Q.For the same marks distribution as Worked Example 4 (50 students; Marks 0-10:5, 10-20:8, 20-30:15, 30-40:12, 40-50:7, 50-60:3), calculate Bowley's coefficient of skewness.

Bowley's coefficient uses only the quartiles and the median of this distribution, and comes out mildly positive — consistent with the Karl Pearson result found using the mean, mode and standard deviation. …

Data: the same 50-student marks distribution — Marks 0-10:5, 10-20:8, 20-30:15, 30-40:12, 40-50:7, 50-60:3 ($N=50$). Cumulative frequencies: 5, 13, 28, 40, 47, 50.

Step 1 — First quartile $Q_1$ (position $\frac{N}{4}=12.5$). The cumulative frequency first reaches 12.5 in the class 10-20 (cf before class = 5, $f=8$, $L=10$, $h=10$):

Step 2 — Median (position $\frac{N}{2}=25$). The cumulative frequency first reaches 25 in the class 20-30 (cf before class = 13, $f=15$, $L=20$, $h=10$):

Step 3 — Third quartile $Q_3$ (position $\frac{3N}{4}=37.5$). The cumulative frequency first reaches 37.5 in the class 30-40 (cf before class = 28, $f=12$, $L=30$, $h=10$):

Step 4 — Bowley's coefficient.

$Sk_B=\dfrac{Q_3+Q_1-2\,\text{Median}}{Q_3-Q_1}=\dfrac{37.92+19.375-56}{37.92-19.375}=\dfrac{1.295}{18.545}=0.07$ …

A common error is using $\frac{N}{4}$ and $\frac{3N}{4}$ against the wrong cumulative frequency column — always find the class where the cumulative frequency first equals or exceeds the required position, not the class containing that midpoint value on the marks axis. Also, do not confuse $cf$ (cumulative fr …