1.03 Associations between categorical variables

Two-way tables allow us to display and examine the relationship between two sets of categorical data. The categories are labelled at the top and the left side of the table, and the frequency of the different characteristics appear in the interior of the table. Often the totals of each row and column are also shown.

The following are the statistics of the passengers and crew who sailed on the Titanic on its fateful maiden voyage in 1912.

Although it is interesting to know that 202 First Class passengers survived, it is far more useful to know the percentage break up of survivors from each class. To do this we need to calculate row percentages. To find the percentage divide the value in the table by the row total and multiply by 100\%. The row percentages are shown in the table below with working shown in some of the cells.

This percentage frequency table now gives us more useful information than the raw data. The first row gives us the percentage breakdown of survivors by class. We can now easily read that 46\% of the people who died were crew members whereas only 8\% of the people who died were in first class.

We can also calculate the percentage in each class type that survived or died. To do this we calculate column percentages. To find the percentage divide the value in the table by the column total and multiply by 100\%. The column percentages are shown in the table below with working shown in some of the cells.

This percentage frequency table now gives us more useful information. The first column gives us the percentage breakdown of survivals and deaths in first class. From the raw data we can see that a similar number of first class (202) and third class (178) passengers survived. However this can be misleading. The percentage frequency table shows us that 62\% of first class passengers survived whereas only 25\% of third class passengers survived.

Examples

Example 1

Maria surveyed a group of people about the type of job they had. She recorded the data in the following graph.

a

Complete the following two-way table displaying the row percentages. Give your answers correct to one decimal place.

	None	Casual	Part-time	Full-time	Total
Men	17.6\%		29.4\%	41.2\%	100\%
Women	22.2\%	27.8\%			100\%

Worked Solution

Create a strategy

Create a table for the data. Then for each missing value, divide the value in the table by the row total and multiply by 100\%.

Apply the idea

From the graph we can make the following two-way table:

	None	Casual	Part-time	Full-time	Total
Men	15	10	25	35	85
Women	20	25	15	30	90
Total	35	35	40	65

From the above table we can see that there were 10 casual men employees, and the row total for men is 85.

\displaystyle \text{Casual Men Percent}	\displaystyle =	\displaystyle \dfrac{10}{85} \times 100\%	Find the row percentage
	\displaystyle \approx	\displaystyle 11.8\%	Evaluate

From the table we can see that there were 15 women part-time employees, and the row total for women is 90.

\displaystyle \text{Part-Time Women Percent}	\displaystyle =	\displaystyle \dfrac{15}{90}\times 100\%	Find the row percentage
	\displaystyle \approx	\displaystyle 16.7\%	Evaluate

From the table we can see that there were 30 full-time women employees.

\displaystyle \text{Full-time women percent}	\displaystyle =	\displaystyle \dfrac{30}{90} \times 100\%	Find the row percentage
	\displaystyle \approx	\displaystyle 33.3\%	Evaluate

So the row percentage table is:

	None	Casual	Part-time	Full-time	Total
Men	17.6\%	11.8\%	29.4\%	41.2\%	100\%
Women	22.2\%	27.8\%	16.7\%	33.3\%	100\%

b

Complete the following two-way table displaying the column percentage. Give your answers correct to one decimal place.

	None	Casual	Part-time	Full-time
Men		28.6\%	62.5\%
Women	57.1\%		37.5\%
Total	100\%	100\%	100\%	100\%

Worked Solution

Create a strategy

Use the table from part (b) and for each missing value, divide the value in the table by the column total and multiply by 100\%.

Apply the idea

	None	Casual	Part-time	Full-time	Total
Men	15	10	25	35	85
Women	20	25	15	30	90
Total	35	35	40	65

From the above table we can see that there were 15 men in the None category, and the None column total is 35.

\displaystyle \text{None Men Percent}	\displaystyle =	\displaystyle \dfrac{15}{35} \times 100\%	Find the column percentage
	\displaystyle \approx	\displaystyle 42.9\%	Evaluate

From the above table we can see that there were 35 full-time men, and the full-time column total is 65.

\displaystyle \text{Full-time Men Percent}	\displaystyle =	\displaystyle \dfrac{35}{65}\times 100\%	Find the column percentage
	\displaystyle \approx	\displaystyle 53.8\%	Evaluate

From the above table we can see that there were 25 casual women employees, and the casual column total is 35.

\displaystyle \text{Casual women Percent}	\displaystyle =	\displaystyle \dfrac{25}{35} \times 100\%	Find the column percentage
	\displaystyle \approx	\displaystyle 71.4\%	Evaluate

From the above table we can see that there were 30 full-time women employees, and the full-time column total is 65.

