Frequency polygons appear regularly on GCSE Maths papers at Foundation and Higher tier. They are a quick way to display grouped data and are especially useful when you need to compare two distributions on the same axes. Unlike bar charts, frequency polygons use a single plotted line, making overlapping comparisons much clearer. This guide explains how to draw and interpret them, works through full exam-style examples and highlights the mistakes that cost marks. For context on where this sits in the specification, see our complete GCSE Maths topics list.
What Is a Frequency Polygon?
A frequency polygon is a line graph drawn from a grouped frequency table. Each class interval is represented by a single point plotted at the midpoint of the class on the horizontal axis and at the frequency on the vertical axis. The points are then joined with straight line segments.
Key Formulas
For example, the midpoint of the class 10 ≤ x < 20 is (10 + 20) ÷ 2 = 15.
Step-by-Step Method
- Calculate the midpoint of each class interval.
- Plot each point at (midpoint, frequency) on a grid.
- Join the points with straight lines in order from left to right.
- Label your axes — the horizontal axis shows the data variable with units; the vertical axis shows frequency.
- Do not join the first or last point back to the horizontal axis unless the question asks you to.
Comparing Two Distributions
To compare, draw both frequency polygons on the same set of axes using different colours or line styles. Then comment on the position (central tendency) and spread of each distribution.
Worked Example 1 — Foundation Level
Question: The table shows the masses (g) of 50 apples.
| Mass (g) | 80–99 | 100–119 | 120–139 | 140–159 | 160–179 |
|---|---|---|---|---|---|
| Frequency | 5 | 12 | 18 | 10 | 5 |
Draw a frequency polygon for this data.
Working:
Midpoints: 89.5, 109.5, 129.5, 149.5, 169.5.
Plot the points (89.5, 5), (109.5, 12), (129.5, 18), (149.5, 10), (169.5, 5) and join them with straight lines.
Answer: A line graph with five plotted points joined in order, peaking at (129.5, 18).
Worked Example 2 — Higher Level
Question: The frequency polygons for the test scores of two classes are drawn on the same axes. Class A peaks at (55, 14) and Class B peaks at (65, 12). Class A has most of its data between 35 and 65, while Class B has most of its data between 45 and 85. Compare the two distributions.
Working:
Average: Class B's peak and general position are further to the right, suggesting Class B scored higher on average.
Spread: Class A's data is concentrated between 35 and 65 (a range of 30 marks), while Class B spans 45 to 85 (a range of 40 marks). Class B's results are more spread out.
Answer: Class B scored higher on average but with more variation in marks.
Worked Example 3 — Exam Style
Question: The grouped frequency table below shows the heights (cm) of 60 seedlings.
| Height (cm) | 0 ≤ h < 4 | 4 ≤ h < 8 | 8 ≤ h < 12 | 12 ≤ h < 16 | 16 ≤ h < 20 |
|---|---|---|---|---|---|
| Frequency | 6 | 15 | 22 | 12 | 5 |
(a) Draw a frequency polygon. (b) Estimate the class interval that contains the median.
Working:
(a) Midpoints: 2, 6, 10, 14, 18. Plot (2, 6), (6, 15), (10, 22), (14, 12), (18, 5) and join with straight lines.
(b) Total = 60, so the median is the 30th value. Cumulative frequencies: 6, 21, 43, 55, 60. The 30th value falls in the class 8 ≤ h < 12.
Answer: (a) Frequency polygon plotted at midpoints. (b) The median lies in the 8 ≤ h < 12 class.
Common Mistakes
- Plotting at the class boundaries instead of midpoints. The point must be plotted at the midpoint of each class interval, not at the start or end.
- Using unequal class widths without adjusting. If class widths vary, you may need frequency density (see histograms) rather than a frequency polygon.
- Forgetting to label axes or give a title. The examiner will deduct marks for missing labels.
Exam Tips
- Always calculate midpoints first and write them in a table column — this avoids plotting errors.
- When comparing two distributions, make at least two statements: one about the centre (e.g. "the peak is further to the right") and one about the spread.
- Use a ruler and plot points as small crosses for accuracy.
- For grouped data averages, see frequency tables and grouped data. For cumulative frequency, see cumulative frequency and box plots.
Practice Questions
Q1 (Foundation): A grouped frequency table for the time (minutes) spent on homework by 40 students has classes 0–9, 10–19, 20–29, 30–39 with frequencies 4, 14, 16, 6. Find the midpoints and plot the frequency polygon.
Q2 (Foundation): From Q1, which class interval contains the mode?
Q3 (Higher): Two frequency polygons are drawn for daily rainfall (mm) in June and December. June's polygon peaks at (8, 12) and December's peaks at (16, 10). June's data spans 0–20 mm and December's spans 6–28 mm. Compare the two distributions.
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Related Topics
Summary
A frequency polygon plots the midpoint of each class interval against its frequency, with points joined by straight line segments. Calculate midpoints using (lower + upper) ÷ 2. Frequency polygons are particularly useful for comparing two distributions on the same axes — comment on both the centre and spread. Always label axes, plot at midpoints (not boundaries), and use a ruler for accuracy. This topic links closely to grouped frequency tables, histograms and cumulative frequency diagrams.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
Further reading from leading academic institutions — free and open-access.
Angle properties and polygon investigations from Cambridge.
University of Cambridge · Free · Open AccessAngle rules, parallel lines, interior and exterior angles.
Corbett Maths · Free · Open AccessCambridge problems exploring averages in context.
University of Cambridge · Free · Open AccessMean, median, mode, range — from tables and lists.
Corbett Maths · Free · Open Access