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I have calculated the skewness of my data. I was wondering what the skewness, when converted to the units of the data, represents in non-mathematical (biological?) terms? For instance, I know that the variance, when "converted" to the standard deviation, represents how far, on average, a single datum deviates from the mean.

Any help would be greatly appreciated!

Alex

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  • $\begingroup$ I'm not aware of any general interpretation of it. Note rhat variance is in squared units, and sd is in the original units of the data. Skewness is unitless, however; it's not immediately clear why you'd take its cube root nor expect it to correspond to something. $\endgroup$
    – Glen_b
    Nov 17 at 22:42
  • $\begingroup$ Skewness is unitless. $\endgroup$
    – Dave
    Nov 17 at 23:47
  • $\begingroup$ As we've both indicated, the premise of the question is false (skewness is not in cubed units of the data). You will need to address that issue in some way unless you want my "Skewness is unitless," or Dave's "Skewness is unitless." for an answer $\endgroup$
    – Glen_b
    Nov 18 at 0:21
  • $\begingroup$ Sorry for being confusing! As you pointed out, we can take the square root of the variance to get a value (standard deviation) that is in the data's units. Suppose now that we do whatever operation is needed (I now know it is not the cube root) such that we get a value (let's call it X), from the skewness, that is in the units of our data. How would we interpret X? $\endgroup$
    – tardigrade
    Nov 18 at 0:26
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    $\begingroup$ @tardigrade note that the moment skewness is $\mu_3/\sigma^3$. If you don't divide by $\sigma^3$ to just have $\mu_3$, it is then in cubed units, but I don't think you're likely to find an answer to "What's an interpretation for $\mu_3^{\frac13}$?" particularly satisfying. $\endgroup$
    – Glen_b
    Nov 18 at 2:00
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By "the skewness" I guess you mean the moment-based skewness calculated from third and second powers of deviations from the mean. But there are other ways of thinking about skewness that perhaps come closer to what you seek. You can measure skewness by a measure using mean, median and standard deviation (SD):

(mean $-$ median) / SD

Like any other similar measure, that ratio is a reduction that allows different shapes to be represented by the same value (and, notably, is necessarily zero when the mean and median are equal, regardless of whether the distribution is symmetric). So the distance between mean and median is pertinent to skewness and is made a measure of skewness (so unitless) by dividing by the SD. The measure is also interesting because it must fall in the interval $[-1, 1]$. This measure can be traced to Karl Pearson, although he did not do much with it.

Other measures start with

[(upper quartile $-$ median) $-$ (median $-$ lower quartile)] / (upper quartile $-$ lower quartile)

which also falls in the same interval $[-1, 1]$. The same idea can be applied to any other symmetrically placed quantiles. As with the first measure, the numerator has the same units as the variable, and is made a measure of skewness by dividing by a reference distance. This measure has been attributed to Francis Galton, who did not propose it, although he did discuss measures in similar spirit; to Arthur Lyon Bowley, who did evidently invent it; and to George Udny Yule, who might have invented it independently but published after Bowley.

That's far from the end of what has been proposed or is possible, but I will stop there.

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