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S3.4StatisticsStretch

Histograms

A histogram displays continuous grouped data, and its defining feature is that AREA represents frequency. With unequal class widths, the height is the frequency density — not the frequency itself.

30 min Representations of Data
Edexcel AS Level Maths: 3.4 HistogramsCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Understand that area represents frequency
  • Calculate frequency density
  • Draw and interpret a histogram
  • Find a frequency from a histogram
1

Frequency density

Because classes can have different widths, the height of each bar is the frequency density\textbf{frequency density}, not the frequency. This ensures the area\textbf{area} of each bar is proportional to its frequency.

frequency density=frequencyclass width\text{frequency density} = \dfrac{\text{frequency}}{\text{class width}}
Height = frequency ÷ width.
Example 1A class of width 5 has frequency 20. Find the frequency density.
1205\dfrac{20}{5}.
Answer44
2

Area = frequency

The key principle: area∝frequency\textbf{area} \propto \textbf{frequency}. To find a frequency from a histogram, multiply the bar’s frequency density by its width (its area). To compare areas you may need a scale factor if total frequency is given.

frequency=frequency density×class width\text{frequency} = \text{frequency density} \times \text{class width}
Recover a frequency by finding the bar’s area.

Tip — Histogram questions are really “area = frequency” questions — always work with areas, not bar heights.

3

Histogram vs bar chart

A histogram is for continuous\textbf{continuous} data with no gaps between bars, and uses frequency density. A bar chart is for categorical\textbf{categorical} data with gaps, where height is just frequency. Do not confuse them.

Formula recap

f.d.=frequencyclass width\text{f.d.} = \dfrac{\text{frequency}}{\text{class width}}
Frequency density (height).
frequency=f.d.×width\text{frequency} = \text{f.d.} \times \text{width}
Area gives frequency.
area∝frequency\text{area} \propto \text{frequency}
The defining principle.

Common mistakes to avoid

Reading the bar height as the frequency.
Height is frequency density; frequency is the AREA (height × width).
Leaving gaps between bars.
Histograms have no gaps — the data is continuous.

Key takeaways

  • Histograms show continuous data with area ∝ frequency.
  • Height = frequency density = frequency ÷ class width.
  • Frequency = frequency density × width (the bar’s area).

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Histograms. Use the feedback to decide what needs another look.

2 past-paper questions · 3 marks

Practise this topic

Related lessons

All 5 lessons

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