Histograms
In a nutshell
Histograms offer a graphical way of representing grouped continuous data. A histogram is a good visual representation of how the data is distributed in terms of location, shape and dispersion. A key feature of a histogram is that the y-axis denotes the frequency density and thus the frequency is proportional to the area of a bar.
Frequency density
In a histogram, the frequency of each interval is proportional to the area of the corresponding bar.
frequency density | |
| class width |
As the area of a rectangle is given by:
Area=length×width
It follows that:
Frequency∝frequency density×class width
Therefore:
Frequency=k×frequency density×class width
Where k is the constant of proportionality.
Note: In the case k=1, the frequency is given by the area of the bar.
Histograms
In order to draw a histogram, the frequency density must be calculated using the above formula for each class interval, or for each row of the grouped frequency table. A histogram can then be plotted with the class interval on the x-axis and frequency density on the y-axis. The completed histogram shows the distribution of data and can be used for further analysis.
Example 1
The table shows the ages of students in an after-school Maths club. Given that the interval 6<x≤10 has frequency density 1, find the other frequency densities and draw a histogram to show the distribution of students.
Age | Frequency |
6<x≤10 | |
10<x≤12 | |
12<x≤13 | |
13<x≤18 | |
First, add columns to show the class width and frequency density. The class width is calculated by subtracting the lower bound from the upper bound for each class.
Age | class width | Frequency | Frequency density |
6<x≤10 | | | |
10<x≤12 | | | |
12<x≤13 | | | |
13<x≤18 | | | |
Use the formula for frequency along with the information given in the first row to find the constant of proportionality.
Frequency8k=k×frequency density×class width=k×1×4=2
Rearrange the formula to make frequency density the subject.
FrequencyFrequencyFrequency density=k×frequency density×class width=2×frequency density×class width=2×class widthfrequency
Use this formula to work out the remaining frequency densities.
Age | class width | Frequency | Frequency density |
6<x≤10 | | | |
10<x≤12 | | | 2×210=2.5 |
12<x≤13 | | | 2×14=2 |
13<x≤18 | | | 2×52=0.2 |
Plot the frequency density against age on a grid: