Topic guide 1 · Unit 3 Topic 1 · Bivariate data analysis 1
A two-way table counts people (or things) by two categories at once, such as year level and travel time, or staff group and opinion. Every question on it comes down to one move: divide a count by the right total. Pick the wrong total and you get a number that is arithmetically fine and answers a different question.
This guide shows you how to find the right total from the wording, how to build and read a percentaged table in either direction, how to justify an association in the shape the marking guides reward, and how to fill an empty table from written clues. Every two-way table question from the 2020 to 2025 externals is here, with mark-by-mark solutions. Tick off each skill once you can do it without help.
Part 1
Every external from 2020 to 2025 had a two-way table question. I count eight: five multiple choice (2020 Paper 1 Question 14, 2021 Question 3, 2022 Question 13, 2023 Question 4 and 2025 Question 9) and three short response (2020 Paper 1 Question 22, 2024 Paper 2 Question 1 and 2025 Paper 1 Question 17). Add 2021 Question 11, a multiple choice question on what bivariate data is, and this sub-topic carried nine questions worth 19 marks.
Bivariate data analysis 1 as a whole is bigger. With its numerical half (explanatory and response variables, scatterplots, direction, form, strength and $r$) it had 16 multiple choice questions and 9 short-response questions worth 39 marks, plus parts of four more. The numerical half has its own guide, Topic guide 2, Scatterplots and numerical association.
These are cheap marks if you know which total to use. From 2021 on, between 71.6% and 77.6% of students got each multiple choice item right, with one exception: only 36.9% got 2022 Question 13, which asks you to read a percentaged table. It has a section of its own in Part 4.
Unit 3 Topic 1, sub-topic “Identifying and describing associations between two categorical variables” (5 hours), has four dot points:
The word “appropriately” in the third dot point does real work. A table percentaged the wrong way round is arithmetically correct and answers a different question. Part 4 shows how to tell which way round a table has been percentaged, and Part 5 which way is appropriate when the question leaves it to you.
The 2019 syllabus gave this sub-topic 4 hours, and its first dot point read “define bivariate data”. The 2025 syllabus gives it 5 hours and says “Understand the meaning of bivariate data”. The other three dot points are word for word the same, so every past question in this guide is still in scope.
Identifying the explanatory and response variables is a dot point of the numerical sub-topic in the 2025 syllabus, so it is taught in Topic guide 2. You still use the idea here, to decide which way to percentage.
| Section | Supplied |
|---|---|
| Data | mean $\bar{x}=\frac{\sum x}{n}$, median $\left(\frac{n+1}{2}\right)^{\text{th}}$ data value, linear equation $y=mx+c$, slope $m=r\frac{s_y}{s_x}$, y-intercept $c=\bar{y}-m\bar{x}$, outliers $Q_1-1.5\times IQR\le x\le Q_3+1.5\times IQR$ |
| Two-way tables | Nothing. You supply $\text{percentage}=\dfrac{\text{count}}{\text{total}}\times 100\%$ and you choose the total. |
None of the Data formulas applies to this sub-topic. What you need is a reliable way to choose the total and a fixed answer shape, and both are below.
Part 2
Bivariate data is two variables recorded for each individual, so you can ask whether they are related. One variable on its own (ratings of a pie, cars through an intersection) is univariate, however many values you collect.
When both variables are categorical (year level, preferred brand, change or keep the uniform), you summarise them in a two-way frequency table. When both are numerical (hours, litres, dollars), you draw a scatterplot, which is Topic guide 2.
Which option is an example of bivariate data?
Look for two variables measured on the same individual. Option (B) records two things for each household: how many people live there and how much water it uses.
(A) is one categorical variable. (C) is one numerical variable (a count). (D) is one numerical variable (a time). Each is univariate. 74.3% of students chose (B).
QCAA usually prints only the inner cells, the counts. The totals are yours to add, and the 2020 report listed row sums and column sums as something to strengthen. Write them on the table before you calculate anything.
Part 3 uses those three totals on a real question, and shows that each one gives a different answer.
Part 3
A “percentage of” question is a count divided by a total. The “percentage of” multiple choice questions from 2020 to 2025 kept offering the same kinds of wrong answer: the count over the grand total, the count over the total in the other direction, and the count itself with a % sign stuck on. The 2020 subject report spelt this out for 2020 Question 14, so it is deliberate.
