Introduction to Bowley's Skewness
In statistics, skewness measures the asymmetry of a probability distribution. When we don't have the mean or standard deviation, we use Bowley's Coefficient of Skewness, which relies solely on the quartiles ($Q_1, Q_3$) and the median ($M_d$ or $Q_2$).
The Formula
Bowley's coefficient ($S_k$) is calculated as: $$S_k = \frac{(Q_3 + Q_1) - 2M_d}{Q_3 - Q_1}$$
Solving the Problem
Given:
- Difference of quartiles: $Q_3 - Q_1 = 20$
- Sum of quartiles: $Q_3 + Q_1 = 70$
- Median: $M_d = 36$
Calculation: Substitute these values into the Bowley formula: $$S_k = \frac{(70) - 2(36)}{20}$$ $$S_k = \frac{70 - 72}{20}$$ $$S_k = \frac{-2}{20}$$ $$S_k = -0.1$$
Interpretation
The result is $-0.1$. Since the value is negative, the distribution is negatively skewed (or left-skewed), meaning the tail is on the left side of the distribution.