Frequency polygons GCSE Questions and Worked Answers
A frequency polygon plots each class frequency at its class midpoint and joins neighbouring points with straight segments. It shows counts across numerical groups, not cumulative totals.
Foundation & Higher6 worked examples10 original questions
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Start with the meaning
What you need to know about frequency polygons
Twenty runners finish a course. Four take up to 10 minutes, nine take more than 10 and up to 20, and seven take more than 20 and up to 30. We no longer know each exact time; we know a count for each interval. A frequency polygon places one dot in the middle of each interval, at the height of its count. Joining the dots makes the pattern of counts easier to see.
See the idea first
One point summarises a whole interval
The first class, 0 < t ≤ 10, is represented at its midpoint 5 with frequency 4. The point (5, 4) does not claim that four people took exactly five minutes. Similarly (15, 9) stands for nine runners somewhere in the second interval.
The groups are 0 < t ≤ 10, 10 < t ≤ 20 and 20 < t ≤ 30. Plot each count at its class midpoint, then join adjacent points with straight lines.
From problem to method
Typical problems you need to be able to solve
These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.
Construct a polygon
What the problem asks: Time classes 0 < t ≤ 10, 10 < t ≤ 20 and 20 < t ≤ 30 minutes have frequencies 4, 9 and 7. Draw the polygon.
How to solve it: Plot (5,4), (15,9), (25,7) and join in increasing midpoint order with straight lines. Label time and frequency.
Read a modal class
What the problem asks: Which class is most frequent in the displayed polygon?
How to solve it: The highest point is at midpoint 15, so the modal class is 10 < t ≤ 20. The mode is not known to be exactly 15.
Compare groups
What the problem asks: One equal-sized group's polygon is concentrated around larger times than another. What might this show?
How to solve it: Its times tend to be longer. Compare centres and spread in context; a single taller point does not describe the entire distribution.
Estimate a mean from grouped values
What the problem asks: Ten journeys are grouped as 0 < t ≤ 10 minutes (6 journeys) and 10 < t ≤ 20 minutes (4 journeys). Estimate the mean time.
How to solve it: Exact times are lost, so use each class midpoint as a representative value: 5 and 15 minutes. The estimated total is 6 × 5 + 4 × 15 = 90 minutes. Divide by all 10 journeys: estimated mean = 9 minutes. The midpoint is an estimate for each observation, not its known time.
A reliable routine
Plot grouped frequency data
Use a class midpoint to give each interval one representative horizontal position. The vertical coordinate is that class's frequency, not a running total.
For each class calculate (lower boundary + upper boundary) ÷ 2.
Plot (midpoint, frequency) using labelled consistent scales.
Join adjacent points with straight lines, not a smooth curve.
Follow any specified endpoint convention; do not invent observed zero-frequency classes.
Check: The examples use equal-width classes. For unequal-width continuous intervals, a histogram with frequency density is the usual GCSE comparison. Edexcel includes frequency polygons in S2 guidance; OCR does not require prior knowledge of them.
Fully worked
Frequency polygons GCSE worked examples
Each solution explains the clue, why the method fits and how to check the result.
Example 1
Find a midpoint
2 marks
Question
Find the midpoint of 20 < t ≤ 30.
The midpoint lies halfway between the boundaries.
(20+30)/2=25
Example 2
Construct from a table
3 marks
Question
Draw the polygon for 0 < t ≤ 10: 4; 10 < t ≤ 20: 9; 20 < t ≤ 30: 7.
Midpoints are 5, 15, 25. Plot (5,4), (15,9), (25,7) and join adjacent points. Label time in minutes and frequency.
The groups are 0 < t ≤ 10, 10 < t ≤ 20 and 20 < t ≤ 30. Plot each count at its class midpoint, then join adjacent points with straight lines.
Example 3
Total frequency
2 marks
Question
Use the displayed polygon to find the total number of runners.
The groups are 0 < t ≤ 10, 10 < t ≤ 20 and 20 < t ≤ 30. Plot each count at its class midpoint, then join adjacent points with straight lines.
Add the class frequencies once.
4+9+7=20
Do not add midpoint values.
Example 4
Most frequent interval
2 marks
Question
State the modal class in the displayed polygon.
