Understand the idea.
Two categories for every individual
A two-way table counts the same individuals by two categorical characteristics. Each person belongs in one internal cell. Row totals add horizontally, column totals vertically, and both routes must give the same grand total.
To compare rates within differently sized groups, divide each relevant cell count by its own group total. A larger number of cyclists can coexist with a smaller cycling rate if that year group contains substantially more students.
State the conditioning group
Read “of” as a guide to the denominator. The percentage of cyclists who are Year 10 uses all cyclists. The percentage of Year 10 students who cycle uses all Year 10 students. The overlap count stays the same, but the reference group changes.
Name the reference group when reporting a percentage. Compare matching conditional rates, then describe any observed association cautiously. A table can show different proportions across categories without explaining why those differences occurred.
See the method in action.
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 |
Worked solution
- 1
Calculate the Year 9 cycling rate.
The denominator must include all Year 9 students in the table.
- 2
Calculate the Year 10 cycling rate.
The Year 10 count must be compared with the larger Year 10 total.
- 3
Compare proportions rather than counts.
Year 9 has the higher cycling rate even though Year 10 has more individual cyclists.
Cycling rates: Year 9 40%, Year 10 30%.
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 |
Worked solution
- 1
Identify the reference group.
The question asks about cyclists, so use the cycling-column total.
- 2
Find the overlap count.
This internal cell belongs to both stated categories.
- 3
Form the percentage within cyclists.
Sixty per cent of the cyclists are Year 10 students, a different statement from the Year 10 cycling rate.
60% of cyclists are Year 10.
Try it yourself.
Check your understanding
There are 8 club members among 20 Year 10 students. What percentage of Year 10 students are members?
Show answer & reasoning
40%
The reference group is the twenty Year 10 students: .
Keep practising
6 worksheet questions · 14 marks
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:; B:.
Yes:; No:.
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.
.
. - 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 .
. - 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 ↗