Bowley's Coefficient of Skewness

Karl Pearson's coefficient needs the mean, the mode (or median) and the standard deviation — all three calculations, including the standard deviation, which is the most laborious of the lot. Bowley's coefficient, proposed by Arthur Bowley, needs only the three quartiles, and is especially useful when:

• the distribution has an open-end class (so the mean and standard deviation cannot be computed at all), or
• you only need a quick measure of skewness without computing the full standard deviation.

Bowley's coefficient (also called the quartile coefficient of skewness) is based on the idea that in a perfectly symmetric distribution, the median lies exactly halfway between $Q_1$ and $Q_3$, i.e. $Q_3 - \text{Median} = \text{Median} - Q_1$. Skewness is present to the extent that this balance fails — that is, to the extent that one quartile is farther from the median than the other.

Interpretation is identical to Karl Pearson's coefficient: $Sk_B = 0$ means symmetric, $Sk_B > 0$ means positively skewed (the upper quartile is farther from the median than the lower quartile is), $Sk_B < 0$ means negatively skewed. Because it is built purely from ratios of quartile distances, Bowley's coefficient always lies between $-1$ and $+1$.

Steps to compute Bowley's coefficient:

1. Compute $Q_1$ (the value below which 25% of observations lie).
2. Compute the Median ($Q_2$).
3. Compute $Q_3$ (the value below which 75% of observations lie).
4. Substitute into $Sk_B = \dfrac{Q_3 + Q_1 - 2\,\text{Median}}{Q_3 - Q_1}$. …