Sheet № 121 · Higher only · AQA · Edexcel · OCR
Cumulative Frequency Diagrams –
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 co
§Key definitions
Question:
The table shows the times (in minutes) 60 students spent on homework. Draw a cumulative frequency diagram and estimate the median.
Answer:
The estimated median homework time is approximately 27 minutes.
Q1 (Foundation):
Build a cumulative frequency table for these test scores and state the position of the median.
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.
§Formulas to memorise
Median position = n ÷ 2
Lower quartile position = n ÷ 4
Upper quartile position = 3n ÷ 4
IQR = Upper quartile − Lower quartile
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
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
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.
⚠ 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.