Karl Pearson's Coefficient of Skewness

Knowing the direction of skewness from the order of the averages is not enough; a business analyst needs a single number that gives its degree as well, and that can be compared across distributions. Karl Pearson's coefficient of skewness does this by measuring how far the mean has been pulled away from the mode, and then scaling that gap by the standard deviation so the result is a pure number.

The coefficient, written $Sk_p$, is defined as $Sk_p = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}} = \frac{\bar{x} - Z}{\sigma},$ where $\bar{x}$ is the mean, $Z$ the mode and $\sigma$ the standard deviation. Because the mean is dragged towards the longer tail while the mode stays at the peak, the sign of $\bar{x} - Z$ is exactly the direction of the skewness: positive when $\bar{x} > Z$, negative when $\bar{x} < Z$, and $0$ when they coincide.

When the mode is not clearly defined. The mode is unreliable when a distribution is irregular or has more than one peak. In that case the empirical relationship $Z = 3\,\text{Median} - 2\,\bar{x}$ is substituted into the definition. Writing $M$ for the median, $\bar{x} - Z = \bar{x} - (3M - 2\bar{x}) = 3\bar{x} - 3M = 3(\bar{x} - M),$ so the coefficient becomes $Sk_p = \frac{3(\text{Mean} - \text{Median})}{\text{Standard Deviation}} = \frac{3(\bar{x} - M)}{\sigma}.$ This median form gives the same value as the mode form whenever the empirical relationship holds, and is the form to use whenever the mode is ill-defined. …