E9.7 Histograms

Learning objectives

E9.7.1Draw and interpret histograms.• Draw and interpret histograms.E9.7.2Calculate with frequency density• Calculate with frequency density. On histograms, the vertical axis is labelled ‘Frequency density’. Frequency density is defined as frequency density = frequency ÷ class width.

A histogram represents grouped continuous data so that each bar's area represents its class frequency. This is why unequal class widths require frequency density on the vertical axis.

Feature	Histogram rule
horizontal axis	mark continuous class boundaries; bar width equals class width
vertical axis	label Frequency density and use a linear scale
bars	draw touching rectangles over the exact class intervals
frequency	compare bar areas, not heights alone

A tall narrow bar can have less frequency than a shorter wide bar. To compare classes, use area $=\text{class width}\times\text{frequency density}$ . The tallest bar identifies the greatest frequency density, while the greatest area identifies the greatest frequency.

If one class has width $10$ and density $4$ , its area and frequency are $40$ . A class of width $25$ and density $2$ has frequency $50$ : it is shorter but contains more observations.

Do not leave gaps between adjacent continuous intervals, plot class midpoints as bar positions, or label the vertical axis Frequency when widths differ. A bar chart compares categories using heights; a histogram encodes grouped frequency through area.

Frequency density adjusts frequency for class width, allowing unequal-width classes to be compared fairly in a histogram.

$\text{frequency density}=\frac{\text{frequency}}{\text{class width}},\qquad \text{frequency}=\text{density}\times\text{class width}$

Class	Width	Frequency	Density
$0<x\leq10$	10	20	$20\div10=2$
$10<x\leq30$	20	30	$30\div20=1.5$
$30<x\leq35$	5	25	$25\div5=5$

If a printed histogram gives bar heights in centimetres rather than a labelled density scale, all heights are proportional to density. Find the common multiplier from one known bar, then apply it to every density. To fill a missing class when total frequency is known, subtract the frequencies represented by the existing bar areas first.

To recover a frequency from the diagram, read the density and multiply by the class width. For only part of a class, the corresponding rectangle area gives an estimate based on the distribution within that class.

Class width is upper boundary minus lower boundary, not the midpoint or number of labels. Do not compare raw frequencies to choose bar height, and do not use a different height multiplier for different bars.

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—Cambridge IGCSE Mathematics (0580)Read