Frequency polygons
A frequency polygon is a line graph for grouped data. You plot the frequency of each class at the midpoint of its class interval, then join the points with straight lines. Frequency polygons show the shape of a distribution at a glance, and because they are just lines, two or more can share one set of axes for comparison.
How to draw a frequency polygon
Forty people (group A) did a puzzle and their times were recorded in this grouped frequency table. The class 5 < t ≤ 10 means more than 5 minutes, up to and including 10.
| Time, t (minutes) | Frequency | Midpoint |
|---|---|---|
| 0 < t ≤ 5 | 4 | 2.5 |
| 5 < t ≤ 10 | 11 | 7.5 |
| 10 < t ≤ 15 | 15 | 12.5 |
| 15 < t ≤ 20 | 7 | 17.5 |
| 20 < t ≤ 25 | 3 | 22.5 |
- Find the midpoint of each class. Add the two class boundaries and halve: (5 + 10) ÷ 2 = 7.5.
- Draw the axes. Put the data (time) along the horizontal axis with a steady scale, and frequency up the vertical axis.
- Plot each point at (midpoint, frequency): (2.5, 4), (7.5, 11), (12.5, 15), (17.5, 7) and (22.5, 3).
- Join the points with straight lines, in order, using a ruler. Don't draw a smooth curve.
Join only the points you plotted unless the question tells you otherwise. Some textbooks close the shape by adding a point with frequency 0 at the midpoint of an empty class at each end, which turns the line into a true polygon.
Reading a frequency polygon
- The highest point shows the modal class. For group A it is 10 < t ≤ 15, with 15 people.
- Add the frequencies to get the total. 4 + 11 + 15 + 7 + 3 = 40 people.
- You can count above or below a class boundary. People who took more than 15 minutes: 7 + 3 = 10, which is 10 ÷ 40 = 25%.
- You can't read exact values. The data is grouped, so the graph doesn't tell you any individual time, and the lines between points don't represent real data.
Estimating the mean
The midpoints you plotted also give an estimate of the mean. Multiply each midpoint by its frequency, add them up and divide by the total frequency:
(2.5 × 4) + (7.5 × 11) + (12.5 × 15) + (17.5 × 7) + (22.5 × 3) = 10 + 82.5 + 187.5 + 122.5 + 67.5 = 470, and 470 ÷ 40 = 11.75 minutes.
It's an estimate because we have assumed everyone in a class took the midpoint time. The mean, median and mode calculator works this out for any table.
Comparing two frequency polygons
This is where frequency polygons beat bar charts. A second group of 40 (group B) did the same puzzle:
| Time, t (minutes) | 0 < t ≤ 5 | 5 < t ≤ 10 | 10 < t ≤ 15 | 15 < t ≤ 20 | 20 < t ≤ 25 |
|---|---|---|---|---|---|
| Group A frequency | 4 | 11 | 15 | 7 | 3 |
| Group B frequency | 1 | 5 | 12 | 14 | 8 |
To compare them, make one point about average and one about spread or shape, and say what it means in context:
- Average: group B's modal class is 15 < t ≤ 20, higher than group A's 10 < t ≤ 15. B's estimated mean is 615 ÷ 40 = 15.4 minutes (1 d.p.), against 11.75 for A. On average, group B took longer.
- Shape: group B's polygon is shifted to the right, and more of its people are in the slower classes: 22 people in B took more than 15 minutes, against 10 in A.
When the totals are different, compare proportions or percentages rather than raw frequencies, or the bigger group's polygon will simply look taller.
Frequency polygons, histograms and frequency diagrams
"Frequency diagram" is a general name for any chart that shows frequencies. The main types:
| Diagram | Best for | How frequency is shown |
|---|---|---|
| Bar chart | Categories, or ungrouped discrete data | Height of separate bars |
| Histogram | Grouped continuous data | Area of bars with no gaps (frequency density for unequal widths) |
| Frequency polygon | Grouped data, especially to compare data sets | Points at class midpoints joined by straight lines |
| Cumulative frequency graph | Medians, quartiles and running totals | Running total plotted at each upper class boundary |
If a histogram has equal class widths, joining the midpoints of the tops of its bars gives the frequency polygon for the same data. For ungrouped discrete data, such as the number of goals scored in each match, you can draw a frequency polygon by plotting each value at its own frequency. The histogram maker draws histograms for grouped data, and the cumulative frequency guide covers running totals.
Common mistakes
- Plotting at the class boundaries. Use the midpoint, not the start or end of each class.
- Drawing bars. A frequency polygon is points joined by lines, with no bars.
- Joining the last point back to the first. The line goes from left to right only.
- Uneven scales. The horizontal axis must go up in equal steps, even if the first class doesn't start at 0.
- Comparing without context. "B is higher" isn't enough. Say "on average, group B took longer to finish the puzzle".
Practise statistics questions alongside other topics in today's GCSE Maths Foundation or GCSE Maths Higher short timed paper.
Common questions
What is a frequency polygon?
How do you draw a frequency polygon?
Do you plot frequency polygons at the midpoint?
What is the difference between a histogram and a frequency polygon?
Do you join the ends of a frequency polygon to the axis?
What is a frequency diagram?
Revision cards
Say the answer first, then flip the card to check it.
Where to go next
- Solve it
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Maths · StatisticsFrequency density formula: how to work it outThe frequency density formula, how to work it out for a histogram with unequal class widths, reading frequencies back, estimating from bars, and practice.