The idea, step by step
Two categories for every individual
A two-way table cross-classifies two categorical variables, such as transport choice and year group. Each individual belongs in one cell. Row totals add across categories; column totals add down categories; the grand totals must agree.
A large cell count does not automatically mean a category is more common within that group. Groups can have different sizes. Use a percentage with the relevant row or column total to make a fair within-group comparison.
State the conditioning group
The percentage of cyclists who are Year 10 uses all cyclists as the denominator. The percentage of Year 10 students who cycle uses all Year 10 students instead. These are different questions even when they use the same numerator.
Check that a conclusion compares like with like. Percentages can reveal an association between the categories, but they do not establish a causal relationship.
A common trap
Use the grand total for every percentage.
Instead: A conditional percentage uses the total of the stated group, not automatically everyone.
See the thinking
Compare cycling rates
Use the table to compare cycling rates in Year 9 and Year 10.
| Year | Cycle | Other | Total |
|---|---|---|---|
| 9 | 12 | 18 | 30 |
| 10 | 18 | 42 | 60 |
| Total | 30 | 60 | 90 |
- Year 9 has 12 cyclists among 30 students:12/30=40%.
- Year 10 has 18 among 60:18/60=30%.
- Year 10 has more cyclists, but the cycling proportion is lower.
Change the denominator
In the same table, what percentage of cyclists are Year 10?
| Year | Cycle | Other | Total |
|---|---|---|---|
| 9 | 12 | 18 | 30 |
| 10 | 18 | 42 | 60 |
| Total | 30 | 60 | 90 |
- There are 30 cyclists altogether.
- 18 of them are Year 10.
- 18/30×100=60%.
Check your understanding
There are 8 club members among 20 Year 10 students. What percentage of Year 10 students are members?
Show answer
40%
8÷20×100=40%.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- Complete all totals in the table.3 marks
Club choices Group Yes No A 8 12 B 10 20 - Using the table, what proportion of group A chose Yes?2 marks
Club choices Group Yes No A 8 12 B 10 20 - Using the table, what percentage of Yes responses are from B?2 marks
Club choices Group Yes No A 8 12 B 10 20 - A has 8 Yes out of 20; B has 10 Yes out of 30. Which has the higher Yes rate?3 marks
- A table's row totals sum to 80 but column totals sum to 78. What does this tell you?2 marks
- Two groups have Yes rates 42% and 43% in small samples. Why should a claim of a large association be cautious?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Complete all totals in the table.3 marks
Club choices Group Yes No A 8 12 B 10 20 A:8+12=20; B:10+20=30.
Yes:8+10=18; No:12+20=32.
Row totals 20 and 30; column totals 18 and 32; grand total 50. - Using the table, what proportion of group A chose Yes?2 marks
Club choices Group Yes No A 8 12 B 10 20 Use group A's total 20.
8÷20=0.4.
8/20=40%. - Using the table, what percentage of Yes responses are from B?2 marks
Club choices Group Yes No A 8 12 B 10 20 There are 18 Yes responses.
Use 10÷18×100.
10/18≈55.6%. - A has 8 Yes out of 20; B has 10 Yes out of 30. Which has the higher Yes rate?3 marks
Use percentages because group sizes differ.
The larger raw Yes count in B is not enough.
A:40% compared with B:33.3%. - A table's row totals sum to 80 but column totals sum to 78. What does this tell you?2 marks
Both sets of totals must count the same individuals.
Recheck cells and classifications.
At least one count or total is incorrect or missing. - Two groups have Yes rates 42% and 43% in small samples. Why should a claim of a large association be cautious?2 marks
A percentage difference alone does not describe uncertainty.
Consider sample sizes and selection.
The difference is small and may reflect sample variation.
Curriculum reference & lesson scope
Australian Curriculum v 9 and SA Curriculum Mathematics
Year 10 continuous distributions, bivariate data, two-way categorical tables, bias and investigations, with the approved percentile and standard-deviation content included in the normal chapter.
Official curriculum source ↗