Two-way tables, segmented bar charts and categorical association: VCE General Mathematics Unit 3 Data analysis

What this dot point is asking

This is the categorical half of the Study Design's "Investigating associations between two variables" topic: "contingency (two-way) frequency tables, their associated bar charts (including percentage segmented bar charts) and their use in identifying and describing associations between two categorical variables". It also relies on the single-variable skills from Investigating data distributions: "data tables, two-way frequency tables and their associated segmented bar charts".

You need to:

1. build a frequency table (with percentages) for one categorical variable;
2. build and read a two-way table, including its row and column totals;
3. decide which variable is explanatory and percentage within its categories;
4. draw and read a percentage segmented bar chart;
5. write an association statement that quotes appropriate percentages.

These are early, high-scoring questions, but one detail keeps costing marks: which total to divide by. In 2023 Examination 2 Question 2b(ii), only half the cohort earned both marks for the "compare two appropriate percentages" sentence.

(When one variable is numerical, such as reaction time by age group, the tools are parallel boxplots and back-to-back stem plots; they have their own page.)

The answer

Categorical data and frequency tables

A categorical variable places each person or object into a group. It is nominal if the groups have no order (colour, sport) and ordinal if they do (small, medium, large). A frequency table lists each category, its count and usually its percentage:

Always include the total row (it was required in 2023 Examination 2 Question 2a). One categorical variable is displayed by a bar chart, or by a single percentage segmented bar that stacks the categories to $100\%$.

Two-way (contingency) tables

A two-way frequency table cross-classifies the same people by two categorical variables. Each cell counts one combination; the margins hold the row totals, column totals and the grand total.

Three different questions use three different totals:

• "What percentage of Year 12 students catch the bus?" uses the Year 12 total: $\frac{70}{140} = 50\%$.
• "What percentage of bus users are in Year 12?" uses the bus total: $\frac{70}{110} \approx 63.6\%$.
• "What percentage of all students are Year 12 bus users?" uses the grand total: $\frac{70}{300} \approx 23.3\%$.

The words straight after "of" name the group you divide by.

Which way to percentage

To investigate whether the response depends on the explanatory variable, compare the distribution of the response within each explanatory category. So:

• if the explanatory variable is across the columns, use column percentages (each column adds to $100\%$);
• if it is down the rows, use row percentages (each row adds to $100\%$).

VCAA often puts the explanatory variable in the columns, but not always: in 2025 Examination 1 Question 7 gender was in the rows, so "the percentage of female buyers" used the female row total, 105. Decide from the context, not the layout.

Why percentages at all? The groups are rarely the same size. Farm A had 44 large oysters and farm B had 46, yet farm A's rate of large oysters ($21\%$) was clearly higher than farm B's ($14\%$), because farm B's sample was bigger.

Percentage segmented bar charts

A percentage segmented bar chart draws one bar for each category of the explanatory variable. Every bar has height $100\%$ and is divided into segments for the response categories, with a key.

Read it by comparing matching segments across the bars:

• Association: the segments differ noticeably between the bars. Here bus use doubles from $25\%$ to $50\%$ while car travel falls from $45\%$ to $30\%$.
• No association: the bars look (almost) the same, because the response is distributed the same way in every group.

A side-by-side bar chart of the same percentages carries the same information. A bar chart of raw counts can mislead when the groups differ in size.

Writing an association statement

Examiners want a conclusion supported by evidence. A complete statement:

1. says whether there is an association (or answers the specific claim with "yes" or "no");
2. quotes two percentages for the same response category, one from each group;
3. states which is larger, in context.

For the transport data: "Yes, transport mode is associated with year level: $50\%$ of Year 12 students catch the bus to school, compared with $25\%$ of Year 7 students."

There is no fixed cut-off for "noticeably different": $50\%$ against $25\%$ is clearly an association; $10.7\%$ against $10.0\%$ is not. When the difference is small, say so and conclude that there is little or no association.

An association between categorical variables is not evidence of causation. The Year 12 students may catch the bus because of where they live, what subjects they take, or their age, not because of their year level as such.