Frequency Diagrams for Grouped Data: KS3 Maths

What is a frequency diagram for grouped data?

A frequency diagram (sometimes called a grouped frequency diagram) displays continuous data that has been sorted into class intervals. Each bar spans one class interval, and the height of the bar shows how many data values fall within that interval.

Key features:

• Bars touch — continuous data has no gaps between groups.
• Class intervals are equal — each bar covers the same width of values.
• Frequency is on the y-axis — the count, not a proportion.
• The variable is on the x-axis — labelled with the scale of the data (e.g., height in cm).

This is different from a bar chart, which is used for categorical (separate) data and has gaps between bars.

How do you set up a grouped frequency table?

Before drawing the diagram, organise the data into a grouped frequency table.

Worked example: 20 students' test scores (out of 50) are: 12, 28, 35, 41, 19, 23, 30, 47, 38, 15, 26, 33, 44, 21, 29, 37, 11, 42, 25, 32.

Using class intervals of width 10 (0–9, 10–19, 20–29, 30–39, 40–49):

Check: The frequencies should sum to the total number of data values (20). ✓

How do you draw a frequency diagram?

Steps:

1. Draw and label the axes. The x-axis shows the variable and scale; the y-axis shows frequency.
2. Choose a suitable scale for the y-axis so that the tallest bar fits.
3. Draw a bar for each class interval. The bar extends from the lower to the upper boundary of the interval, and its height equals the frequency.
4. Ensure bars touch — do not leave gaps.
5. Give the diagram a title.

Using the test-score data above, the bars would be:

The two tallest bars are the middle two, showing most scores were in the 20–39 range.

How do you read values from a frequency diagram?

Reading a single bar: Find the bar for the class interval in question, and read off its height on the y-axis.

Estimating the total frequency: Add all bar heights.

Estimating the modal class: This is the class interval with the tallest bar. In the example, both 20–29 and 30–39 are joint modal classes.

Estimating a value within a class: You cannot tell exactly where individual values fall inside a class interval. To estimate a quantity such as the median, you often assume values are evenly spread within each class.

What mistakes should you avoid?

Mistake 1 — Leaving gaps between bars. A bar chart has gaps; a frequency diagram for continuous data does not. Gaps suggest the data is categorical when it is not.

Mistake 2 — Using unequal class intervals. If the class widths vary (e.g., 0–10, 10–30, 30–40), comparing bar heights is misleading — a wider bar does not mean a higher frequency per unit. Equal-width class intervals are required at KS3. (GCSE histograms handle unequal widths using frequency density.)

Mistake 3 — Mislabelling the x-axis. Mark the boundaries of the class intervals (10, 20, 30, …), not the midpoints or the interval labels inside each bar.

Mistake 4 — Forgetting the title and axis labels. Every statistical diagram must have a descriptive title, a labelled x-axis (including units), and a labelled y-axis.

How is a frequency diagram different from a GCSE histogram?

At KS3, frequency diagrams always use frequency on the y-axis with equal class widths. At GCSE, a histogram uses frequency density (frequency ÷ class width) on the y-axis, allowing unequal class widths to be shown fairly. The area of each bar, not its height, represents the frequency in a GCSE histogram. At KS3 you only need the simpler frequency diagram.

Frequently asked questions

Do I always need to start the x-axis at 0?

Not necessarily — start the x-axis at the lower boundary of your first class interval. If all scores are between 10 and 50, start at 10 rather than 0 (you may draw a small zigzag on the axis to show it does not start at zero). The y-axis for frequency should start at 0.

What if two class intervals have the same frequency?

Draw both bars at the same height. This is completely fine and simply shows that those two groups contain the same number of data values. You describe this as "joint modal classes" if they are the tallest bars.

How do I choose a good class width?

Aim for 5 to 8 class intervals — too few and you lose information; too many and the diagram becomes hard to read. A class width that divides the data range into roughly equal groups is usually best. For 20 data values, class intervals of width 10 are typical.

Can a frequency diagram have a bar of height 0?

Yes. If no data values fall in a class interval, that bar has a height of 0. You still draw the interval on the x-axis to show it exists; you simply draw no bar (or a bar of zero height) for that class. Omitting the interval from the axis would suggest the scale jumps, which is misleading.

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