The groups are 0 < t ≤ 10, 10 < t ≤ 20 and 20 < t ≤ 30. Plot each count at its class midpoint, then join adjacent points with straight lines.
The highest frequency is 9 at midpoint 15. It represents 10 < t ≤ 20 minutes, not exactly 15 minutes.
Example 5
An estimated mean
4 marks
Question
Estimate mean time using the displayed grouped data.
The groups are 0 < t ≤ 10, 10 < t ≤ 20 and 20 < t ≤ 30. Plot each count at its class midpoint, then join adjacent points with straight lines.
Use each midpoint as a representative value.
5×4+15×9+25×7=330Estimated mean=330/20=16.5 min
Exact individual times were not recorded, so this is an estimate.
Example 6
Compare distributions
3 marks
Question
For the same three time intervals, group A has frequencies 8, 8, 4 and B has 3, 7, 10. Each group has 20 runners. Which tends to take longer?
B has more runners in the longest interval and fewer in the shortest. Midpoint estimates confirm it: A mean = (40 + 120 + 100)/20 = 13 min; B mean = (15 + 105 + 250)/20 = 18.5 min. B tends to take longer; not every B runner is slower.
10 original questions · total 22 marks
Frequency polygons GCSE exam-style questions
Try each question before opening its fully worked answer. The difficulty rises through the set.
Before you startAllow about 27 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
1
Midpoint
2 marks
Find the midpoint of 30 < x ≤ 40.
Show worked answer
(30+40)/2=35
2
Coordinate
2 marks
The class 40 < x ≤ 50 has frequency 7. Which point is plotted?
Show worked answer
Midpoint 45, frequency 7: (45, 7).
3
Read frequency
1 mark
A point (25, 11) represents class 20 < t ≤ 30. What is its frequency?
Show worked answer
11. The horizontal 25 locates the interval; the vertical 11 counts observations.
4
Total
2 marks
Class frequencies are 5, 12, 8 and 3. Find total frequency.
Show worked answer
5+12+8+3=28
5
Not cumulative
2 marks
Classes have frequencies 4, 6 and 3. Should polygon heights be 4, 10 and 13?
Show worked answer
No. Use 4, 6, 3. Heights 4, 10, 13 are running totals for cumulative frequency.
6
Missing frequency
2 marks
Total frequency is 30; the other three classes have 6, 9 and 8. Find the missing frequency.
Show worked answer
30−(6+9+8)=7
7
Zero class
2 marks
An actual class 10 < t ≤ 20 has zero observations. What point represents it?
Show worked answer
Plot (15, 0). This zero comes from supplied data, not an invented endpoint.
8
Mean estimate
4 marks
Classes 0 < x ≤ 10 and 10 < x ≤ 20 have frequencies 6 and 4. Estimate mean.
Show worked answer
Midpoints are 5 and 15.
xˉ≈(5(6)+15(4))/10=90/10=9
9
Lost exact values
2 marks
A class 20 < x ≤ 30 has five observations. Does point (25, 5) prove that all five equal 25?
Show worked answer
No. It represents five values somewhere in that interval. The midpoint is a plotting convention, not recovered individual data.
10
Unequal sample sizes
3 marks
The same class has frequency 12 out of 20 in A and 15 out of 50 in B. Which has the larger proportion there?
Show worked answer
12/20=60%15/50=30%
A has the larger proportion despite its lower raw frequency.
Examiner-style feedback
Common frequency polygons mistakes
Plotting upper boundaries
Use midpoints for a frequency polygon; upper boundaries belong to cumulative-frequency plotting.
Joining smoothly
A frequency polygon uses straight segments between plotted points.
Reading the midpoint as an exact observed mode
The highest point identifies a class, not the original individual mode.
30-second recap
Choose, carry out, check
Translate the wording before you reach for a rule.
Use class midpoints.
Use separate class counts.
Join with straight lines.
Treat grouped means as estimates.
Quick answers
Frequency polygons FAQ
Must I close the polygon to the axis?
Follow the question or teacher's specified convention. Do not invent observed classes; any added zero-frequency endpoints are only a drawing convention.
Does OCR require frequency polygons?
OCR states that prior knowledge is not required; Edexcel explicitly includes them.
S2/S4 grouped-data representations and summaries; Edexcel inclusion and OCR prior-knowledge exception explicitly distinguished. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.
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