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I am aware that measures of skewness based on the mean are affected by outliers, and just one outlier can significantly shift the mean. However, in case of a distribution with a very long tail, should I use a mean-based measure of skewness or a median-based measure of skewness, to measure the lack of symmetry of the distribution? Indeed, I am thinking that, probably, long tails might have an effect similar to outliers, by shifting considerably the mean, but I am not sure.

Ommo
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What you "should" do depends on what you want to find out. It's true that (as you write) a mean based measure of skewnesss is affected by outliers (if they are all on one end). Do you want that? Or not?

This is similar to the case with deciding among mean, median, mode, trimmed mean, etc. It's not that any of them are right or wrong in any particular case (although some elementary stats books may give that idea). It's a question of what suits your particular purposes in your particular case.

It's also similar to choosing a measure of dispersion.

There are also quintile based measures, where you can pick any symmetric quintiles. While quartiles are the most common choice, you could choose any -- again, to suit your case.

Peter Flom
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  • Thanks for your comment @PeterFlom! I understood that any measure is relatively useful to the needs or questions we want to investigate. However, I am still wondering if "long tails might have an effect similar to outliers" when calculating the skewness. In other words, I wonder if mean-based skewness coefficients are biased in a similar way by outliers or long tails? (Indeed, if we have many outliers close to each others, I would see them as a discontinuous long tail) This is something I did not find in the papers I read so far.. – Ommo Dec 06 '23 at 15:22
  • To evaluate whether some estimator is biased, we need to know biased against what? You can generate various distributions and check various estimators of skewness and see which one does what in what circumstances. There is a big literature on skewness that you can search as well.

    However, it might be that, in your case, you really need a graphical measure.

     – Peter Flom Dec 06 '23 at 15:31