The group named after the word “of” (or after “who”) is the group whose total goes on the bottom of the fraction. Read the question as a sentence about one group, then circle that group’s total.
| The question says | Divide by |
|---|---|
| “What percentage of university students prefer to drive their own car?” | the Students row total |
| “The percentage of field athletes who prefer brand Y” | the Field athletes row total |
| “Of the students who preferred comedy, what percentage were in Year 9 or higher?” | the Comedy column total |
| “What percentage of all students …” | the grand total |
The first three are quoted from 2020, 2023 and 2021 papers. The last is the shape a grand-total question would take.
A sample of university staff and students was asked whether they preferred catching public transport or driving their own car to university. The data collected is shown in the table.
| Public transport | Drive own car | |
|---|---|---|
| Staff | 2 | 18 |
| Students | 48 | 12 |
What percentage of university students prefer to drive their own car?
The group is university students, so divide by the Students row total, $48+12=60$:
$$\frac{12}{60}\times 100\% = 20\%$$
The 2020 subject report explained each distractor. (A) states the raw count 12 as a percentage. (B) is $\frac{12}{80}$, the grand total. (D) is $\frac{12}{30}$, the Drive own car column total. The 2020 report published no multiple choice percentages.
Athletes were surveyed about their preferred shoe brand: X, Y or Z. The results are shown in the frequency table.
| X | Y | Z | Total | |
|---|---|---|---|---|
| Field athletes | 26 | 12 | 2 | 40 |
| Track athletes | 14 | 18 | 8 | 40 |
| Total | 40 | 30 | 10 | 80 |
The percentage of field athletes who prefer brand Y is
The group is field athletes, and this time the totals are printed. The Field athletes row total is 40:
$$\frac{12}{40}\times 100\% = 30\%$$
The distractors are the 2020 set again. (A) is the count 12 with a % sign. (B) is $\frac{12}{80}$, the grand total. (D) is $\frac{12}{30}$, the column total for brand Y, and 16.6% of students chose it. 71.6% chose (C).
The table shows the school start-time preferences of students in different year levels.
| 8 am | 9 am | |
|---|---|---|
| Years 7–9 | 124 | 74 |
| Years 10–12 | 82 | 120 |
What percentage of students in Years 10–12 prefer a start time of 8 am?
The group is students in Years 10 to 12, so divide by that row total, $82+120=202$:
$$\frac{82}{202}\times 100\% = 40.6\%$$
(A) is $\frac{82}{400}$, the grand total. (B) is $\frac{82}{206}$, the 8 am column total, and it sits less than one percentage point from the key, so a wrong total here still looks plausible. (D) is $\frac{82}{124}$, a comparison with the other row’s 8 am count. 77.6% of students chose (C).
The table shows the results of a student survey about their preferred movie genre.
Of the students who preferred comedy, what percentage were in Year 9 or higher?
“Of the students who preferred comedy” names the group, so the total is the Comedy column, $20+24+36=80$. “Year 9 or higher” takes in two rows, so the count is $24+36=60$:
$$\frac{60}{80}\times 100\% = 75\%$$
(A) 25% is $\frac{20}{80}$, the Years 7 to 8 share, and also $\frac{60}{240}$, the count over all 240 students. (B) 30% is $\frac{24}{80}$, Years 9 to 10 only. (C) 60% is the count 60 with a % sign. 74.5% of students chose (D).
Part 4
A percentaged two-way table replaces every count with a percentage of a total. There are two ways to do it.
If the question says which totals to use, use those, whatever you would have chosen. If you are handed a percentaged table, add one row and one column. Whichever direction makes 100% is the direction it was percentaged, and that tells you which group each percentage belongs to.
A store asked its junior and senior staff whether or not they would like to change the store uniform. The results are in the frequency table.
| Change uniform | Do not change uniform | |
|---|---|---|
| Junior staff | 92 | 28 |
| Senior staff | 23 | 67 |
a) Convert the two-way table into a percentaged two-way frequency table using column totals. [2 marks]
b) Explain whether there is an association between staff groups and a desire to change the store uniform. [2 marks]
a) The instruction says column totals, so each column must come to 100%. Change uniform: $92+23=115$. Do not change uniform: $28+67=95$.
1 mark · correctly determines column totals (may be implied by subsequent working)
| Change uniform | Do not change uniform | |
|---|---|---|
| Junior staff | $\frac{92}{115}=80\%$ | $\frac{28}{95}=29.5\%$ |
| Senior staff | $\frac{23}{115}=20\%$ | $\frac{67}{95}=70.5\%$ |
| Total | 100% | 100% |
1 mark · correctly represents the data in a percentaged two-way table
b) The marking guide’s sample: “There does appear to be an association between the staff groups and wanting to change the uniform. The data suggests that junior staff want to change the uniform (80% as opposed to 20% of senior staff) and senior staff do not want to change (70.5% compared with 29.5% of junior staff).”
