Cumulative frequency

#### Tier: #Foundation #Higher

Description

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:

1. Add up the frequencies progressively (running total)
2. Plot points at the upper class boundary of each group
3. 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.

Links

Median Interquartile range Boxplots