Nonparametric Skew of Data

Question

Recently in my studies of statistics, I have come across the second skewness coefficient to determine the skewness of the set of data. The formula is given by: $$ \frac{3(\mu - \nu)}{\sigma}$$ However, as a positive skew is indicated by a larger positive number, and a negative one by a larger negative number, I do not see the need for the factor of 3. Pearson thought it important enough to adopt this new way, but now most people simply drop the factor of 3. The reason on Wikipedia is "..the difference between the mean and the mode for many distributions is approximately three times the difference between the mean and the median", which I do not understand. Please can someone explain why the factor of 3 is there? Thanks for the help

Answer 1

For a convenient example of a slightly skewed probability distribution, we may take the random variable $Y = \exp(\alpha X)$ where $X \sim N(0,1)$ and $\alpha$ is small. You can compute: the mean of $Y$ is $\exp(\alpha^2/2)$, the median is $1$ and the mode is $\exp(-\alpha^2)$. As $\alpha \to 0$, we do indeed have

$$ \dfrac{mean - mode}{mean - median} = \dfrac{e^{\alpha^2/2} - e^{-\alpha^2}}{e^{\alpha^2/2} - 1} = 1 + e^{-\alpha^2/2} + e^{-\alpha^2} \to 3 \ \text{as}\ \alpha \to 0$$