Cumulative frequency is a running total of frequencies. It allows you to estimate medians, quartiles, and interquartile ranges from grouped data.
To draw a cumulative frequency graph:
- Add up the frequencies progressively (running total)
- Plot points at the upper class boundary of each group
- Join with a smooth S-shaped curve (not straight lines)
Reading off values:
- Median: read off at $\frac{n}{2}$ on the $y$-axis
- Lower quartile (Q1): read off at $\frac{n}{4}$
- Upper quartile (Q3): read off at $\frac{3n}{4}$
- IQR $= Q3 - Q1$
Example: For $n = 80$ data values, the median is at cumulative frequency $40$, Q1 at $20$, Q3 at $60$.
$$\text{IQR} = Q3 - Q1$$
Common error: plotting at the midpoint of a class rather than the upper class boundary, or reading the quartiles off the wrong axis.