Cumulative frequency diagrams are a Higher-tier topic that appears regularly on GCSE Maths exams. They allow you to estimate the median, quartiles and interquartile range from grouped data, and are often paired with box plots for comparing distributions. This guide covers how to plot them, how to read key values, and how to use them in comparison questions.
What Is a Cumulative Frequency Diagram?
A cumulative frequency diagram (also called a cumulative frequency graph or ogive) shows the running total of frequencies up to each class boundary. Instead of plotting individual class frequencies, you plot the accumulated total at the upper end of each class interval, then join the points with a smooth curve.
Cumulative frequency is useful because it lets you estimate the median (the middle value), the lower quartile (LQ, the value one quarter of the way through), the upper quartile (UQ, three quarters of the way through), and the interquartile range (IQR = UQ − LQ). These measures are difficult to find from a standard frequency table when the data is grouped.
The shape of the curve also reveals information about the distribution. A steep section means data values are concentrated there. A flat section means few data values fall in that range.
Key Formulas
Step-by-Step Method
- Add a cumulative frequency column — for each class, add its frequency to the running total.
- Plot points at the upper class boundary against the cumulative frequency. The first point is at (lower boundary of first class, 0).
- Join the points with a smooth S-shaped curve (not straight lines between every pair of points).
- Read off the median at the n/2 position, LQ at n/4, and UQ at 3n/4 by drawing horizontal lines to the curve, then vertical lines down to the x-axis.
Worked Example 1 — Foundation Level
Question: The table shows the times (in minutes) 60 students spent on homework. Draw a cumulative frequency diagram and estimate the median.
| Time (min) | 0-9 | 10-19 | 20-29 | 30-39 | 40-49 |
|---|---|---|---|---|---|
| Frequency | 4 | 10 | 22 | 16 | 8 |
Working: Cumulative frequencies: 4, 14, 36, 52, 60. Plot: (9.5, 4), (19.5, 14), (29.5, 36), (39.5, 52), (49.5, 60). Start at (−0.5, 0). Median position = 60 ÷ 2 = 30th value. Draw across at CF = 30 and read down: approximately 27 minutes.
Answer: The estimated median homework time is approximately 27 minutes.
Worked Example 2 — Higher Level
Question: Using the diagram from Worked Example 1, find the interquartile range.
Working: LQ position = 60 ÷ 4 = 15th value. Reading from the curve: approximately 20 minutes. UQ position = 3 × 60 ÷ 4 = 45th value. Reading from the curve: approximately 35 minutes. IQR = 35 − 20 = 15 minutes.
Answer: The interquartile range is approximately 15 minutes.
Worked Example 3 — Exam Style
Question: A second class has a median of 32 minutes and an IQR of 10 minutes for the same homework. Compare the two classes.
Working: Class 1: median ≈ 27 min, IQR ≈ 15 min. Class 2: median = 32 min, IQR = 10 min. On average, Class 2 spent longer on homework (higher median). Class 1's times are more spread out (larger IQR), meaning homework times were less consistent.
Answer: Class 2 spent longer on average (median 32 vs 27) but their times were more consistent (IQR 10 vs 15).
Common Mistakes
- Plotting at the midpoint instead of the upper boundary. Cumulative frequency points must be plotted at the upper class boundary because the running total is complete only at the end of each class.
- Using straight ruled lines instead of a smooth curve. The examiner expects a smooth S-shaped curve through the plotted points.
- Reading the axes the wrong way round. To find the median, start on the cumulative frequency axis (vertical), draw across to the curve, then down to the data axis (horizontal) — not the other way.
Exam Tips
- Always label both axes clearly: the horizontal axis shows the data values, and the vertical axis shows "Cumulative frequency".
- When comparing two distributions, make two clear comparison statements — one about an average (median) and one about spread (IQR).
- If asked to draw a box plot from your cumulative frequency diagram, you need: minimum, LQ, median, UQ, and maximum.
Practice Questions
Q1 (Foundation): Build a cumulative frequency table for these test scores and state the position of the median.
| Score | 0-19 | 20-39 | 40-59 | 60-79 | 80-100 |
|---|---|---|---|---|---|
| Frequency | 5 | 12 | 18 | 10 | 5 |
Q2 (Foundation): Using the cumulative frequency table from Q1, state the class interval containing the lower quartile.
Q3 (Higher): Data set A has median 45 and IQR 20. Data set B has median 50 and IQR 8. Compare the two data sets.
Practise cumulative frequency questions with instant feedback — completely free on GCSEMathsAI.
Related Topics
Summary
- A cumulative frequency diagram plots running totals at the upper class boundaries.
- The median is read at the n/2 position, LQ at n/4, and UQ at 3n/4.
- IQR = UQ − LQ measures the spread of the middle 50% of the data.
- When comparing distributions, comment on both an average (median) and spread (IQR).
- Plot points at upper boundaries, join with a smooth curve, and label axes clearly.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
Further reading from leading academic institutions — free and open-access.
Cambridge problems exploring averages in context.
University of Cambridge · Free · Open AccessMean, median, mode, range — from tables and lists.
Corbett Maths · Free · Open AccessCambridge data interpretation and representation tasks.
University of Cambridge · Free · Open AccessHistograms, cumulative frequency, box plots, scatter graphs.
Corbett Maths · Free · Open Access