Histograms

Maths · Statistics. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A histogram shows grouped continuous data. It looks like a bar chart, but there are no gaps between the bars and the bars can have different widths, because the classes can have different widths.

When the widths differ, the height of a bar cannot be the frequency: a wide class would look too important. Instead the height is the frequency density, which is the frequency for each unit of class width.

The idea to remember is that the area of each bar represents the frequency. So to find a frequency from a histogram, multiply the height of the bar by its width.

Rules to remember

Frequency density = frequency ÷ class width.
Frequency = frequency density × class width = area of the bar.
Class width = upper class boundary − lower class boundary, e.g. 10 < x ≤ 25 has width 15.
The vertical axis is frequency density; the horizontal axis is a continuous scale with no gaps between bars.
Bars of equal height have equal frequency density, not equal frequency.
For part of a class: frequency ≈ frequency density × width of that part.

Common mistake

Reading the height of a bar as the frequency, or dividing by the upper boundary instead of the class width. The height is the frequency density; the frequency is height × width.

Worked example

In a histogram, the bar for the class 0 < x ≤ 20 has height 1.5. The class 20 < x ≤ 50 has frequency 24. Find the frequency of the first class and the height of the bar for the second class.

First class: width = 20 − 0 = 20, so frequency = 1.5 × 20 = 30.
Second class: width = 50 − 20 = 30.
Height = frequency density = 24 ÷ 30 = 0.8.

Answer: Frequency of the first class = 30; height of the second bar = 0.8

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