Investigating Associations Between Categorical Variables (VCE SSCE General Mathematics): Revision Notes

Investigating Associations Between Categorical Variables

What is an association?

When two variables are related or connected in some way, we say they are associated. For example, a person's attitude to a particular issue might be associated with where they live, or a student's future plans might be associated with their gender.

To investigate whether an association exists between two categorical variables, we use a two-way frequency table (also called a contingency table). This special table summarises data from two variables simultaneously, allowing us to examine whether changes in one variable correspond to changes in the other.

Constructing a two-way frequency table

Understanding the data structure

Imagine we want to investigate whether people's attitudes to gun control depend on where they live (city or country). We could collect data from a sample of people, recording two pieces of information about each person:

• Their residence (city or country)
• Their attitude to gun control (for or against)

Both of these are categorical variables, as they sort people into distinct categories rather than measuring quantities.

From separate tables to a two-way table

We could present each variable separately in simple frequency tables. For example, from a sample of 100 people:

These separate tables tell us about each variable individually, but they don't reveal the relationship between them. To investigate the connection between residence and attitude, we need to combine this information into a single two-way frequency table.

Identifying explanatory and response variables

Before constructing our two-way table, we must identify which variable is which:

• The explanatory variable (EV) is the variable that might explain or influence the other variable
• The response variable (RV) is the variable that might depend on or be influenced by the explanatory variable

In our gun control example, we think a person's attitude might depend on where they live, but where someone lives doesn't depend on their attitude. Therefore:

• Residence is the explanatory variable
• Attitude to gun control is the response variable

Building the table

Following this convention, we create a table where:

• Columns represent residence (country and city)
• Rows represent attitude to gun control (for and against)

We then complete the table by adding row and column totals:

The shaded cells in the table contain the core data. The row sums show how many people overall had each attitude, while the column sums show how many people lived in each type of area.

Worked example: constructing a table from raw data

Converting to percentaged two-way frequency tables

Why use percentages?

Looking at the gun control table, we see that more country people (32) favour gun control than city people (30). But does this really mean country people are more supportive? Not necessarily - there were more country people in the sample overall (58 versus 42). To make a fair comparison, we need to express the data as percentages.

Calculating column percentages

When the explanatory variable labels the columns (as it should), we calculate column percentages by dividing each cell frequency by its column sum, then multiplying by 100.

For the gun control example:

• Percentage of country people for gun control: $\frac{32}{58} \times 100 = 55.2\%$
• Percentage of country people against gun control: $\frac{26}{58} \times 100 = 44.8\%$
• Percentage of city people for gun control: $\frac{30}{42} \times 100 = 71.4\%$
• Percentage of city people against gun control: $\frac{12}{42} \times 100 = 28.6\%$

The percentaged table now looks like this:

Using percentages to identify associations

Evidence of association

The percentaged table reveals something the frequencies concealed: city people are actually more supportive of gun control than country people (71.4% versus 55.2%). The difference between these percentages indicates that attitude to gun control is associated with residence.

Writing a report

When reporting findings, quote relevant percentages and state your conclusion clearly:

In this sample of 100 people, a higher percentage of city people were for gun control than country people: 71.4% to 55.2%. This indicates that a person's attitude to gun control is associated with their place of residence.

Example of no association

Consider this table about whether mobile phones should be banned in cinemas, comparing Year 10 and Year 12 students:

The percentages in favour are very similar (87.9% versus 86.8%). We would report:

In this sample of 100 students, the percentage of Year 10 and Year 12 students supporting a ban on mobile phones in cinemas is similar: 87.9% to 86.8%. This indicates that support for banning mobile phones in cinemas is not associated with year level.

Worked example: identifying association from a percentaged table

Working with larger tables

The same principles apply when variables have more than two categories. For example, this table shows smoking status by education level:

To identify an association, compare percentages across any row. Looking at the "Smoker" row:

The percentage of smokers steadily decreases with education level, from 34.0% for Year 9 or below to 18.4% for university graduates. This indicates that smoking is associated with level of education.

Here's another example showing interest in sport by age group:

Identifying the explanatory variable: Age could explain interest in sport, but interest in sport cannot explain age. Therefore, age group is the explanatory variable.

Finding the association: Looking at the "High" interest row, we see high interest decreases with age (from 56.5% for under 18s to 35.0% for 36-50 year olds). This indicates an association between age and interest in sport.

Segmented bar charts

What is a segmented bar chart?

A segmented bar chart provides a visual way to display the information from a percentaged two-way frequency table. It consists of:

• One bar for each category of the explanatory variable
• Each bar divided into coloured segments representing the response variable categories
• Each bar reaching 100% in height

Constructing a segmented bar chart

To create a segmented bar chart for interest in sport by age group:

1. Label the horizontal axis with the explanatory variable (age group)
2. Scale the vertical axis from 0% to 100%
3. Draw one bar for each age category
4. Divide each bar into segments based on the percentages in the table
5. Colour each segment consistently and add a legend

Interpreting segmented bar charts

Segmented bar charts make it easier to spot associations visually. Look for differences in the heights of corresponding segments across bars.

For example, this chart shows preferred holiday type by age:

The coastal holiday segment (yellow) is noticeably larger for the under 40 group than for the 40 and over group. We can conclude:

There is an association between holiday preference and age. Those aged under forty are more likely to choose a coastal holiday (75%) than those aged forty or over (60%).

Exam tips

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