1 mark · suggests the presence of an association
1 mark · provides reasons to support conclusion (follow-through allowed, alternative reasoning accepted)
To say exactly what a column percentage means, read it down its column: 80% of the staff who want a change are junior, and 70.5% of the staff who don’t are senior. If you percentage within each staff group instead (row totals), 76.7% of junior staff want a change against 25.6% of senior staff. Either view shows a large difference, and the guide accepts alternative reasoning, so the conclusion is the same.
The two-way table summarises the semester 1 results for students enrolled in two courses, Machinery and Electrical. Students achieved either satisfactory (S) or unsatisfactory (U).
The 10% cell in the table indicates that
Test which way the table adds to 100%. The columns do: $80+20=100$ and $10+90=100$. The rows don’t: $80+10=90$ and $20+90=110$. So these are column percentages, and each cell is a percentage of its Machinery group.
The 10% cell is in the Machinery U column and the Electrical S row. Of the students who got U in Machinery, 10% got S in Electrical.
(C) reads the cell along its row, as a row percentage. It was the most popular answer, chosen by 46.5% of students. Only 36.9% chose (D). (A) and (B) treat the cell as a percentage of all students, which a percentaged table never shows unless it was percentaged by the grand total.
Most people read a table left to right, so “Electrical S … 10%” sounds like “of the Electrical S students, 10% …”. That reading only works if the rows add to 100%. Spend five seconds adding before you choose. The direction that makes 100% is the group after “of the”.
Try it yourself
This is the 2020 Paper 1 Question 22 uniform table. Choose which totals to divide by, then select a cell to read it as a sentence. Watch which way the table adds to 100%: that direction names the group after “of”. Change the counts to see that the rule does not depend on the numbers.
Divide each count by
| Change | Do not change | Total | |
|---|---|---|---|
| Junior staff | |||
| Senior staff | |||
| Total |
Each cell shows the count and its percentage. Select a cell to read it. The 100% in orange shows which way the table adds up.
The selected cell
Part 5
Two categorical variables are associated when the percentages in one category differ noticeably from group to group. If year level made no difference to travel time, the share of short trips would be about the same in each year level. A large gap suggests an association.
Raw counts can’t be compared when the groups are different sizes. Percentage within each group of the explanatory variable (the one that may explain the other, here year level), so each group totals 100%. Then compare the same response category across the groups. If the question tells you which percentages to compare, as 2025 Question 17 does, that settles it. Choosing the explanatory variable is taught in Topic guide 2.
The 2025 report is direct about step 2: “Stating there is a difference in percentages is not the same as stating there is a large difference in percentages. Percentages that differ but are similar (e.g. 45% and 55%) suggest no association.”
A conclusion of no association uses the same three steps. If the percentages are close, quote them, say they are similar, and conclude there is little evidence of an association between the two variables. A small difference is still a finding, and it needs the same evidence.
Students responded to a survey about their travel time to school.
| Travel less than 15 minutes | Travel 15 minutes or more | |
|---|---|---|
| Year 6 students | 11 | 6 |
| Year 10 students | 3 | 9 |
a) Determine the total number of students in each year level who responded to the survey. [1 mark]
The survey results suggest there is an association between a student’s year level and their travel time.
b) Justify the suggested association by comparing the percentages of Year 6 and Year 10 students who travel less than 15 minutes. [3 marks]
a) Add along each row. Year 6: $11+6=17$. Year 10: $3+9=12$.
1 mark · correctly identifies both totals
b) The question names the groups (Year 6, Year 10) and the category (less than 15 minutes), so percentage within each year level.
Year 6:
$$\frac{11}{17}\times 100\% = 64.7\%$$
1 mark · calculates percentage of Year 6 students who travel less than 15 minutes (accept a value between 64 and 65 inclusive, follow-through allowed)
Year 10:
$$\frac{3}{12}\times 100\% = 25\%$$
1 mark · calculates percentage of Year 10 students who travel less than 15 minutes
The guide’s sample conclusion: “The large difference in the percentages for the different year levels (64.7% is more than double 25%) suggests there is an association between a student’s year level and their travel time.”
