Plot frequencies at class midpoints and join them with straight lines, then use the polygon to compare two sets of data. It is a Foundation topic, aimed at grade 2. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A frequency polygon plots frequency against the midpoint of each class, then joins the points with straight lines. For 30<x≤45 the midpoint is 230+45=37.5. Plot (37.5,frequency). Do not join the first point down to the origin — the origin is not a class.
The class width does not have to be equal, but at this grade it usually is. Read the inequality signs: 30<x≤45 does not include 30 and does include 45. The midpoint is still halfway between the printed bounds.
Frequency polygons are for comparing two distributions on the same axes. The higher polygon in a region means more of the data sat in those classes.
Objectives
For a class 0 to 10, the midpoint is 5.
The lines between points are straight.
The height of a point is the frequency of that class.
Key points
Find the midpoint of a class interval such as 10<t≤20.
Plot each frequency at its class midpoint and join the points with straight ruled lines.
Read frequencies and the modal class from a frequency polygon.
Compare two frequency polygons drawn on the same axes using their peaks and spread.
Worked examples
Example 1
A class on a frequency polygon runs from 5 to 9. What is the midpoint of the class?
2 marks
The ends of the class are 5 and 9, and they add to 5 + 9 = 14.
Halve it: 14 ÷ 2 = 7.
The midpoint is 7.
Answer: 7
Common slips
Slip: The ends of the class are 7 and 9, and they add to 5 + 9 = 14. That changes a number from the question, so it answers a different question.
Slip: Halve it: 17 ÷ 2 = 7. That changes a number in the working, so the step no longer follows from the one before.
Slip: The midpoint is 11. That does not match the working above it.
Example 2
A class runs from 10 to 16 and has frequency 5. What are the coordinates of its point on a frequency polygon?
2 marks
The midpoint of 10 to 16 is (10 + 16) ÷ 2 = 13.
The frequency 5 is the height of the point.
The point is (13, 5).
Answer: (13, 5)
Common slips
Slip: The midpoint of 12 to 16 is (10 + 16) ÷ 2 = 13. That changes a number from the question, so it answers a different question.
Slip: The frequency 8 is the height of the point. That changes a number in the working, so the step no longer follows from the one before.
Slip: The point is (17, 5). That does not match the working above it.
Example 3
A frequency polygon has points with frequencies 5, 9 and 6. What is the total frequency?
2 marks
The total frequency is the sum of the heights: 5 + 9 + 6.
5 + 9 = 14, then 14 + 6 = 20.
The total frequency is 20.
Answer: 20
Common slips
Slip: The total frequency is the sum of the heights: 7 + 9 + 6. That changes a number from the question, so it answers a different question.
Slip: 8 + 9 = 14, then 14 + 6 = 20. That changes a number in the working, so the step no longer follows from the one before.
Slip: The total frequency is 24. That does not match the working above it.
Questions and answers
GCSE Maths frequency polygons questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Two frequency polygons use the same classes. Group A has frequencies 11 and 8. Group B has frequencies 10 and 6. How many more people are in group A than in group B?
Show answer
Answer: 3
The totals are the sums of the heights: A is 11 + 8 = 19, and B is 10 + 6 = 16.
Subtract the totals: 19 − 16 = 3.
M1 — The totals are the sums of the heights: A is 11 + 8 = 19, and B is 10 + 6 = 16.
A1 — Answer 3
Question 2 Easy2 marks
A class on a frequency polygon runs from 25 to 37. What is the midpoint of the class?
Show answer
Answer: 31
The ends of the class are 25 and 37, and they add to 25 + 37 = 62.
Halve it: 62 ÷ 2 = 31.
M1 — The ends of the class are 25 and 37, and they add to 25 + 37 = 62.
A1 — Answer 31
Question 3 Medium2 marks
Two frequency polygons use the same classes. Group A has frequencies 11 and 9. Group B has frequencies 9 and 6. How many more people are in group A than in group B?
Show answer
Answer: 5
The totals are the sums of the heights: A is 11 + 9 = 20, and B is 9 + 6 = 15.
Subtract the totals: 20 − 15 = 5.
M1 — The totals are the sums of the heights: A is 11 + 9 = 20, and B is 9 + 6 = 15.
A1 — Answer 5
Question 4 Medium2 marks
A class on a frequency polygon runs from 10 to 20. What is the midpoint of the class?
Show answer
Answer: 15
The ends of the class are 10 and 20, and they add to 10 + 20 = 30.
Halve it: 30 ÷ 2 = 15.
M1 — The ends of the class are 10 and 20, and they add to 10 + 20 = 30.
A1 — Answer 15
Question 5 Hard2 marks
Two frequency polygons use the same classes. Group A has frequencies 11 and 10. Group B has frequencies 8 and 8. How many more people are in group A than in group B?
Show answer
Answer: 5
The totals are the sums of the heights: A is 11 + 10 = 21, and B is 8 + 8 = 16.
Subtract the totals: 21 − 16 = 5.
M1 — The totals are the sums of the heights: A is 11 + 10 = 21, and B is 8 + 8 = 16.
A1 — Answer 5
Question 6 Hard2 marks
A class on a frequency polygon runs from 30 to 38. What is the midpoint of the class?
Show answer
Answer: 34
The ends of the class are 30 and 38, and they add to 30 + 38 = 68.
Halve it: 68 ÷ 2 = 34.
M1 — The ends of the class are 30 and 38, and they add to 30 + 38 = 68.
A1 — Answer 34
Question 7 Exam-style2 marks
A class runs from 10 to 14 and has frequency 10. What are the coordinates of its point on a frequency polygon?
Show answer
Answer: (12, 10)
The midpoint of 10 to 14 is (10 + 14) ÷ 2 = 12.
The frequency 10 is the height of the point.
The point is (12, 10).
M1 — The midpoint of 10 to 14 is (10 + 14) ÷ 2 = 12.
A1 — Answer (12, 10)
Common mistakes
Most lost marks on this topic are one slip.
Plotting at the end of the class instead of the midpoint.
The point is halfway along the class.
Adding the midpoints to get a total frequency.
Add the frequencies, which are the heights.
Treating the joining line as more frequencies.
Only the plotted points are data.
Spot the slip
A frequency polygon is drawn by joining
the midpoints of the bar tops
the starts of each bar
the axes
Show the slip
the midpoints of the bar tops Each point sits at the middle of the class.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
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.
Add a midpoint column to the table before you plot anything — it is a visible method step.
Use a ruler for the lines and small crosses for points; points that are a square out lose accuracy marks.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.