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The Wikipedia article on Skewness indicates that the median of a distribution can be estimated from the mean, standard deviation, and skeweness with an error term that goes as $O(skewness^2)$. Specifically:

$$skewness = \frac{3(mean-median)}{\sigma}+\mathcal{O}(skewness^2)$$

Unfortunately the article indicates that a citation is needed, and I've been unable to track one down. The article suggests that it follows from a cumulative expansion, but the cumulative generating function does not involve the median so it's not clear how this relates. This leads to my two part question:

  1. Is this formula correct? If so how is it derived/where can I find a citation?
  2. If the formula is correct, then is there a similar formula for the first and third quartiles as well, possibly involving higher moments?
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    $\begingroup$ Using (mean $-$ median) / SD to measure skewness is a choice (in my view an underused one) but either way the name skewness is just a label. Here, and more widely, the definition should be read from right to left, not right to left. The multiplier of 3 was introduced by Karl Pearson to make this measure closer in practice to (mean $-$ mode) / SD. I don't understand this definition of skewness to be a basis to estimate medians. How could you estimate skewness in this way without knowing the median? Note that the moment-based measure of skewness is utterly different in principle. $\endgroup$ – Nick Cox Dec 11 '19 at 12:34
  • $\begingroup$ @NickCox I think you have a typo. Right now you say "from right to left, not right to left". $\endgroup$ – Peter Flom Dec 12 '19 at 11:03
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    $\begingroup$ @PeterFlom-ReinstateMonica Indeed, and thanks. The sense intended was that in definitions of the form label := details the details on the right are primary and the label is secondary. In my childhood bananas came with sticky labels saying "bananas" and we found it entertaining to peel off the labels and stick them on others, saying "You are a banana!". Even in this very childish game we knew that what you call something (notation, terms) is secondary. In short, names (e.g. skewness) are just sticky labels. See also: reification, hypostatization, fallacy of misplaced concreteness. $\endgroup$ – Nick Cox Dec 12 '19 at 11:18
  • $\begingroup$ @NickCox, there is no definition in the post; there is an equation described by the phrase “can be estimated”, with an error term. $\endgroup$ – Matt F. Dec 12 '19 at 14:53
  • $\begingroup$ An overarching question is: What is skewness? Some people say: skewness is to be calculated from the mean and the second and third moments about the mean. Others, from mean, mode and SD. Others, from mean, median and SD. And I've seen other definitions too. So, the point is that skewness is just a label shared by these definitions. $\endgroup$ – Nick Cox Dec 12 '19 at 15:42
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I think the Wikipedia article is wrong here, and it should read $$\text{skewness} = \frac{6(\text{mean} - \text{median})}{\sigma} + O(\text{skewness}^2).$$

As an example, consider $Y=a+be^{cX}$, where $X$ is a standard normal variable, so that $Y$ is a shifted log-normal variable. We abbreviate $C=\exp(c^2/2)$ and then have the following properties of $Y$:

\begin{align} \text{median} &: a+b\\ \text{mean} &: a+bC\\ \text{variance} &: b^2C^2(C^2-1) \\ E[(Y-\text{mean})^3] &: b^3C^3(C^2-1)^2(C^2+2) \\ \text{skewness}\, \cdot \text{sd} &: bC(C^2-1)(C^2+2) \\ \text{mean}\, - \text{median} &: b(C-1)\\ \text{quartiles} &: a + be^{\pm 0.6745c} \\ \end{align} So the ratio between $\text{skewness}$ and $(\text{mean} - \text{median})/\text{sd}$ is $(C+1)(C^2+2)$. In the limit of small $c$, where $C$ is close to $1$, this goes to $6$.

More generally, for $Y={\large\sum} b_j e^{jcX}$, where all $jb_j$ are non-negative, \begin{align} \text{median} &: {\small\sum}\, b_j \\ \text{mean} &: {\small\sum}\, b_j(1 + j^2c/2) + O(c^3)\\ \text{variance} &: {\small\sum}\, j^2 b_j^2 c^2 + 2 {\small\sum_{i<j}}\, ijb_i b_j c^2+ O(c^3) \\ E[(Y-\text{mean})^3] &: (\text{long expression}) c^4+ O(c^5) \\ \text{skewness}\, \cdot \text{sd} &: \frac{1}{2}{\sum}\, j^2b_j c^2 + O(c^3)\\ \text{mean}\, - \text{median} &: \,3{\sum}\, j^2b_j c^2 + O(c^3) \\ \text{quartiles} &: \sum b_je^{\pm 0.6745jc} \\ \end{align}

This class of perturbations of the normal distribution can approximate many of the skewed perturbations of interest. Within this class the mean and median are related with the corrected formula, and the quartiles have a simple formula too.

Update: Haldane (1942) provides a similar analysis, and Hall (1980) provides a more general and rigorous analysis; both are freely available online.

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    $\begingroup$ I can't follow this at all. Skewness measured in terms of mean, median and SD is just a different choice from moment-based skewness. At most, values will sometimes coincide (e.g. all symmetric distributions will have zero skewness on both measures), but the principles are different. $\endgroup$ – Nick Cox Dec 11 '19 at 12:38
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    $\begingroup$ There seems to be some notational equivocation here between the definition of skewness and something else; I may be missing something basic there. But I do know that (mean $-$ median) / SD falls within $[-1, 1]$ with evident implications for any multiple of it. Simply put, what is the definition of "skewness" that you are using? $\endgroup$ – Nick Cox Dec 11 '19 at 14:17
  • $\begingroup$ @NickCox, what do you think is the best way to correct the claim in Wikipedia? For perturbations of a normal, I think correcting the 3 to 6 is enough. But perhaps you think some further conceptual correction is required also. $\endgroup$ – Matt F. Dec 11 '19 at 14:23
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    $\begingroup$ I don't have recommendations about Wikipedia. I use it often but as a point of principle as well as in practice I never edit content or propose changes. The ideal for CV is to be intelligible directly as well as correct; references are more than welcome, and I often give them myself, but neither questions nor answers should depend on anyone studying an external source. That doesn't really answer your query and is mostly personal. $\endgroup$ – Nick Cox Dec 11 '19 at 14:41
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    $\begingroup$ @LBogaardt, I derived these in Mathematica; all the formulas are the results of standard integrals. I think Pearson knew much of this, but I don't have a reference. $\endgroup$ – Matt F. Dec 14 '19 at 15:38

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