1 mark · compares percentages to justify suggested association (only awarded if prior working supports the comparison)
The difference is 39.7 percentage points. The 2025 report praised answers that called it large, significant, substantial or more than double. Dividing by a column total ($\frac{11}{14}=78.6\%$) or by the grand total ($\frac{11}{29}=37.9\%$) answers a different question, and the comparison mark depends on the working before it.
Try it yourself
These are the 2025 Paper 1 Question 17 counts. Each year level is percentaged within itself, so each bar is out of 100%. Change the counts and watch the two percentages, and the gap between them.
Conclude
QCAA sets no cut-off for a large difference. Its one benchmark is that 45% and 55% are similar.
Part 6
The hardest shape so far gives you sentences instead of counts. The order matters, because each entry depends on the one before.
Each of the 60 performers in a music and dance concert is either a Year 11 or Year 12 student and either a musician or a dancer.
There are four more Year 11 students than Year 12 students. One quarter of the Year 11 students are dancers and half of the Year 12 students are dancers.
Complete the two-way frequency table to calculate the percentage of students who are musicians.
| Year 11 | Year 12 | Total | |
|---|---|---|---|
| Musician | |||
| Dancer | |||
| Total | 60 |
Step 1 · Year level totals. Let Year 12 have $n$ students, so Year 11 has $n+4$:
$$\begin{aligned} n + (n+4) &= 60 \\ 2n &= 56 \\ n &= 28 \end{aligned}$$
Year 12 has 28 students and Year 11 has 32. A quick check: $28+32=60$ and $32-28=4$.
1 mark · correctly calculates the frequencies for total Year 11 students and total Year 12 students
Step 2 · Dancers. One quarter of 32 is 8. Half of 28 is 14.
1 mark · calculates frequencies for dancers in Year 11 and dancers in Year 12
Step 3 · Musicians, by subtraction. $32-8=24$ and $28-14=14$.
1 mark · calculates frequencies for musicians in Year 11 and musicians in Year 12
Step 4 · Row totals. Musicians $24+14=38$, dancers $8+14=22$, and $38+22=60$ checks against the grand total.
| Year 11 | Year 12 | Total | |
|---|---|---|---|
| Musician | 24 | 14 | 38 |
| Dancer | 8 | 14 | 22 |
| Total | 32 | 28 | 60 |
1 mark · calculates frequencies for total musicians and total dancers
Step 5 · The percentage. “The percentage of students who are musicians” means all 60 students, so use the grand total:
$$\frac{38}{60}\times 100\% = 63.\dot{3}\%$$
That is 63.3% to one decimal place.
1 mark · calculates percentage of students who are musicians
Follow-through marks are allowed throughout. Get the year totals wrong and you can still earn the later marks if each step follows correctly from them.
Part 7
The three short-response marking guides give one mark per step, so working is worth more than the final answer.
| Question | Where the marks went |
|---|---|
| 2020 P1 Q22 (4) | column totals (may be implied) · the percentaged table · suggests an association · reasons that support it |
| 2024 P2 Q1 (5) | year level totals · dancers · musicians · row totals · the percentage |
| 2025 P1 Q17 (4) | both totals · Year 6 percentage · Year 10 percentage · a comparison that justifies the association |
| Checkpoint | How to run it |
|---|---|
| Which total? | Find the group after “of” or “of the … who”. Its total goes on the bottom. |
| Do the totals agree? | Row sums and column sums both add to the grand total. |
| Rows or columns? | Follow the instruction. With none, percentage within the groups you are comparing. |
| Does it make 100%? | Each row (or each column) you percentaged must add to 100%. On a printed percentaged table, this tells you how to read a cell. |
| Both percentages quoted? | One percentage on its own is not a comparison. |
| Size of the difference? | Large, substantial or more than double suggests an association. Similar suggests none. |
| Conclusion in context? | Name both variables, using the question’s words. |
Game plan
In the Game plans
Practice
I wrote these four to the QCAA format and checked every answer in Python. The marking shown is my suggested marking, in the style of the QCAA guides. Question 4 has the shape of 2024 Paper 2 Question 1, so attempt it as one question with no parts and give yourself about 12 minutes.
A library recorded whether its borrowers chose print books or e-books.
| Print books | E-books | |
|---|---|---|
| Adults | 84 | 36 |
| Teenagers | 27 | 54 |
Of the borrowers who chose e-books, what percentage were teenagers?
“Of the borrowers who chose e-books” names the group, so the total is the E-books column, $36+54=90$:
$$\frac{54}{90}\times 100\% = 60.0\%$$
(A) is $\frac{54}{201}$, the grand total. (B) is the count 54 with a % sign. (D) is $\frac{54}{81}$, the Teenagers row total, which answers “what percentage of teenagers chose e-books”.
