Draw and interpret a box plot from its five-number summary
Place minimum, quartiles, median and maximum on one scale · Read IQR from the box and range from extrema · Avoid treating horizontal lengths as observation counts
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
- Ordered data
- Quartiles and range
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
- Place minimum, quartiles, median and maximum on one scale
- Read IQR from the box and range from extrema
- Avoid treating horizontal lengths as observation counts
State the summary convention
For the ordered illustrative list 2,6,10,10,14,18,22,30, take the median of the full list, then the medians of its lower and upper halves. This gives minimum 2, Q1=8, median 12, Q3=20, maximum 30. For odd lists in this lesson, exclude the full-list median when forming halves. Other questions can state another quartile convention; follow it.
Draw a box from Q1 to Q3, a median line inside it, and whiskers to the minimum and maximum for this simple min–max box plot. If a chart separately plots outliers and uses another whisker rule, its legend controls the meaning.
Position describes value, not count density by itself
IQR=20−8=12 and range=30−2=28. The box describes the middle part of the ordered distribution. Equal quarter shares need not occupy equal horizontal lengths. Ties and finite-sample conventions can affect exact proportions at boundaries; a box plot does not reveal every observation or an exact mean.
Sources for this lesson
Pause and explain
Can the mean be determined exactly from a five-number summary alone?
Worked example
Find IQR and range from the illustrative summary 2,8,12,20,30.
Show the worked solution
- Subtract Q1 from Q3: 20−8.
- Subtract minimum from maximum: 30−2.
Answer IQR 12; range 28.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
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 2,6,10,10,14,18,22,30, use the stated half-list convention to find Q1, median and Q3.
Give me a hint
Average the middle pair in each relevant list.
Compare my reasoning
- Median=(10+14)/2=12; lower-half median=(6+10)/2=8.
- Upper-half median=(18+22)/2=20.
Q1=8; median=12; Q3=20.
Look for these in your work
- Use the stated convention
- Keep full-list and half-list centers distinct
A simple min–max box plot has summary 3,7,10,15,21. Find IQR and range.
Give me a hint
Use the quartiles for IQR and extremes for range.
Compare my reasoning
- IQR=15−7=8.
- Range=21−3=18.
IQR 8; range 18.
Look for these in your work
- Choose the correct pair of summary values
- Keep the two spread measures separate
In the illustrated box plot, the upper half of the box is longer than the lower half. Does this prove more observations lie above the median?
Give me a hint
Horizontal length measures value differences.
Compare my reasoning
- The upper section spans a larger interval of values.
- It does not establish a larger observation count above the median.
No; the length concerns spread, not a greater count above the median.
Look for these in your work
- Distinguish interval width from frequency
- Allow ties and finite-sample conventions
Common mistakes
- A long whisker is not automatically more observations.
- Check whether whiskers are extrema or follow an outlier rule.
What can you explain now?
Can the mean be determined exactly from a five-number summary alone?
Compare with the explanation
No; the observations inside the sections are not fully specified.
A box plot preserves selected order statistics, not every value needed for the total.
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Make it stick
Can the mean be determined exactly from a five-number summary alone?
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.