The idea, step by step
Use comparable measures and a common scale
A comparison should say which group has the higher typical value and which is more variable. Use median with IQR for skewed data or data with strong outliers; mean and standard deviation are useful for reasonably symmetric distributions without dominant outliers. Show units and use the same axis scale for graphical comparisons. A larger range alone may reflect just one unusual observation.
Separate description from explanation
A sample can show a difference without revealing its cause. Describe direction and size, mention overlap and sample sizes, and consider selection bias or different conditions. Avoid saying every member of one group exceeds every member of the other when the plots overlap. An observational comparison does not prove that group membership caused the difference.
A common trap
Comparing centres while ignoring variability or outliers.
Instead: A centre describes typical level, not consistency. Compare suitable spread and shape as well.
See the thinking
Compare the supplied box plots
For the supplied journey-time box plots, compare typical time and consistency.
- Medians: A 13 min, B 16 min.
- IQRs: A 18−10=8 min; B 21−11=10 min.
- Ranges: A 23−5=18 min; B 29−6=23 min.
- The plots overlap; this does not mean every B journey is longer. No causal conclusion follows from the summaries alone.
An outlier changes the preferred summary
Set A is 4,5,5,6,30; set B is 7,8,8,9,10. Which group has the higher typical score?
- A has sum 50 and mean 10, but median 5.
- B has sum 42 and mean 8.4, median 8.
- Report the outlier and use a resistant summary rather than choosing a measure only to favour A.
Check your understanding
A has median 12/IQR 4 and B median 15/IQR 9. Which is more consistent in its central half?
Show answer
A.
Its smaller IQR means the central half spans 4 rather than 9 units.
Your turn
6 questions · 15 marks · Work at your own pace.
Read the questions online
- Group A has median 18 and IQR 6. Group B has median 22 and IQR 4. Compare both features.2 marks
- Two groups have equal means but standard deviations 2 and 7. What differs?2 marks
- Why is a common horizontal scale essential for two box plots?2 marks
- Compare 1,2,2,3,12 with 4,4,5,5,6 using medians. Mention a reason not to rely only on means.3 marks
- A study samples 12 volunteers in A and 100 randomly selected people in B. Give two cautions about comparing the results.3 marks
- Class A's score range is 40 and Class B's is 25. A student says every A score is more variable than every B score. Explain the error.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Group A has median 18 and IQR 6. Group B has median 22 and IQR 4. Compare both features.2 marks
B is typically higher and its central half is less spread out.
B has a 4-unit higher median and a 2-unit smaller IQR. - Two groups have equal means but standard deviations 2 and 7. What differs?2 marks
Equal centres do not imply equal variation.
The second group has greater spread around its mean. - Why is a common horizontal scale essential for two box plots?2 marks
Compare numerical positions and lengths on one consistent unit scale.
Different scales can make equal spreads appear unequal or reverse a visual comparison. - Compare 1,2,2,3,12 with 4,4,5,5,6 using medians. Mention a reason not to rely only on means.3 marks
The 12 in the first group is an outlier that pulls its mean upwards.
Medians 2 and 5, so the second group has a higher central value. - A study samples 12 volunteers in A and 100 randomly selected people in B. Give two cautions about comparing the results.3 marks
A observed difference may reflect selection rather than the groups themselves.
The different sample sizes affect stability; volunteers in A may differ systematically from non-volunteers. - Class A's score range is 40 and Class B's is 25. A student says every A score is more variable than every B score. Explain the error.3 marks
A larger range only describes the two endpoints.
It does not determine IQR, standard deviation or all observations.
Variation describes a distribution, not each individual value.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 Mathematics: counting, binomial expansion, data, samples, summary measures and introductory normal distributions. These lessons provide the Counting and Statistics strand of the approved Methods preparation route.
Official curriculum source ↗