Introduction
Histograms are one of the most commonly tested representations of continuous data in Paper 5 (Probability & Statistics 1). Unlike a bar chart — which you will have met when studying Representing Data and Choosing a Diagram — a histogram has no gaps between bars and encodes information in area, not height. This distinction is the engine of virtually every histogram exam question: if you confuse height with frequency, you will lose marks. The syllabus requires you to both draw and interpret histograms, including those with classes of unequal width, so both skills are tested here.
Core Concept
Why area, not height?
When all classes have equal width, height and area are proportional, so a simple bar height looks like frequency and no ambiguity arises. But when class widths differ — as they almost always do in exam data — a tall bar over a narrow class would visually exaggerate its frequency. To give every class a fair visual representation, we insist:
Because , solving for height gives the quantity plotted on the vertical axis:
The vertical axis is therefore labelled Frequency Density (often abbreviated fd in workings).
Reading a histogram
To recover frequency from a drawn histogram, reverse the process:
This means you can compare frequencies across unequal classes simply by comparing areas — a narrow but tall bar may represent fewer data values than a short but wide bar.
Class boundaries and widths
For continuous data, class boundaries are precise values at which one bar ends and the next begins (no gap). The class width is the difference between the upper and lower boundaries of that class. Always identify boundaries before calculating frequency density.
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