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Histograms

HigherHigher tier onlyAQAEdexcelOCR

This free Higher GCSE Maths worksheet on histograms helps you revise histograms with unequal class widths. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. Frequency is the area of the bar, found using frequency density.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR 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

A histogram displays grouped continuous data with unequal class widths. Unlike a bar chart, the area of each bar represents the frequency, not its height.

The vertical axis shows frequency density, calculated as frequency divided by class width. This correction is what allows classes of different widths to be compared fairly.

Because area represents frequency, a wide bar of modest height can contain more data than a narrow tall one. Reading the height as the frequency is the single most common error in the topic.

Revision notes

Frequency density

Frequency density is frequency divided by class width.

A class of width \(10\) with frequency \(30\) has frequency density \(3\). This goes on the vertical axis.

Area represents frequency

Multiply frequency density by class width to recover the frequency.

So a bar of density \(4\) spanning a width of \(5\) represents \(20\) items. There are no gaps between bars, because the data is continuous.

Comparing with bar charts

A bar chart has equal widths and gaps, and height shows frequency.

A histogram has no gaps, allows unequal widths, and uses area. Confusing the two is a frequent source of lost marks.

Key points

  • Histograms show grouped continuous data.
  • Area represents frequency, not height.
  • Frequency density is frequency divided by class width.
  • Frequency density goes on the vertical axis.
  • Multiply density by width to recover frequency.
  • There are no gaps between the bars.

Worked examples

Example 1

A class of width \(5\) has frequency \(20\). Find the frequency density.

Working

\[20 \div 5\]divide frequency by class width
\[= 4\]state the frequency density

Example 2

A bar has frequency density \(6\) and width \(15\). Find the frequency.

Working

\[6 \times 15\]multiply density by width
\[= 90\]state the frequency

Example 3

A class of width \(20\) has frequency \(50\). Find the frequency density.

Working

\[50 \div 20\]divide frequency by class width
\[= 2.5\]state the frequency density

Common mistakes

  • Reading the height as the frequency.

    In a histogram the area gives the frequency, not the height.

  • Leaving gaps between the bars.

    The data is continuous, so the bars touch.

  • Multiplying instead of dividing for density.

    Frequency density is frequency divided by class width.

  • Using a histogram for categorical data.

    That needs a bar chart, with gaps and equal widths.

Exam tips

  • Add a frequency density column to the table before drawing.
  • Label the vertical axis frequency density, not frequency.
  • Multiply density by width to recover a frequency.
  • Leave no gaps between the bars.

Key terms

Histogram
A diagram where area represents frequency.
Frequency density
Frequency divided by class width.
Class width
The size of the interval a bar covers.
Continuous data
Data that can take any value in a range.

Related topics

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