📈 Mastering Cumulative Frequency Graphs: Find the Median & More!

Cumulative frequency graphs are a **powerful visual tool** in statistics. They help O Level students quickly estimate the median, quartiles, inter–quartile range (IQR), and percentiles, all of which appear frequently in exam questions. Mastering this skill boosts your confidence, speeds up problem-solving, and ensures accuracy in data-handling tasks.

Understanding the Core Idea

A cumulative frequency graph plots the upper class boundary on the

x

-axis against the cumulative frequency on the

y

-axis. **By reading off specific cumulative percentages (50%, 25%, 75%, etc.) we can estimate the median, quartiles, and percentiles with ease.** Remember that

\text{cumulative frequency} = \text{running total of frequencies up to a class boundary}.

Median lies at the 50\% level: find $\tfrac{1}{2}\,N$ on the $y$ -axis, then read across and down.
$Q_1$ (lower quartile) at 25\% level: $0.25N$ on the $y$ -axis.
$Q_3$ (upper quartile) at 75\% level: $0.75N$ on the $y$ -axis.
Inter-quartile range: $IQR = Q_3 - Q_1$ .
The $k$ -th percentile is at $\tfrac{k}{100}N$ on the $y$ -axis.

Worked Example: Exam Score Analysis

Problem: The table shows the distribution of 80 students' maths test scores. Draw a cumulative frequency graph and estimate (i) the median, (ii) the lower quartile, (iii) the inter-quartile range, and (iv) the 90th percentile.

Step 1: Form a cumulative frequency table and plot points at **upper class boundaries** such as
$\text{scores }0\!\text{–}10,\;10\!\text{–}20,\;\dots,\;90\!\text{–}100.$
Step 2: Draw a smooth cumulative curve (ogive). Locate
$\tfrac{1}{2}\times80 = 40$
on the $y$ -axis; read across to meet the curve, then down. The corresponding $x$ -value is the **median**.
Step 3: For $Q_1$ , use
$0.25\times80 = 20$
; for $Q_3$ ,
$0.75\times80 = 60$
. Read off their $x$ -values and compute
$IQR = Q_3 - Q_1.$
Step 4: 90th percentile lies at
$0.90\times80 = 72$
on the $y$ -axis. Read across to the curve, then down to estimate the score.

With practice, you'll interpret cumulative frequency graphs in seconds. **Sketch neatly, label axes clearly,** and always double-check scale readings. Keep solving past-paper questions to cement these techniques!