QCE General Mathematics - Unit 3 - Bivariate data analysis 1
Associations Between Two Categorical Variables | QCE General Mathematics
Learn how QCE General Mathematics uses two-way frequency tables, row percentages and column percentages to describe associations between categorical variables.
Updated 2026-05-18 - 7 min read
QCAA official coverage - General Mathematics 2025 v1.3
Exact syllabus points covered
- Understand the meaning of bivariate data.
- Construct two-way frequency tables and determine the associated row and column sums and percentages.
- Use an appropriately percentaged two-way frequency table to identify patterns that suggest the presence of an association.
- Understand an association in terms of differences observed in percentages across categories in a systematic and concise manner, and interpret this in the context of the data.
Categorical bivariate data appears when each person, object or event is classified in two different ways. A student might be grouped by year level and transport method, a household by region and internet access, or a customer by membership type and preferred product. The mathematics is not about calculating a single average. It is about asking whether the pattern in one category seems to change when we look at another category.
Original Sylligence diagram for general two way frequency table.
Building a two-way table
A two-way frequency table has one variable in the rows and the other in the columns. The inside cells show joint frequencies. The row totals and column totals are called marginal totals because they sit on the margins of the table.
| Commute method | Year 11 | Year 12 | Total | |---|---:|---:|---:| | Bus | 38 | 29 | 67 | | Car | 24 | 41 | 65 | | Walk or cycle | 18 | 10 | 28 | | Total | 80 | 80 | 160 |
The table shows raw counts. Raw counts are useful, but they can be misleading if the groups are different sizes. Percentages let you compare like with like.
Choosing the right percentage
Use the percentage that matches the comparison you want. If the question asks whether commute method differs between Year 11 and Year 12, compare within each year level, so use column percentages. If it asks which year level is more common within each commute method, use row percentages.
| Commute method | Year 11 percentage | Year 12 percentage | |---|---:|---:| | Bus | $38/80=47.5\%$ | $29/80=36.25\%$ | | Car | $24/80=30\%$ | $41/80=51.25\%$ | | Walk or cycle | $18/80=22.5\%$ | $10/80=12.5\%$ |
This suggests Year 12 students in this sample were more likely to travel by car, while Year 11 students were more likely to use the bus or walk/cycle. The wording matters: say "in this sample" unless the data collection justifies a broader claim.
Worked example
Common traps
Use concise statistical language: "higher percentage", "lower percentage", "similar percentage" and "difference of ___ percentage points". Avoid causal language. A two-way table can show a pattern, not prove why the pattern exists.
Completing and percentaging tables
Many exam questions give a partly completed two-way table. Treat it like a consistency puzzle: each row must add to its row total, each column must add to its column total, and the grand total must agree from both directions.
| Preferred revision method | Year 11 | Year 12 | Total | |---|---:|---:|---:| | Flashcards | 34 | 26 | 60 | | Practice questions | 22 | 42 | 64 | | Summary notes | 24 | 12 | 36 | | Total | 80 | 80 | 160 |
From this table you can form three different percentage tables:
| Percentage type | Denominator | Best use | |---|---|---| | Whole-table percentage | grand total | describes the overall sample composition | | Row percentage | row total | compares the column variable within each row category | | Column percentage | column total | compares the row variable within each column category |
Whole-table percentages are rarely the best way to decide whether two variables are associated, because they mix the group sizes with the category pattern. Row or column percentages are usually more informative.
Writing association conclusions
A strong conclusion has three parts:
- State the comparison being made.
- Quote percentages, not only raw counts.
- Interpret the pattern in context.
For the table above, column percentages compare the two year levels:
| Revision method | Year 11 | Year 12 | |---|---:|---:| | Flashcards | $34/80=42.5\%$ | $26/80=32.5\%$ | | Practice questions | $22/80=27.5\%$ | $42/80=52.5\%$ | | Summary notes | $24/80=30\%$ | $12/80=15\%$ |
This suggests an association between year level and preferred revision method: Year 12 students in this sample were much more likely to prefer practice questions, while Year 11 students were more likely to prefer flashcards or summary notes.
Depth: deciding the direction of comparison
The most common error in categorical association questions is using the wrong denominator. A useful way to decide is to turn the question into a sentence with the word "among".
| Question wording | Natural sentence | Use | |---|---|---| | Is method different between year levels? | Among Year 11 students, what percentage chose each method? | column percentages if year level is in columns | | Is year level different between methods? | Among bus users, what percentage are in each year level? | row percentages if method is in rows | | What fraction of the whole sample are Year 12 car users? | Among all students, what percentage are Year 12 car users? | whole-table percentage |
Once the denominator is chosen, keep it consistent across the comparison. Do not compare a row percentage for one group with a column percentage for another group. They answer different questions.
Strength of association
An association is stronger when the conditional percentages are very different across groups. It is weaker when the conditional percentages are similar. There is no single cutoff in General Mathematics, so use comparative language that matches the size of the difference.
| Percentage-point difference | Careful wording | |---:|---| | 0 to about 5 | little or no evidence of association in this sample | | about 5 to 15 | some evidence of association | | more than about 15 | a noticeable or strong association |
These are writing guides, not fixed rules. A small sample can make a large percentage difference less reliable. For example, a change from 1 out of 4 to 3 out of 4 is a 50 percentage-point difference, but it is based on only eight people in total.
Completing a missing-cell table
When a table has missing values, fill the values that have only one possible answer first.
Marking-quality wording
A full-mark written answer usually avoids vague phrases such as "more people" unless the sample sizes are equal. Prefer:
- "The percentage of laptop users who use a planner is about $68.6\%$."
- "This is about $34.3$ percentage points higher than for phone users."
- "This suggests an association between device type and planner use in this sample."
Do not write "laptops cause planner use". A table of observed categories only describes a pattern.