HistogramsEdexcel IGCSE Maths: Revision notes

A histogram displays continuous grouped data, and unlike a bar chart, the vertical axis is NOT frequency — it is frequency density.

This matters because histograms are often used when class widths are unequal. If you plotted frequency directly, a wide class with more data points just because it spans a bigger range would look misleadingly tall. Frequency density corrects for this by scaling frequency to the width of each class.

$\text{Frequency density} = \frac{\text{Frequency}}{\text{Class width}}$

The key rule: in a histogram, it is the area of each bar that represents the frequency, not the height.

Work along each row of the table: find the class width, then divide frequency by that width.

Notice the class widths are unequal (10, 10, 20, 30) — this is exactly when frequency density is essential, otherwise the wider bars would be drawn too tall relative to their true frequency.

Since area represents frequency:

$\text{Frequency} = \text{Frequency density} \times \text{Class width}$

This is the reverse of the density calculation and is examined constantly — you are given a drawn histogram (with a scale on the frequency density axis) and asked to find how many data values fall in a class, or in part of a class.

Worked example: A bar has frequency density 1.2 over the class $20 \le t < 40$ (width 20). $\text{Frequency} = 1.2 \times 20 = 24$

If a question asks for a partial class (e.g. how many values lie in $20 \le t < 30$), use the SAME frequency density across that sub-width, since density is constant within one bar: $1.2 \times 10 = 12$

Sometimes one bar's frequency is given (e.g. 'this bar represents 24 students') but the frequency density axis has no numbers. Use that one known bar to work out the scale, then apply it to the rest.

Method:

1. Use the known bar: $\text{frequency density} = \text{frequency} \div \text{class width}$.
2. This tells you what one unit of height on the axis is worth (e.g. "1 small square = 0.2").
3. Measure the heights of the other bars using this scale, then multiply each by its own class width to get frequency.

This is the classic 'find the missing frequency' histogram question.

With equal class widths, frequency density is just proportional to frequency, so bar heights look 'normal' and mistakes are less costly. With unequal widths, examiners specifically test whether you:

• Remember to divide by class width before plotting (not just frequency).
• Remember to multiply by class width to recover frequency (not just read the height).
• Handle a much wider or narrower final class correctly (e.g. an 'open' class like $70 \le t < 100$, width 30).