Associations between two categorical variables: QCE General Mathematics Unit 3 Bivariate data analysis 1

Syllabus dot point

“Understand bivariate data, construct two-way frequency tables with row and column sums and percentages, and use an appropriately percentaged table to identify and describe an association between two categorical variables in context”

QCEGeneral MathematicsUnit 3: Bivariate data and time series…14 min readUpdated 23 Sep 2026

Quick answer

A two-way frequency table shows the counts for two categorical variables, with row and column sums adding to the same grand total. Convert the counts to percentages within each category of the explanatory variable (row percentages if it is in the rows). If the percentages for a response category change across the groups, the variables are associated; describe the change systematically, quoting percentages for every group, and do not claim causation.

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What this dot point is asking
The answer
Practice questions

What this dot point is asking

Bivariate data records two variables for each individual. When both variables are categorical (year level and exercise habit, age group and news source, delivery on time or late and customer satisfied or not), the tool for investigating them is the two-way frequency table. The QCAA General Mathematics 2025 syllabus asks you to construct these tables with their row and column sums, convert them to percentages, use an appropriately percentaged table to see whether the variables are associated, and describe the association "in a systematic and concise manner" in context.

This is the first sub-topic of Topic 1 (Bivariate data analysis 1) in Unit 3. It is examinable in the Unit 3 internal examination (IA2, if your school's paper samples this topic), in the problem-solving and modelling task (IA1), and in the external assessment, where short-response questions often ask you to complete a table and justify a conclusion with percentages.

The answer

Bivariate data and types of variables

A categorical variable places each individual in a category (yes or no; Year 10, 11 or 12; online, television or print). A numerical variable is a count or measurement. With two numerical variables you use a scatterplot and correlation; with two categorical variables you use a two-way table. When exactly one of the variables might explain the other, call it the explanatory variable; the other is the response variable. For example, age group may explain news source, so age group is explanatory.

Constructing a two-way frequency table

A two-way frequency table has one variable's categories as rows and the other's as columns. Each cell holds the number of individuals in that combination. The row sums (totals) run down the right-hand side and the column sums along the bottom; they must both add to the same grand total.

A survey of 400 adults recorded their age group and their main source of news:

Age group	Online	Television	Print	Total
Under 25	96	18	6	120
25 to 54	110	64	26	200
55 and over	24	38	18	80
Total	230	120	50	400

Missing cells can always be found by subtraction from a total. That is a common short-response question: "complete the table".

Choosing the right percentages

Counts are hard to compare because the groups are different sizes: 110 online readers aged 25 to 54 sounds like more than 96 under 25, but the 25 to 54 group is much bigger. Percentages put every group on the same footing. The key decision is which percentages to calculate.

Key fact

Calculate percentages within each category of the explanatory variable. If the explanatory variable is in the rows, use row percentages (each row adds to 100%); if it is in the columns, use column percentages (each column adds to 100%). Then compare the percentages for the same response category across the explanatory groups. If they differ noticeably, the variables are associated.

For the news data, age group is explanatory and sits in the rows, so divide each cell by its row total:

Age group	Online	Television	Print	Total
Under 25	80.0%	15.0%	5.0%	100%
25 to 54	55.0%	32.0%	13.0%	100%
55 and over	30.0%	47.5%	22.5%	100%

For example, $\frac{96}{120} \times 100\% = 80.0\%$ of under 25s get their news mainly online.

Deciding whether there is an association

Two categorical variables are associated if the percentage distribution of the response variable changes across the categories of the explanatory variable. If the percentages are the same (or nearly the same) for every group, there is no association.

News source and age: online falls from 80.0% to 30.0% across the age groups. Strong association.
Handedness and gender (45 of 300 males and 42 of 280 females left-handed): 15% in both groups. No association.

There is no fixed cut-off for "noticeably different", but a difference of a few percentage points in a sample is weak evidence, while differences of 10 or more percentage points that follow a consistent pattern are good evidence of an association.

Describing an association systematically and concisely

The syllabus wording, "in a systematic and concise manner", describes the answer the markers want:

State that there is (or is not) an association between the two variables, named in context.
Pick one response category and describe how its percentage changes across all the explanatory groups, in order, quoting the percentages.
If useful, add a second category that changes the other way.

"Main news source is associated with age. As age increases, the percentage of adults whose main news source is online decreases, from 80.0% (under 25) to 55.0% (25 to 54) to 30.0% (55 and over), while the percentage relying on television increases from 15.0% to 32.0% to 47.5%."

That is systematic (every group, in order) and concise (two sentences, only the percentages needed).

Worked examples: building, percentaging and describing

Completing a table from partial information

250 customers were surveyed. 160 deliveries were on time, and 144 of those customers were satisfied. 36 customers whose delivery was late were satisfied. Complete the table.

