Skip to content
VirtusAcademy

Histograms with Unequal Class Widths

HigherHigher tier onlyAQAEdexcel

Understand Histograms with Unequal Class Widths for GCSE Statistics with this free worksheet and full mark scheme — Higher tier exam-style questions with worked answers for AQA and Edexcel. When class widths are unequal, histograms use frequency density (frequency divided by class width) so that area represents frequency.

Free downloads

These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.

This is a Higher tier only topic, so there's no Foundation paper — only the Higher worksheet and mark scheme below.

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

When class widths are unequal, the area of each bar must represent the frequency, not its height. This is a Higher-only topic.

The vertical axis therefore shows frequency density, calculated as \(\text{frequency density} = \frac{\text{frequency}}{\text{class width}}\). Rearranged, frequency \(= \text{frequency density} \times \text{class width}\), which is the area of the bar.

Using frequency as the height would be misleading, because a wide class would produce a bar of much greater area than a narrow class with the same frequency, exaggerating its importance. Reading a frequency from a histogram means calculating an area, not reading a height off the axis.

Revision notes

Frequency density

\(\text{Frequency density} = \frac{\text{frequency}}{\text{class width}}\).

This is plotted on the vertical axis. Rearranged, frequency \(= \text{frequency density} \times \text{class width}\), so the frequency is the area of the bar.

Why area, not height

If frequency were used as the height, a wide class would give a bar of much greater area than a narrow class with the same frequency.

The eye judges area, so this would exaggerate the wide class. Using frequency density makes the area proportional to frequency for every bar.

Reading from a histogram

To find a frequency, multiply the frequency density by the class width — that is, calculate the area.

For part of a class, find the fraction of the class width covered and take that fraction of the frequency, assuming values are evenly spread within the class.

Key points

  • Area represents frequency, not height.
  • Frequency density = frequency ÷ class width.
  • Frequency density goes on the vertical axis.
  • Frequency = frequency density × class width.
  • Using height alone would be misleading.
  • Reading a frequency means calculating an area.

Worked examples

Example 1

A class \(10 \leq x < 30\) has frequency 40. Work out the frequency density. [2 marks]

Working

Class width \(= 30 - 10 = 20\)work out the class width
\(\frac{40}{20} = 2\)divide frequency by class width

Example 2

A bar has frequency density 3.5 and covers the class \(20 \leq x < 28\). Work out the frequency. [3 marks]

Working

Class width \(= 28 - 20 = 8\)work out the class width
Frequency \(= 3.5 \times 8\)multiply frequency density by class width
\(= 28\)work out the frequency

Example 3

Explain why frequency density is used rather than frequency when class widths are unequal. [2 marks]

Working

Using frequency as the height would give a wide class a much larger area than a narrow class with the same frequencystate the problem
Frequency density makes the area of every bar proportional to its frequency, so the diagram is not misleadingexplain the solution

Common mistakes

  • Reading frequency from the vertical axis.

    The axis shows frequency density; frequency is the area.

  • Forgetting to calculate the class width.

    It is needed for every frequency density calculation.

  • Multiplying when you should divide.

    Frequency density = frequency ÷ class width.

  • Using frequency as the bar height.

    That is only valid when all class widths are equal.

Exam tips

  • Work out the class width as a separate first step.
  • Remember frequency is the area of the bar.
  • Divide to find density, multiply to find frequency.
  • Remember this is a Higher-only topic.

Key terms

Frequency density
Frequency divided by class width.
Class width
The difference between the upper and lower class boundaries.
Area
What represents frequency on a histogram.
Unequal class widths
Classes of differing sizes, requiring frequency density.

Related topics

Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.