The idea, step by step
Use comparable proportions
Raw counts are not enough for unequal sample sizes. Convert each count to a fraction or percentage of its own sample. Forty of fifty is 80%, which exceeds sixty-four of one hundred, or 64%, despite the smaller raw count. Check that populations, questions and collection periods are comparable.
Size is not everything
Two fair samples can differ because different units are selected: this is sampling variation. Larger well-chosen random samples usually give more stable estimates, but not an exact guarantee. Increasing a biased convenience sample does not repair its frame. Report both results and sample sizes so readers can judge the evidence.
A common trap
Different sample estimates prove the population changed.
Instead: Different fair samples can give different estimates from an unchanged population.
See the thinking
Worked example 1
Compare 40 of 50 with 64 of 100 as proportions.
- 40/50=0.8=80%.
- 64/100=64%.
- The first observed proportion is higher.
Worked example 2
Compare samples 10,12,14 and 11,12,13.
- Both totals are 36, so both means equal 12.
- Ranges are 14−10=4 and 13−11=2.
- Matching centres do not imply matching spread.
Check your understanding
Which proportion is larger: 18 of 30 or 25 of 50?
Show answer
18 of 30.
They are 60% and 50% respectively.
Your turn
6 questions · 15 marks · Work at your own pace.
Read the questions online
- Compare 12 of 20 with 42 of 100 as percentages.3 marks
- A random sample of 25 from 500 students includes 10 cyclists. Estimate the population count and state a limitation.3 marks
- Why might 1000 responses from sports-club members still misrepresent all students’ exercise?2 marks
- An unusual value is included in samples of 10 and 100, with otherwise similar observations. In which does it influence the mean more?2 marks
- Name two details to match when comparing samples as evidence of a change over time.2 marks
- Which usually produces more stable proportions: random samples of 20 or 200? Is exact agreement guaranteed?3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Compare 12 of 20 with 42 of 100 as percentages.3 marks
12÷20×100=60; 42÷100×100=42.
60% and 42%; the first proportion is larger. - A random sample of 25 from 500 students includes 10 cyclists. Estimate the population count and state a limitation.3 marks
10/25=0.4.
0.4×500=200.
About 200; sampling variation may affect the estimate. - Why might 1000 responses from sports-club members still misrepresent all students’ exercise?2 marks
Large size cannot remove systematic under-coverage.
The frame excludes non-members. - An unusual value is included in samples of 10 and 100, with otherwise similar observations. In which does it influence the mean more?2 marks
Each observation has weight 1/10 rather than 1/100 in the mean.
The sample of 10. - Name two details to match when comparing samples as evidence of a change over time.2 marks
A changed group or question could explain the result instead of time.
For example, the target population and measurement definition. - Which usually produces more stable proportions: random samples of 20 or 200? Is exact agreement guaranteed?3 marks
Larger random samples usually reduce variation, but chance differences remain.
200; exact agreement is not guaranteed.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum
Population, census and sample; observational versus experimental data; random and non-random sampling; primary/secondary data; reviewing numerical displays and summary statistics; comparing samples and sample sizes; bias, variation and uncertainty; planning, conducting and reporting a fair investigation.
Official curriculum source ↗