variable by a factor of $r$ will also scale your "mean deviation from mean" by a factor of $r$.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.