Compare distributions using center, spread and the question’s context
Make separate comparisons of median and IQR · Use the meaning of the variable to interpret better or worse · Avoid conclusions about every individual or an unknown mean
Section Paper 2 · Historical Extended · non-calculator
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Box plots
- Mean and median
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Make separate comparisons of median and IQR
- Use the meaning of the variable to interpret better or worse
- Avoid conclusions about every individual or an unknown mean
Compare at least center and spread
The illustrative journey-time group A has summary 2,8,12,20,30 minutes; B has 5,10,13,14,19. A median is lower, 12 versus 13 minutes, while B has smaller IQR, 4 versus 12, indicating less spread among the central values. In journey times a lower central time can be desirable, while consistency is a different consideration.
Choose statistics appropriate to shape
An extreme upper tail can pull a mean above the median. For the supplied raw A list, mean is 14; for the supplied raw B list 5,9,11,12,14,14,14,19, mean is 12.25. Thus mean and median comparisons need not agree. Use the measure that answers the stated question and explain its sensitivity.
Box plots alone do not establish a mean or prove every B journey is shorter. Data sets can overlap, and a smaller IQR concerns the middle spread rather than all possible variation. Sampling and population differences also limit a wider inference.
Sources for this lesson
Pause and explain
Does a smaller IQR mean every observation in that group is smaller?
Worked example
Compare the illustrated groups’ median journey time and IQR.
Show the worked solution
- Medians are A=12, B=13.
- IQRs are A=12, B=4.
Answer A has the lower median; B has less central spread.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For summaries A=(2,8,12,20,30) and B=(5,10,13,14,19), calculate both IQRs and compare medians.
Give me a hint
Use Q3−Q1.
Compare my reasoning
- A IQR=12 and B IQR=4.
- A median 12 is below B median 13.
A has lower median; B has smaller IQR.
Look for these in your work
- Compare center separately
- Interpret spread from quartiles
Groups have medians 20 and 24, with IQRs 8 and 3. If the variable is delivery time, which is typically faster by median and which is more consistent centrally?
Give me a hint
Smaller time and smaller spread answer different questions.
Compare my reasoning
- The median-20 group is faster by the median comparison.
- The IQR-3 group has less central spread.
Median-20 group is typically faster; IQR-3 group is more consistent centrally.
Look for these in your work
- Use the time context
- Keep the two desirable properties distinct
Given only the illustrated box plots, can you claim B has the lower mean or that every B journey is shorter than every A journey?
Give me a hint
The box plots omit most observations and the groups overlap.
Compare my reasoning
- A five-number summary does not determine the mean.
- The displayed value intervals overlap, so there is no universal individual ordering.
Neither claim follows from the box plots alone.
Look for these in your work
- Avoid an unsupported mean
- Limit group comparisons
Common mistakes
- State both the center and spread comparison.
- A group-level comparison does not order every individual observation.
What can you explain now?
Does a smaller IQR mean every observation in that group is smaller?
Compare with the explanation
No; IQR describes spread, not the ordering of all observations.
Groups can overlap and have different centers.
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Make it stick
Does a smaller IQR mean every observation in that group is smaller?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.