Histograms

Histograms are a way of representing data. They are like bar charts, but show the frequency density instead of the frequency. They can be used to determine information about the distribution of data.

Frequency density

A histogram is drawn like a bar chart, but often has bars of unequal width. It is the area of the bar that tells us the frequency in a histogram, not its height.

Instead of plotting frequency on the y-axis, we plot the frequency density. To calculate this, you divide the frequency of a group by the width of it.

Example

Look at the following table:

Time taken t seconds	Frequency
0 < t ≤ 20	4
20 < t ≤ 35	9
35 < t ≤ 50	6
50 < t ≤ 60	6
60 < t ≤ 80	5

In order to draw a histogram to represent this data, we need to find the frequency density for each group.
- If we look at the first group, we can see it has a frequency of 4 and a width of 20, because 20 - 0 = 20.

Frequency density = frequency ÷ group width

= 4 ÷ 20

= 0.2

So we need to draw a bar which goes from 0 to 20 on the x-axis and up to 0.2 on the y–axis.
- Looking at the second group, we have a frequency of 9 and a width of 15.

Frequency density = frequency ÷ group width

= 9 ÷ 15

= 0.6

So we need to draw a bar which goes from 20 to 35 on the x-axis and up to 0.6 on the y–axis.

Calculating similarly for the remaining groups we get:

Time taken t seconds	Frequency	Width	Frequency density
0 < t ≤ 20	4	20	0.2
20 < t ≤ 35	9	15	0.6
35 < t ≤ 50	6	15	0.4
50 < t ≤ 60	6	10	0.6
60 < t ≤ 80	5	20	0.25

Plotting this data, our histogram will look like this:

Histogram to show the time taken

Histogram labelled 'Frequency density' on the y-axis and 'Time taken (t seconds)' on the x-axis.

Finding the frequency

Sometimes a histogram will already be drawn for us. We can then use this to find the frequency of each group, and hence the total frequency for the distribution.