Fill in by subtraction. Late $= 250 - 160 = 90$ . On time and not satisfied $= 160 - 144 = 16$ . Late and not satisfied $= 90 - 36 = 54$ . Column totals: satisfied $144 + 36 = 180$ , not satisfied $16 + 54 = 70$ .

	Satisfied	Not satisfied	Total
On time	144	16	160
Late	36	54	90
Total	180	70	250

Check. $180 + 70 = 250$ . Correct.

Marker's note: show the subtraction for each missing cell; a completed table with no working risks losing marks if one number is wrong.

Choosing and calculating percentages

For the delivery table, decide which percentages to use and whether there is an association.

Explanatory variable: Whether the delivery was on time may explain satisfaction, so timeliness (in the rows) is explanatory. Use row percentages.
Calculate: On time: $\frac{144}{160} = 90\%$ satisfied, $10\%$ not. Late: $\frac{36}{90} = 40\%$ satisfied, $60\%$ not.
Conclude: Satisfaction is associated with delivery timeliness: $90\%$ of on-time customers were satisfied compared with only $40\%$ of late customers.

Marker's note: quote percentages from both groups. "Most on-time customers were satisfied" does not show a comparison.

A table with no association

Among 300 males, 45 are left-handed; among 280 females, 42 are left-handed. Is handedness associated with gender?

Percentages. Males $\frac{45}{300} = 15\%$ ; females $\frac{42}{280} = 15\%$ .

Conclude. No association: the percentage of left-handers is the same for both genders.

Marker's note: a conclusion of "no association" also needs percentages to justify it.

The column-percentage trap

A student says most television viewers are 25 to 54 ( $\frac{64}{120} \approx 53.3\%$ ), so television is most popular with that group. Evaluate this.

What the 53.3% measures. It is a column percentage: the share of television viewers who are 25 to 54. It is large mainly because half the sample is 25 to 54.

Correct comparison. Within each age group, television is the main source for 15.0%, 32.0% and 47.5%. Television is most popular with people aged 55 and over.

Marker's note: this is the classic "evaluate the reasonableness" question in the complex unfamiliar style. Percentage within the explanatory groups is always the fair comparison.

Common traps

Comparing counts instead of percentages: Groups of different sizes cannot be compared by counts.
Percentaging the wrong way: Percentages must be calculated within the categories of the explanatory variable. Column percentages when the explanatory variable is in the rows answer a different question.
Describing one group only: An association is a comparison; quote percentages from every group.
Unsystematic descriptions: Jumping between categories and groups in no order makes the pattern unclear. Pick a category and follow it across the groups in order.
Forgetting that totals must agree: Row totals and column totals must add to the same grand total.
Implying causation: An association in a two-way table does not show that one variable causes the other (see the association and causation page).

Exam technique

Before calculating anything, write "explanatory variable: ..." and "percentages within each ...". In a table-completion question, fill the totals first, then the cells by subtraction, then check the grand total both ways. For "Is there an association? Justify" questions, answer yes or no in the first sentence, then give at least two percentages from different groups for the same response category. With a scientific calculator only, keep percentages to one decimal place unless the question says otherwise.

Note

A two-way table sorts people by two things at once, like their age group and where they get their news. Because some groups are bigger than others, raw counts can be misleading, so you turn each group into percentages that add up to 100. Then you compare: if 80% of young people but only 30% of older people read news online, age and news source are linked. If every group had about the same percentages, there would be no link at all.

Practice questions

Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.

foundation2 marks

For each pair of variables, state whether the data is bivariate and whether each variable is categorical or numerical. (a) A student's year level (Year 11 or Year 12) and whether they have a part-time job (yes or no). (b) The height of each student.

Show worked solution →

(a) Bivariate: two variables are recorded for each student. Both are categorical (year level and job status are categories, not measurements). (1 mark)

(b) Not bivariate: only one variable (height, which is numerical) is recorded, so it is univariate data. (1 mark)

foundation3 marks

Complete the two-way frequency table. Year 10 students: exercise 0 to 2 days a week 42, 3 to 5 days 58, 6 to 7 days 20. Year 12 students: 0 to 2 days 30, 3 to 5 days 45, 6 to 7 days 45. Find (a) the row totals, (b) the column totals, (c) the grand total.

Show worked solution →

	0 to 2 days	3 to 5 days	6 to 7 days	Total
Year 10	42	58	20	120
Year 12	30	45	45	120
Total	72	103	65	240

(a) Row totals: Year 10 $42 + 58 + 20 = 120$ ; Year 12 $30 + 45 + 45 = 120$ . (1 mark)

(b) Column totals: 72, 103 and 65. (1 mark)

(c) Grand total: $120 + 120 = 240$ , which equals $72 + 103 + 65$ . (1 mark) Checking that the row and column totals give the same grand total catches arithmetic slips.

foundation2 marks

Using the exercise table from the previous question, calculate the percentage of Year 12 students who exercise 6 to 7 days a week, and the percentage of Year 10 students who do.

