3.3 Histograms with unequal width bars

The real advantage of a histogram or frequency polygon over a grouped bar chart is how it represents data with unequal class intervals. When you were first introduced to the play park investigation, the age demographic data focused on the age of younger village residents. The data looked like this.

Table 7 Village age demographic showing distribution of under 18s

Let’s organise that data in a grouped frequency table.

Table 8 Village age demographic (continuous groups) showing distribution of under 18s

If you plot this data as a grouped bar chart in the usual way, remembering the values on the axes should be evenly spaced and then the bars fitted to them, your graph will look something like this.

Figure 15 Grouped bar chart, unequal groups

Does this seem a reasonable representation of the data presented in Table 8? There are clearly a lot more adults (over 18s) than children, but do you think the graph in Figure 15 represents the proportional comparison accurately?

The second group (5 ≤ a < 11) represents 107 people and the over 18s group (18 ≤ a < 100) represents about twelve times this amount (approximately 1300 people). So, proportionally, it seems logical that the over 18s rectangle should be about twelve times the size of the second rectangle on the graph.

The image below shows twelve of the second rectangle copied into the large rectangle however there is a lot of unused space left in the large bar. When we look at the size of rectangles, we intuitively compare areas and not just heights. The area of the large rectangle seems at least a hundred times bigger. This could be very misleading.

Figure 16 Grouped bar chart, unequal groups, comparing proportions

The way round this misrepresentation is to represent the frequency of each bar with its area rather than its height; this is what we do for histograms where data is collated into groups of unequal class widths. Doing so for this data, gives the following histogram.

Figure 17 Histogram, with proportional unequal groups, village age demographic

Here you can see the large rectangle for over 18s represents about twelve times the number of people than the second bar and its area is about twelve times the size.

This brings us back to the earlier definition of a histogram stating that the area of each rectangle is proportional to the frequency of a variable, and its width is proportional to the class interval. Some people find it hard to understand that area and height can be very different ways of measuring size. This is why children need to have experiences of shape and space in the curriculum, as well as number. You may have noticed the vertical axis is now labelled ‘frequency density’ rather than just ‘frequency’ and has a different scale. In geography, population density is the amount of people per unit area of land. Similarly, frequency density is the amount of frequency per unit of the data, here those units are years.

Figure 18 Rectangle representing one bar of a histogram

The area of a rectangle is found by multiplying its length by its width, and the frequency of a bar in a histogram can be found by multiplying the class width by the frequency density.

Frequency=Class width ×Frequency density

This can be rearranged to find the frequency density.

Activity 9

Timing: Allow 5 minutes

Here is the grouped frequency data for the histogram in Figure 17

Table 9 Village age demographic (continuous groups) showing distribution of under 18sadd image here

Complete the class width and frequency density columns. Give the frequency density to one decimal place. The first one is done for you.

Class width=upper bound of the class interval-lower bound of the class interval

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Answer

The completed table should look like this

Table 10 Village age demographic (continuous groups) showing distribution of under 18s (completed)

The numbers in the final column correspond to the heights of the bars on the histogram shown in Figure 17.

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Activity 10

Timing: Allow 10 minutes

Here is the frequency data from Week 2 about the length of time in minutes it would take some village residents to walk to the site of the proposed play park.

Table 11 Time taken by a sample of village residents to walk to the play park site

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Complete the table by calculating the class width and frequency density for each group

Plot a histogram to represent this data – remember the frequency density should be on the vertical axis here because the data is collated in unequal groups

The following points still apply: • Values on the axes should be evenly spaced and clearly labelled including a title showing what information each axis represents, with units of measure if applicable. • The chart should have a title describing what it shows.

Answer

The completed table is shown below

Table 12 Time taken by a sample of village residents to walk to the play park site (completed)

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You may have rounded this last frequency density value to 0.1 – it depends on the vertical scale used on your histogram how accurately this value can be plotted.

Your histogram will depend on the scale you have used but, proportionally, it should look the same as this one.

Figure 19 Histogram of walking times

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