Q.Using the ogive of the distribution in Worked Example 5 (classes $0\text{–}50$, frequencies $5, 8, 15, 12, 10$, $N = 50$), explain how to locate $Q_1$, $Q_2$ and $Q_3$ graphically and state the values read.

Plot the less-than ogive, mark $\frac{N}{4}, \frac{N}{2}, \frac{3N}{4}$ on the $y$-axis, go across to the curve and drop to the $x$-axis.

Step 1 — plot the ogive. Plot each class's upper boundary against its less-than cumulative frequency, starting at the lower boundary of the first class with height $0$:

and join them with a smooth rising curve.

Step 2 — mark the locating figures on the $y$-axis. For quartiles these are $\dfrac{N}{4} = 12.5$, $\dfrac{N}{2} = 25$ and $\dfrac{3N}{4} = 37.5$.

Step 3 — read across and down. From each marked height draw a horizontal line to the ogive, then a vertical line down to the $x$-axis:

• $y = 12.5$ meets the curve at about $x = 19.4$ → $Q_1 \approx 19.4$. …

A single ogive also gives deciles and percentiles: to read $P_{90}$, mark $\frac{90N}{100} = 45$ on the $y$-axis and drop from the curve — it lands at $x = 45$, matching Worked Exa …

Plotting class mid-points instead of upper boundaries for a less-than ogive, or marking the locating figure on the $x$-axis instead of the $y$-axis. The cumulative frequency is al …