1.02 Categorical data

In the world of data, often there is an interest in the number of times, or frequency, that something occurs. It could be the number of road accidents caused by drink driving, the number of hot days in a year, or the number of visits to a website in a month.

In situations like these, where the same data value can occur multiple times, the data can be organised into a frequency table.

Categorical data and frequency tables

As an example, say that the colour of every car that passed though a given intersection was recorded over a ten minute period:

green, white, yellow, white, black, green, black, blue, blue, silver, white, black, green, blue, blue, white, black, silver, silver, red, red, red, black, white, blue, white, black, silver, silver, white, blue, white, black, yellow, blue, white, white, red, green, silver, black, white, black, white.

The same colours are occurring multiple times, so it makes sense to organise the data using a frequency table.

Vehicle colour Tally Frequency Black |||| |||| $9$9 White |||| |||| || $12$12 Blue |||| || $7$7 Green |||| $4$4 Silver |||| | $6$6 Yellow || $2$2 Red |||| $4$4

Notice that the frequency table has three columns:

• The first column shows the subgroups within the data
• The tally column (optional) uses tally marks to record the frequency of each subgroup
• The final column sums the tally marks and records the frequency as a number

The sum of the frequencies is equal to the total number of data values. In this case, the colours of $44$44 vehicles were recorded.

The graph on the right is a column graph, which will be discussed further below. Column graphs are well suited for displaying frequencies for categorical data. The table and graph allow us to analyse the frequency distribution–how the frequency of outcomes is spread across the different categories.

Practice questions

Question 1

Column graphs

A column graph (or bar graph) is a graph used to display relative quantities of data in different categories. There are a wide range of topics where this type of graph can be useful. For example, the number of tourists visiting different destinations in a month, the number of people with certain eye colours in a class or the popularity of the top ten YouTube channels.

Eye colour in a class

 

Column graph features to remember

• Title the graph clearly
• Label each axis
• Label each category and give the vertical axis a clear scale
• Make sure the columns are equal widths and equally spaced
• Make sure that the height of the column matches the value it represents which is often the frequency of the category (how often that outcome occurred)

Worked example

Example 1

As we can see, the scale of a column graph is important for reading values. However, there are some cases where the scale may be misleading.

Consider the column graph below:

If we look only at the columns and ignore the scale, it certainly seems like the number of albums sold doubled from 2014 to 2015. However, if we look at the scale, we can see that the number of albums sold in 2014 and 2015 were $25$25 thousand and $30$30 thousand respectively, so it is clearly not the case that the sales doubled.

The reason why the heights of the columns are so misleading is because the scale doesn't start at zero on this graph. Because it starts at $20$20 thousand, the height of each column actually indicates how much greater than $20$20 thousand the value is.

So we should always check the scale to find actual values before making any conclusions about the data.

Practice questions

Question 2

Question 3