\displaystyle \text{Full-time women Percent}	\displaystyle =	\displaystyle \dfrac{30}{65} \times 100\%	Find the column percentage
	\displaystyle \approx	\displaystyle 46.2\%	Evaluate

So the column percentage table is:

	None	Casual	Part-time	Full-time
Men	42.9\%	28.6\%	62.5\%	53.8\%
Women	57.1\%	71.4\%	37.5\%	46.2\%
Total	100\%	100\%	100\%	100\%

To find if there is an association between the variables in the Titanic table we can ask the question "Is survival rate dependent on the class of the passenger?"

In order to find this we must first identify the explanatory variable. In this problem it is the class of passenger. The explanatory variable forms the heading of each column, therefore the column percentage frequency table will best indicate any patterns.

When we read across the first row in the column percentage frequency table and look at the numbers we can see a clear difference in the percentages of passengers that lived or died in each class. This suggests there is an association between the class of passenger and the rate of survival. It appears that the higher the class of passenger, the higher the rate of survival.

Which percentage frequency table to use?

If the explanatory variable forms the column headings then we use the column percentage frequency table to look for association.

If the explanatory variable forms the row headings then we use the row percentage frequency table to look for association.

How do we tell if there is an association?

If it's a column percentage table then look across the rows for differences in the values. If the values are similar then we say there is NO clear association.

If it's a row percentage table then look down the columns for differences in the values. If the values are similar then we say there is NO clear association.

How to describe the association?

First state if there is or is not an association apparent. For example: There appears to be an association between the variables.

Next describe the association. For example: The class of passenger affects the survival rate of the passengers.

Finally give an example. For example: The higher the class of passenger, the more likely the passenger was to survive.

An overview of the percentages in two-way tables can bring to light clear associations. The presence of more subtle associations and an objective measure of the significance of such associations requires additional analysis and methods from further studies in statistics.

Note: The term 'association' is used to describe a relationships between variables. An association does not mean one variable causes the other variable to change but that a change in one variable appears to affect the other.

Examples

Association between variables can often be seen more clearly in a stacked column graph. Below is a stacked column graph (also called segmented column graph) for the data from the Titanic table earlier. When we look at each column we can see the proportion of death to survival in each column is different. This indicates there is an association between the variables.

If there is no association then the proportion of the sections in each column are the same. When we look at the graph below we can see that each column is divided into similar size sections. This indicates there is NO clear association between household composition and distribution of money.

How to draw a stacked column graph?

• Label the horizontal axis with the explanatory variables.

• Label the vertical axis with percentages from 0\% to 100\%.

• Draw a column for each explanatory variable that reaches the height of 100\% on the vertical axis.

• To divide each column into the percentages as shown in the frequency table start from the bottom of the column, count to the first percentage and draw a horizontal line to mark it off, then count up to the second percentage from the horizontal line and then mark off again, until all sections are complete

• Write the label and percentage in each section of the columns which indicates the response variables displayed or provide a key.

Examples

Example 3

A group of year 12 students surveyed their class and recorded the hair colour and eye colour for each student. The results are displayed in the 100\% stacked column chart shown below.

a

What is the explanatory variable for this chart?

A

Eye colour

B

Hair colour

Worked Solution

Create a strategy

In a stacked column graph, the explanatory variable is placed along the horizontal axis.

Apply the idea

The stacked column graph has the eye colour placed along the horizontal axis. So it is the explanatory variable, option A.

b

Does the chart suggest an association between eye colour and hair colour?

A

Yes, as the corresponding segments are similar in size.

B

Yes, as the corresponding segments are of different sizes.

C

No, as the corresponding segments are similar in size.

D

No, as the corresponding segments are of different sizes.

Worked Solution

Create a strategy

Check whether the proportion of the segments in each column are similar sizes.

Apply the idea

The stacked column graph segments differ in sizes which means that it does suggests an association between eye colour and hair colour, option B.

c

Can we say that having blue eyes causes a high chance of having blonde hair?

A

Yes. The data shows that students with blue eyes are more likely to have blonde hair.

B

No. There appears to be an association, but we cannot say that one causes the other.

Worked Solution

Create a strategy

Consider the difference between association and causation.

Apply the idea

An association does not mean one variable causes the other variable to change but that a change in one variable appears to affect the other.

The chance of having blonde hair can be caused by many factors. Having blue eyes may affect the chance of having one but we cannot say that it causes such. So the correct answer is option B.