Bowley's (Quartile) Coefficient of Skewness

Karl Pearson's coefficient needs the standard deviation and the mean, both of which are badly disturbed by a single extreme value and cannot even be computed for an open-ended distribution (one whose first or last class has no definite boundary). For such data Bowley's coefficient of skewness, built only from the three quartiles, is preferred. It is also called the quartile coefficient of skewness.

In a symmetric distribution the median $Q_2$ sits exactly midway between the first quartile $Q_1$ and the third quartile $Q_3$, so the two quartile gaps are equal: $Q_3 - Q_2 = Q_2 - Q_1$. Skewness stretches one of these gaps. Bowley's coefficient measures the imbalance between them and scales it by the total inter-quartile range: $Sk_b = \frac{(Q_3 - Q_2) - (Q_2 - Q_1)}{(Q_3 - Q_2) + (Q_2 - Q_1)} = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1}.$

The two written forms are identical — expanding the first numerator gives $Q_3 - Q_2 - Q_2 + Q_1 = Q_3 + Q_1 - 2Q_2$, and the first denominator gives $Q_3 - Q_1$. The compact form $\dfrac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1}$ is the one normally used in computation.

Its sign carries the direction of skewness. If the upper gap exceeds the lower gap ($Q_3 - Q_2 > Q_2 - Q_1$) the numerator is positive and the distribution is positively skewed; if the lower gap is larger the numerator is negative and the distribution is negatively skewed; if the two gaps are equal the numerator is $0$ and the distribution is symmetric. …