GCSE Maths Histograms: Why Frequency Density Matters
Learn why histogram bars use frequency density, how to calculate it for unequal class widths, and how to find frequencies from bar areas.
By Math School Editorial Team
The key idea: area, not height
A histogram displays numerical data grouped into intervals. Unlike an ordinary bar chart, the horizontal widths of histogram bars can represent different-sized intervals. When the class widths are unequal, comparing the heights alone is misleading.
The rule is:
Frequency density = frequency ÷ class width.
Equivalently:
Frequency = frequency density × class width.
That means the area of a bar represents its frequency. The AQA GCSE Mathematics specification includes histograms with both equal and unequal class intervals.
Example 1: Build a histogram table
Suppose 75 students record their journey time to school.
| Journey time (minutes) | Frequency | Class width | Frequency density |
|---|---|---|---|
| 0 to less than 10 | 12 | 10 | 1.2 |
| 10 to less than 20 | 18 | 10 | 1.8 |
| 20 to less than 40 | 30 | 20 | 1.5 |
| 40 to less than 70 | 15 | 30 | 0.5 |
| Total | 75 |
For 20–40 minutes, the width is 40 − 20 = 20, so the density is 30 ÷ 20 = 1.5. For 40–70, the width is 30, so the density is 15 ÷ 30 = 0.5.
To draw the histogram, each rectangle covers its stated interval and rises to its density. The 20–40 rectangle is twice as wide as the 10–20 rectangle.
Which interval has the most students?
The tallest bar is 10–20 minutes, with density 1.8, but it represents only 18 students.
The 20–40 minute bar is shorter, at density 1.5, but wider. Its area is:
20 × 1.5 = 30 students.
Therefore 20–40 minutes has the highest frequency, even though its bar is not the tallest. The essential question is: “Are we comparing bar heights or bar areas?”
Example 2: Recover a missing frequency
Suppose a histogram bar covers 15–35 minutes and has frequency density 2.4.
Class width = 35 − 15 = 20.
Frequency = 20 × 2.4 = 48.
A common error is to report 2.4 as the number of observations. It is the density, not the frequency.
Example 3: Calculate a proportion
In the journey-time table, 30 of the 75 students travelled for 20 to less than 40 minutes.
Proportion = 30 ÷ 75 = 0.4, or 40%.
If a student is chosen at random from these 75 records, the probability of falling into this class is 0.4. However, grouped data do not reveal the exact journey time of every student.
Histogram versus bar chart
A bar chart might compare categories such as “bus”, “car” and “walking”. Each category is distinct, and the bar height typically represents its count.
A histogram uses numerical intervals, such as 0–10 minutes and 10–20 minutes. The intervals sit next to one another. When class widths differ, bar area represents frequency. Drawing all the classes at equal widths would misrepresent the data.
Be consistent with boundaries. A journey of exactly 10 minutes belongs in “10 to less than 20”, not “0 to less than 10”.
Three mistakes to avoid
Mistake 1: Plotting frequency as height when class widths differ. Divide frequency by width to find density.
Mistake 2: Assuming the tallest bar has the largest frequency. Compare areas, not heights.
Mistake 3: Treating grouped data as exact individual values. A histogram summarises a distribution; it does not identify every measurement.
Your turn: an original GCSE-style question
A histogram bar extends from 30 to 50 on the horizontal axis. Its frequency density is 1.6.
(a) What is the class width? (b) How many observations does the bar represent? (c) If there are 80 observations altogether, what percentage lies in this class?
Answers: (a) 50 − 30 = 20. (b) 20 × 1.6 = 32. (c) 32 ÷ 80 × 100 = 40%.
The strongest explanation is not simply quoting a formula. It is recognising that the width and height of a rectangle multiply to its area, which represents the frequency.
How parents can help
Give your child four unequal class intervals and their frequencies. Ask them to calculate densities before drawing the histogram. Next, hide the frequency column and see whether they can reconstruct it from the bar widths and heights.
Ask why the tallest bar might not contain the most observations. If they can answer in terms of area, they understand the core idea.
Official references
-
AQA GCSE Mathematics 8300: Statistics — histograms with equal and unequal class intervals.
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AQA GCSE Statistics 8382: Formula appendix — density formula; this is a separate GCSE Statistics qualification, cited for the formula only.
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OCR: Mathematical Skills Handbook — histogram bar widths and densities.
