Frequency Polygons: KS3 Maths Explained

A frequency polygon is a line graph drawn from a grouped frequency table. Each point is plotted at the midpoint of the class interval, at the corresponding frequency. The points are then joined in order with straight lines. Frequency polygons make it easy to compare two distributions on the same axes.

What is a frequency polygon used for?

Frequency polygons display the shape of a distribution — whether it is roughly symmetrical, skewed to one side, or has a clear peak. Because they are drawn as lines rather than bars, two or more frequency polygons can be overlaid on the same graph, making them ideal for comparing groups (for example, the test scores of two different classes).

They are commonly used in KS3 statistics alongside bar charts, histograms, and stem-and-leaf diagrams.

How do you find the midpoints of class intervals?

The midpoint of a class interval is the average of its lower and upper boundaries.

Midpoint = (lower boundary + upper boundary) ÷ 2

Class interval | Lower | Upper | Midpoint
0 ≤ t < 10 | 0 | 10 | (0 + 10) ÷ 2 = 5
10 ≤ t < 20 | 10 | 20 | (10 + 20) ÷ 2 = 15
20 ≤ t < 30 | 20 | 30 | 25
30 ≤ t < 40 | 30 | 40 | 35
40 ≤ t < 50 | 40 | 50 | 45

Equal-width class intervals all have midpoints spaced the same distance apart — in the example above, every 10 units.

How do you draw a frequency polygon step by step?

Worked example: The table below shows the time (in minutes) taken by 30 students to complete a puzzle.

Time (minutes) | Frequency
0 ≤ t < 10 | 3
10 ≤ t < 20 | 8
20 ≤ t < 30 | 11
30 ≤ t < 40 | 6
40 ≤ t < 50 | 2

Step 1 — Find the midpoints: 5, 15, 25, 35, 45.

Step 2 — Write out the coordinate pairs (midpoint, frequency): (5, 3), (15, 8), (25, 11), (35, 6), (45, 2).

Step 3 — Draw a set of axes. Label the horizontal axis "Time (minutes)" and the vertical axis "Frequency." Mark a suitable scale.

Step 4 — Plot each coordinate pair as a point.

Step 5 — Join the points in order with straight lines.

Step 6 — Do not join the first and last points to the origin or to each other unless the question specifically asks you to close the polygon.

How do you interpret a frequency polygon?

Peak: the class with the highest frequency. In the example, the peak is at t = 25, meaning most students took between 20 and 30 minutes.

Shape: the distribution rises to a peak at 25 minutes then falls — this is a roughly bell-shaped (symmetric) distribution.

Total frequency: add all the frequencies. Here 3 + 8 + 11 + 6 + 2 = 30 students — a useful check.

How do you compare two frequency polygons?

Draw both on the same pair of axes using the same scale. Use different colours or a key to distinguish them.

When comparing:

Compare the peaks — which group has a higher modal class? Which is faster/slower, heavier/lighter, etc.?

Compare the spread — which polygon is wider (more varied) or narrower (more consistent)?

Compare the shapes — is one distribution skewed? Does one have a higher frequency at the extremes?

Always refer back to the context — for example, "Class B generally completed the puzzle faster, with a peak at 15–25 minutes compared to Class A's peak at 25–35 minutes."

What mistakes should you avoid?

Plotting at the class boundaries instead of the midpoints. Points go at the midpoint (e.g. 25), not at the boundary (e.g. 20 or 30).

Joining points with a curve. Frequency polygons are drawn with straight lines between each pair of points — not smooth curves.

Not labelling the axes. Always label both axes, including units.

Forgetting to add a key when comparing two polygons on the same graph.

Frequently asked questions

What is the difference between a frequency polygon and a histogram?

A histogram uses bars whose areas represent frequencies (or frequency densities). A frequency polygon connects the midpoints of the tops of those bars with straight lines. Frequency polygons make it easier to compare groups because you can overlay lines, whereas overlapping bars become confusing.

Do I need to close the frequency polygon into a complete shape?

Not usually at KS3. Some textbooks ask you to close the polygon by joining the first and last points to the x-axis (at zero frequency), creating a proper polygon shape. Follow the instructions in the question — if it says "draw a frequency polygon," joining the outer midpoints to the axis is good practice but not always required.

Can frequency polygons be used with unequal class widths?

Technically, if class widths are unequal, frequency density (not frequency) should be plotted on the vertical axis — and that becomes a histogram rather than a standard frequency polygon. At KS3, frequency polygons are almost always drawn from tables with equal class widths, so frequency is plotted directly.

How do frequency polygons link to the mean of grouped data?

The midpoints you use to plot a frequency polygon are the same midpoints used to estimate the mean from a grouped frequency table. Estimated mean = Σ(midpoint × frequency) ÷ total frequency. So mastering midpoints helps with two important statistics skills at once.

For Socratic KS3 statistics and data practice, see aitutors.me.