The idea, step by step
Make comparisons fair
Use the same variable, units and collection conditions for each group. Graphs should share a numerical scale. State sample sizes and selection methods; volunteer data and a class census may support different conclusions.
Compare shape, centre, spread and unusual observations. Explain the context: shorter waiting times may be desirable, whereas higher test scores may be desirable.
Support the sentence with numbers
A strong sentence identifies groups and evidence: B’s median wait was 9 min versus A’s 13 min, and B’s IQR was 4 min versus 8 min. This describes B as typically faster and more consistent for those observations.
Do not claim a method caused the difference unless the design supports it. Overlapping distributions mean a better group median does not imply a better value for every individual.
A common trap
Largest range always means the worst group.
Instead: Desirability depends on context, and range is sensitive to extremes.
See the thinking
Compare service
A median 13/IQR 8; B median 9/IQR 4, all minutes. Compare.
- B median is 4 min lower.
- B middle-half spread is 4 min smaller.
- Limit the comparison to measured visits.
Group and individual
A median score 72; B median 68, with overlapping ranges. Must every A score be higher?
- Medians describe middle positions.
- Overlap allows individual orderings to reverse.
Check your understanding
Equal medians; IQRs 3 and 9. Which is tighter centrally?
Show answer
IQR 3 group.
Its middle half spans less.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Scores A median 25/IQR 6; B median 29/IQR 10. Compare.2 marks
- Why should two travel histograms share a horizontal scale?2 marks
- Compare one class before exams with another during holidays. Name a limitation.2 marks
- Groups have means 10, 12 and SDs 4, 1. Compare.2 marks
- Does a random sample of 80 necessarily give a closer mean than a sample of 8?2 marks
- A report says only “A is better”. Improve the approach.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Scores A median 25/IQR 6; B median 29/IQR 10. Compare.2 marks
Use centre and spread separately.
B has median 4 points higher but wider middle-half spread. - Why should two travel histograms share a horizontal scale?2 marks
The same minutes should occupy the same width.
Otherwise equal time differences can appear different. - Compare one class before exams with another during holidays. Name a limitation.2 marks
Use comparable periods or qualify the inference.
Time conditions differ and may affect sleep/study observations. - Groups have means 10, 12 and SDs 4, 1. Compare.2 marks
The SD comparison is not a sample-size statement.
Second group has higher mean and tighter concentration around it. - Does a random sample of 80 necessarily give a closer mean than a sample of 8?2 marks
Larger samples generally reduce random variation but cannot guarantee exactness.
No, not for every realised sample. - A report says only “A is better”. Improve the approach.2 marks
A value judgement needs a defined criterion and evidence.
Name the outcome and compare specific medians/IQRs or justified summaries.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: statistical investigation, univariate data, sampling, displays, summaries and interpretation.
Official curriculum source ↗