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 Video by Zeeshan Zamurred Representations of Data

Edexcel AS Level Maths: 3.4 HistogramsWatch the full walkthrough 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

Frequency density

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

frequency density = \frac{frequency}{class width}

Height = frequency ÷ width.

\frac{20}{5}

Answer

4

Area = frequency

The key principle: $area \propto frequency$ . 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 \times 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.

Histogram vs bar chart

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

Formula recap

f.d. = \frac{frequency}{class width}

Frequency density (height).

frequency = f.d. \times width

Area gives frequency.

area \propto 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

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