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Let's say I have a distribution of some weights in kg. I want to estimate the skewness of this distribution using a robust statistic, the medcouple [ G. Brys; M. Hubert; A. Struyf (2004). A robust measure of skewness. Journal of Computational and Graphical Statistics 13(4), 996-1017.]

Could you help me with the units of the medcouple measure? In this example, are they kg or 1/kg or kg$^2$ or something else, or does it have no units?

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    $\begingroup$ The medcouple has no units. Certain complications aside, which don't produce an exception to that statement, it is a ratio of quantities on the original scale, so the units cancel. It could hardly be a (useful) measure of skewness otherwise. $\endgroup$ – Nick Cox Mar 14 '19 at 8:48
  • $\begingroup$ It's always worth remembering that (mean $-$ median) / SD is bounded in $[-1, 1]$ and is easy to explain and calculate. It's not particularly robust. Personally I often value sensitivity in measures of skewness and kurtosis, as they alert me to what may be a problem, or even a feature as high skewness may be one signal that I would be better off working on some transformed scale. $\endgroup$ – Nick Cox Mar 14 '19 at 8:54
  • $\begingroup$ @NickCox Thank you for clarification. Unfortunately, I should measure skewness ~3, so the transformation has no sense for me. I use medcouple, as my results are much more reproducible in this case. $\endgroup$ – zlon Mar 14 '19 at 8:58
  • $\begingroup$ Sorry, but I really don't understand that comment. First, what measure of skewness are you using? Second, why "should" you measure it? That may be just a choice of word that's not quite right. Third, why do you say that transformation makes no sense in your problem? Fourth, reproducible in what sense? With any measure of skewness, the same recipe produces the same answer on the same data; what other sense do you have in mind? To get a better reply, you may need to back up, tell us much more about your problem, your data and why you think the medcouple is useful. $\endgroup$ – Nick Cox Mar 14 '19 at 9:00
  • $\begingroup$ @NickCox I try to use just usual skewness as the third moment of the distribution (more precisely skewness function in the Matlab). It is really noisy and not reproducible (the results from experiment repeats have a big relative error). If I would use medcouple for estimation how skewed is my distribution I have 2 times smaller relative error. In my case transformation has no sense, as "skewness" is a measure of what I want. In other words, I need to compare how different are distributions obtained in different conditions in terms of the skewness. $\endgroup$ – zlon Mar 14 '19 at 9:16
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The medcouple has no units. Certain complications aside, which don't produce an exception to that statement, it is a ratio of quantities on the original scale, so the units cancel. It could hardly be a (useful) measure of skewness otherwise.

Personally I often value sensitivity in measures of skewness (and kurtosis, which otherwise we will leave on one side here), as they alert me to what may be a problem, or even a feature: high skewness may be one signal that I would be better off working on some transformed scale.

For those in search of alternatives to moment-based skewness:

  1. It's always worth remembering that (mean $−$ median) / SD is bounded in $[−1,1]$ and is easy to explain and calculate. It's not particularly robust, but as a bounded measure it is less explosive than the moment-based measure.

  2. L-moments offer a well-based set of ideas. The Wikipedia entry is a fair start, with several key references.

The thread Robust analogues of Mean, CV and Skewness also touches on these issues.

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