How do you investigate the association between two categorical variables, or between a numerical and a categorical variable, using tables and grouped displays?
Construct and interpret a two-way frequency table, convert it to percentages to investigate association between two categorical variables, use a segmented or side-by-side bar chart, and compare a numerical variable across categories with parallel boxplots
A focused answer to the VCE General Mathematics Unit 3 Data analysis key-knowledge point on categorical association. Two-way frequency tables, column percentages, segmented bar charts, judging association, and comparing a numerical variable across groups with parallel boxplots.
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What this dot point is asking
VCAA wants you to investigate association between variables that are not both numerical. For two categorical variables you build a two-way frequency table, convert it to percentages, and decide whether the variables are associated. For a numerical variable measured across several categories, you compare groups using parallel boxplots or back-to-back displays. This complements the scatterplot and correlation work, which only handles two numerical variables.
Building and reading a two-way table
Suppose people are classified by whether they exercise regularly (Yes or No) and by age group (Under 30, 30 or over). The table records the count in each cell. The explanatory variable conventionally goes across the columns, so we work out percentages down each column to compare the groups fairly.
| Under 30 | 30 or over | |
|---|---|---|
| Exercises: Yes | 70 | 50 |
| Exercises: No | 30 | 50 |
| Total | 100 | 100 |
Converting to percentages
Always percentage by the explanatory variable. Here the columns are age groups, so divide each count by its column total.
Displaying categorical association
A segmented bar chart stacks the percentages within each category to , so you compare the coloured segments across bars. A side-by-side (clustered) bar chart places the bars next to each other. In both, association shows up as different segment heights between the bars. If the segments are the same height across all bars, the variables are not associated.
A numerical variable across categories
When one variable is numerical and the other categorical (for example, reaction time by age group), you compare the groups with parallel boxplots or a back-to-back stem plot. You look at how the medians, spreads (IQR) and shapes differ between groups. If a higher category tends to have a higher median, the numerical variable is associated with the category.
Why this matters for the exams
Two-way tables and segmented bar charts are reliable early-exam questions and feature in the statistics SAC. The marks come from percentaging correctly by the explanatory variable and from writing a clear association sentence that quotes the differing percentages. The parallel boxplot comparison links directly to your univariate boxplot work, now used to compare groups rather than describe one.
Exam-style practice questions
Practice questions written in the style of VCAA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
2025 VCAA1 marksThe two-way frequency table shows the preferred car colour for a sample of buyers. Female buyers: black 28, silver 42, white 35. Male buyers: black 55, silver 32, white 42. From this table, the percentage of female car buyers whose preferred car colour is silver is closest to A. 33% B. 40% C. 42% D. 57%Show worked answer →
To find a percentage within the female group, divide the relevant count by the female column total, not the grand total.
Total female buyers = 28 + 42 + 35 = 105.
Female buyers preferring silver = 42.
percentage = 42 / 105 x 100 = 40%, so the answer is B. Using the column total keeps the comparison within the female group.
2023 VCAA2 marksA two-way table shows oyster size by farm. Farm A: small 42, medium 124, large 44, total 210. Farm B: small 114, medium 160, large 46, total 320. The farmer believes that farm A has a greater capacity to grow larger oysters than farm B. Does the information support the farmer's belief? Explain your conclusion by comparing the values of two appropriate percentages, rounded to the nearest whole number.Show worked answer →
To compare the two farms fairly, work out the percentage of large oysters within each farm, using each farm's own total as the denominator.
Farm A: 44 / 210 x 100 = 21% large.
Farm B: 46 / 320 x 100 = 14% large (1 mark for the two correct percentages).
Since 21% is greater than 14%, a higher proportion of Farm A's oysters are large, so the data does support the farmer's belief that Farm A has a greater capacity to grow larger oysters (1 mark for the comparison and conclusion).