Show worked solution →

Year 12: $\frac{45}{120} \times 100\% = 37.5\%$ . (1 mark)

Year 10: $\frac{20}{120} \times 100\% \approx 16.7\%$ . (1 mark)

The percentages are calculated within each year level (row percentages), because year level is the explanatory variable.

core3 marks

For the exercise table, (a) identify the explanatory variable, (b) calculate the row percentages for both year levels, and (c) state, with reasons, whether there is an association between year level and exercise frequency.

Show worked solution →

(a) Year level is the explanatory variable; it may help explain how often a student exercises (the response). (1 mark)

(b)

	0 to 2 days	3 to 5 days	6 to 7 days	Total
Year 10	35.0%	48.3%	16.7%	100%
Year 12	25.0%	37.5%	37.5%	100%

(1 mark)

(c) Yes, there is an association. A larger percentage of Year 12 students exercise 6 to 7 days a week (37.5%) than Year 10 students (16.7%), and a smaller percentage of Year 12 students exercise only 0 to 2 days (25.0% compared with 35.0%). (1 mark for a comparison quoting percentages from both groups.)

core2 marks

In a survey, 45 of 300 males and 42 of 280 females were left-handed. Is there an association between gender and handedness? Justify your answer.

Show worked solution →

Percentage left-handed: males $\frac{45}{300} \times 100\% = 15\%$ ; females $\frac{42}{280} \times 100\% = 15\%$ . (1 mark)

The percentages are the same, so there is no association between gender and handedness in this sample: knowing a person's gender does not change the likelihood that they are left-handed. (1 mark)

core3 marks

A two-way table records whether 250 customers were satisfied with a delivery (yes or no) and whether the delivery was on time (on time or late). 160 deliveries were on time, and 144 of these customers were satisfied. Of the late deliveries, 36 customers were satisfied. Construct the complete table and find the percentage of late-delivery customers who were not satisfied.

Show worked solution →

Late deliveries: $250 - 160 = 90$ . Not satisfied (on time): $160 - 144 = 16$ . Not satisfied (late): $90 - 36 = 54$ .

	Satisfied	Not satisfied	Total
On time	144	16	160
Late	36	54	90
Total	180	70	250

(2 marks for the completed table.)

Late customers not satisfied: $\frac{54}{90} \times 100\% = 60\%$ . (1 mark) Compare with $\frac{16}{160} \times 100\% = 10\%$ for on-time deliveries: a strong association between delivery timeliness and satisfaction.

exam4 marks

A survey of 400 adults recorded age group and main source of news. Under 25: online 96, television 18, print 6. Age 25 to 54: online 110, television 64, print 26. Age 55 and over: online 24, television 38, print 18. (a) Explain why row percentages (within each age group) are appropriate. (b) Calculate them. (c) Describe the association in a systematic and concise way.

Show worked solution →

(a) Age group is the explanatory variable and the age groups have different sizes (120, 200 and 80). Percentages within each age group put the groups on the same scale, so we can compare how news source changes with age. (1 mark)

(b)

	Online	Television	Print	Total
Under 25	80.0%	15.0%	5.0%	100%
25 to 54	55.0%	32.0%	13.0%	100%
55 and over	30.0%	47.5%	22.5%	100%

(1 mark)

(c) Main news source is associated with age. As age increases, the percentage of adults whose main source is online decreases (from 80.0% to 55.0% to 30.0%), while the percentages relying on television (15.0% to 32.0% to 47.5%) and print (5.0% to 13.0% to 22.5%) both increase. (2 marks: 1 for the trend across all three groups in one category, 1 for supporting percentages in a second category.)

exam3 marks

A student uses the news table and writes: "Most television viewers are aged 25 to 54, so television is more popular among 25 to 54 year olds than among people aged 55 and over." (a) Calculate the percentage of television viewers who are aged 25 to 54. (b) Explain the flaw in the student's reasoning.

Show worked solution →

(a) $\frac{64}{120} \times 100\% \approx 53.3\%$ of television viewers are aged 25 to 54. (1 mark)

(b) This is a column percentage; it mostly reflects the fact that the 25 to 54 group is the largest group in the sample (200 of 400 people). (1 mark) To compare popularity across age groups you must use percentages within each age group: television is the main source for 32.0% of 25 to 54 year olds but 47.5% of people aged 55 and over, so television is actually more popular among the older group. (1 mark)

Practise this

Flashcards General Mathematics deck

Sources & how we know this

General Mathematics General senior syllabus 2025 — QCAA