A cinema surveyed its members about whether they preferred morning or evening sessions.
| Morning | Evening | |
|---|---|---|
| Under 30 | 18 | 62 |
| 30 or over | 51 | 24 |
a) Determine the total number of members surveyed in each age group. [1 mark]
b) Justify whether there is an association between age group and session preference, by comparing the percentages of each age group who prefer evening sessions. [3 marks]
a) Under 30: $18+62=80$ members. Aged 30 or over: $51+24=75$ members.
1 mark · correctly determines both totals
b) The groups are different sizes, so percentage within each age group.
Under 30:
$$\frac{62}{80}\times 100\% = 77.5\%$$
1 mark · calculates the percentage of under-30 members who prefer evening sessions
30 or over:
$$\frac{24}{75}\times 100\% = 32.0\%$$
1 mark · calculates the percentage of members aged 30 or over who prefer evening sessions
77.5% of members under 30 prefer evening sessions, compared with 32.0% of members aged 30 or over. This large difference, about 45 percentage points, suggests there is an association between age group and session preference.
1 mark · compares the percentages, describing the difference as large, to justify an association between age group and session preference
A council surveyed 150 commuters about how they usually travel to work.
| Car | Public transport | |
|---|---|---|
| City residents | 30 | 60 |
| Regional residents | 48 | 12 |
a) Convert the two-way table into a percentaged two-way frequency table using row totals. [2 marks]
b) A council officer claims that where a commuter lives is not associated with how they travel to work. Evaluate the reasonableness of the officer’s claim. [2 marks]
a) Row totals: city $30+60=90$ and regional $48+12=60$. Each row must come to 100%. For example, city residents by car: $\frac{30}{90}\times 100\% = 33.3\%$.
1 mark · correctly determines row totals (may be implied)
| Car | Public transport | |
|---|---|---|
| City | 33.3% | 66.7% |
| Regional | 80.0% | 20.0% |
Each row adds to 100%.
1 mark · correctly represents the data in a percentaged two-way table
b) 80.0% of regional residents travel by car, compared with 33.3% of city residents. That is a large difference of about 47 percentage points (more than double).
1 mark · compares percentages across the two groups and describes the difference as large
The large difference suggests there is an association between where a commuter lives and how they travel to work, so the officer’s claim is not reasonable.
1 mark · evaluates the claim as not reasonable, with the conclusion in context
A sports club has 90 members, each of whom plays either singles or doubles, and each of whom is either a junior or a senior member. There are twice as many senior members as junior members. Two thirds of the junior members play singles, and 25% of the senior members play singles.
Construct a two-way frequency table and use it to provide a justified decision about whether there is an association between membership category and preferred format.
Group totals. Let there be $j$ juniors, so $2j$ seniors: $j+2j=90$, $j=30$. So 30 juniors and 60 seniors.
1 mark · correctly determines the totals for junior and senior members
Singles. Two thirds of 30 is 20 juniors. 25% of 60 is 15 seniors. Doubles, by subtraction: $30-20=10$ and $60-15=45$.
1 mark · correctly calculates the singles and doubles frequencies for each category
| Junior | Senior | Total | |
|---|---|---|---|
| Singles | 20 | 15 | 35 |
| Doubles | 10 | 45 | 55 |
| Total | 30 | 60 | 90 |
1 mark · completes the table with row totals that check against 90
Percentage within each membership category. The categories are different sizes, so compare the percentage of each that plays singles:
Juniors:
$$\frac{20}{30}\times 100\% = 66.7\%$$
Seniors:
$$\frac{15}{60}\times 100\% = 25.0\%$$
1 mark · calculates the percentage of juniors and of seniors who play singles
66.7% of junior members play singles, compared with 25.0% of senior members. This large difference of about 42 percentage points suggests there is an association between membership category and preferred format.
1 mark · provides a justified decision, comparing the percentages and concluding in context
Sources: QCAA General Mathematics external assessment question books, multiple choice books and marking guides, 2020 to 2025. General Mathematics 2025 v1.3 general senior syllabus, Unit 3 Topic 1, and the 2019 v1.2 syllabus for comparison. QCAA General Mathematics subject reports, 2020 to 2025. General Mathematics formula book. Every percentage on this page was recomputed in Python from the published tables.
Past questions are from the 2020–2025 QCAA General Mathematics external assessments, © State of Queensland (QCAA) 2020–2025, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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