Estimating Quartiles with Moments 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:


*

*Is this formula correct? If so how is it derived/where can I find a citation?

*If the formula is correct, then is there a similar formula for the first and third quartiles as well, possibly involving higher moments?

 A: 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.
