The idea, step by step
Make a balanced comparison
A useful comparison describes centre, spread, shape and unusual values, then relates these to the context. Name both groups and give numbers where possible. Do not simply say one group is better without identifying the measured quantity.
A higher test-score median may be desirable, while a higher delay median may not be. Similar centres do not imply similar variability, and a broad range may come from one extreme result.
Use comparable data
Check the units, measurement methods and population groups before comparing. Sample sizes and the way people were selected affect how confidently the results can be generalised. A box plot is excellent for quartiles, but does not show every feature of the distribution.
Use language such as 'in these samples' when the evidence is sampled. A statistically sensible conclusion can acknowledge that the available summaries cannot answer a particular question.
A common trap
Equal medians mean the data sets are identical.
Instead: The centre can match while the spreads, shapes, sample sizes and individual values differ.
See the thinking
Compare delivery times
Sample A has median 18 min, IQR 8 min; sample B has median 22 min, IQR 4 min. Compare delivery times.
- A has the smaller median by 4 minutes.
- B has the smaller IQR by 4 minutes, so its middle half is more consistent.
- Faster typical time and consistency favour different groups.
Avoid a range-only judgement
A has summary 4, 8, 12, 16, 24; B has 6, 10, 11, 13, 20. Compare centres and spreads.
- A median 12 versus B median 11: B's typical value is lower.
- A IQR 8 versus B IQR 3: B's central spread is smaller.
- A range 20 versus B range 14: B also has the smaller full span.
Check your understanding
Two samples both have median 30. Their IQRs are 4 and 12. What clear difference can you state?
Show answer
The second sample has a wider middle-half spread.
Its IQR is 8 units larger; the shared median says nothing about that spread.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- Class A has mean 72, SD 6; class B has mean 72, SD 11. Compare.2 marks
- Bus A has median delay 5 min, IQR 2; Bus B median delay 3 min, IQR 6. State two differences.3 marks
- One sample is in centimetres and another in metres. What must you do before comparing spreads?2 marks
- A class has one unusually high score. Which is generally less affected: mean or median? Explain.2 marks
- A sample of 10 volunteers has a lower median than a random sample of 200. Can you conclude its population median is lower?2 marks
- Write a comparison using both plots.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Class A has mean 72, SD 6; class B has mean 72, SD 11. Compare.2 marks
Centres coincide.
6<11 compares SD.
Same mean; A's scores are less spread around the mean. - Bus A has median delay 5 min, IQR 2; Bus B median delay 3 min, IQR 6. State two differences.3 marks
3<5 for centre.
2<6 for central spread.
B has lower typical delay; A has more consistent middle-half delays. - One sample is in centimetres and another in metres. What must you do before comparing spreads?2 marks
The numerical scales differ.
Use one unit for centres and spreads.
Convert to a common unit. - A class has one unusually high score. Which is generally less affected: mean or median? Explain.2 marks
The median depends on ordered position.
An extreme value can substantially change the total used for the mean.
Median. - A sample of 10 volunteers has a lower median than a random sample of 200. Can you conclude its population median is lower?2 marks
The volunteer sample may be biased and is small.
The observations describe the samples; generalisations need more evidence.
Not from these facts alone. - Write a comparison using both plots.3 marks
Read medians and quartiles on the common scale.
Keep the conclusion within the sampled groups.
B has median 11 min rather than 12 min and IQR 3 min rather than 8 min; B is slightly quicker and more consistent in these samples.
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 ↗