Build a running total from a grouped table, draw the curve, and read off the median, quartiles and interquartile range. It is Higher only, aimed at grade 6. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Cumulative frequency is a running total of frequencies. Plot it against the upper class boundary, join with a smooth curve (or straight sections if asked). The median is read at 2n on the cf axis, across to the curve, down to the data axis. Lower quartile at 4n, upper at 43n. IQR =UQ−LQ.
A comparison of two cf curves: further right means generally larger values; a shallower middle means more spread. Always read from the curve, not from a table guess at the midpoint. n is the final cumulative frequency, the total number of people.
Plot upper class boundaries against running totals, join with a smooth curve. Median at 2n (or 2n+1), LQ at 4n, UQ at 43n. Read across then down.
Objectives
Read the median at half the total frequency.
Read the lower and upper quartiles at a quarter and three quarters.
Compare two medians and say which set is higher.
Key points
Add frequencies as you go down the table to get cumulative frequencies.
Plot each cumulative frequency at the upper end of its class and join with a smooth curve.
Find the median at 2n, the lower quartile at 4n and the upper quartile at 43n on the vertical axis.
Calculate the interquartile range (IQR) = UQ − LQ.
Worked examples
Example 1
The total frequency is 24. The curve reads 20 at cumulative frequency 12. Find the median.
2 marks
Calculate half the total frequency: 24 ÷ 2 = 12.
Locate 12 on the vertical cumulative frequency axis and read across to the curve.
Read down to the horizontal axis to get 20.
Answer: 20
Common slips
Slip: Take the total frequency 24 as the position. The median is located at the 50% cumulative frequency.
Slip: Read 12 directly as the answer. 12 is only the frequency rank, not the variable value.
Slip: Stop at the vertical axis. The median data value is read on the horizontal axis.
Example 2
The total frequency is 24. The curve reads 14 at cumulative frequency 6. Find the lower quartile.
2 marks
Find one quarter of the total frequency: 24 ÷ 4 = 6.
Locate 6 on the vertical cumulative frequency axis and trace across.
Read down to the horizontal axis to find 14.
Answer: 14
Common slips
Slip: Halve 24 to get 12. 12 is the median position. The lower quartile sits at the quarter mark, 24 ÷ 4 = 6.
Slip: Use 6 as the answer. 6 is the position, not the data value.
Slip: Write 6. The lower quartile is 14.
Example 3
Which set has the higher median? Set A has median 20 and set B has median 28.
3 marks
Compare the two median values: 20 and 28.
Identify that 28 > 20.
State that Set B has the higher median.
Answer: B
Common slips
Slip: Compare the spreads of the sets. The question specifically asks about the medians.
Slip: Select Set A because 20 is smaller. A higher median corresponds to the greater number.
Slip: Choose Set A. Set B has the higher median.
Questions and answers
GCSE Maths cumulative frequency questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
The total frequency is 40. The curve reads 18 at cumulative frequency 20. Find the median.
Show answer
Answer: 18
Half of 40 is 20.
The curve reads 18 at cumulative frequency 20.
M1 — Half of 40 is 20.
A1 — The curve reads 18 at cumulative frequency 20.
Question 2 Easy2 marks
The total frequency is 20. The curve reads 15 at cumulative frequency 10. Find the median.
Show answer
Answer: 15
Half of 20 is 10.
The curve reads 15 at cumulative frequency 10.
M1 — Half of 20 is 10.
A1 — The curve reads 15 at cumulative frequency 10.
Question 3 Medium2 marks
The total frequency is 40. The curve reads 11 at cumulative frequency 10. Find the lower quartile.
Show answer
Answer: 11
A quarter of 40 is 10.
The curve reads 11 at cumulative frequency 10.
M1 — A quarter of 40 is 10.
A1 — The curve reads 11 at cumulative frequency 10.
Question 4 Medium2 marks
The total frequency is 20. The curve reads 8 at cumulative frequency 5. Find the lower quartile.
Show answer
Answer: 8
A quarter of 20 is 5.
The curve reads 8 at cumulative frequency 5.
M1 — A quarter of 20 is 5.
A1 — The curve reads 8 at cumulative frequency 5.
Question 5 Hard2 marks
Which set has the higher median? Set A has median 15 and set B has median 19.
Show answer
Answer: B
Compare medians 15 and 19.
19 is greater than 15, so set B is higher.
M1 — Compare medians 15 and 19.
A1 — 19 is greater than 15, so set B is higher.
Question 6 Hard2 marks
Which set has the higher median? Set A has median 40 and set B has median 30.
Show answer
Answer: A
Compare medians 40 and 30.
40 is greater than 30, so set A is higher.
M1 — Compare medians 40 and 30.
A1 — 40 is greater than 30, so set A is higher.
Question 7 Exam-style4 marks
The total frequency is 60. The curve reads 18 at cumulative frequency 15 and 41 at cumulative frequency 45. Work out the interquartile range.
Show answer
Answer: 23
A quarter of 60 is 15, so the lower quartile is 18.
Three quarters of 60 is 45, so the upper quartile is 41.
Interquartile range = 41 - 18 = 23.
M1 — Positions 15 and 45
A1 — Lower quartile 18 and upper quartile 41
M1 — 41 - 18
A1 — 23
Common mistakes
Most lost marks on this topic are one slip.
Writing the cumulative frequency instead of the curve reading
Read across from half the total, then down to the value
Using half the total for a quartile
Quartiles use one quarter and three quarters
Choosing the set with the smaller median as the higher set
The higher median is the larger number
Spot the slip
The total frequency is 24. The curve reads 20 at cumulative frequency 12. What is the median?
20
24
12
Show the slip
20 Half of 24 is 12, and the curve reads 20 there.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Draw your reading lines on the graph in pencil and leave them there — they show the marker exactly how you got your numbers.
Write the positions (4n, 2n, 43n) before reading; this earns method marks even if your reading is slightly off.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.