The idea, step by step
The box measures the middle half
The interquartile range is Q3−Q1. It describes the width occupied by the middle 50% of ordered observations and is less affected by a single extreme result than the full range.
A box plot marks Q1 and Q3 as the box ends, with the median inside. In this lesson whiskers go to the minimum and maximum. Some plots use an outlier rule instead; always check the stated convention.
Compare on a common scale
When comparing two box plots, use the same units and scale. First compare medians for a typical result, then compare IQRs for the spread of the middle half. A larger median is not automatically better: for waiting time, a smaller median may be preferable.
The diagram does not show the number of observations unless that is stated separately. Equal-width sections do not prove equal counts of different score values, and the shape of a box does not tell you the mean.
A common trap
The longer half of a box contains more people.
Instead: Each side of the median represents roughly one-quarter of ordered positions, even when the widths differ.
See the thinking
Read a box plot
For this waiting-time box plot, find the median, IQR and full range.
- The median line is at 15 minutes.
- The box runs from 10 to 20, so IQR=20−10=10 minutes.
- The whiskers run from 5 to 25, so range=20 minutes.
Compare two groups
Group A has summary 4, 8, 12, 16, 24; Group B has 6, 10, 11, 13, 20. Compare typical times and spread.
- A's median is 12; B's is 11, so B has a slightly lower typical time.
- A's IQR is 16−8=8; B's is 13−10=3.
- B's central half is less spread out.
Check your understanding
Q1=14 and Q3=23. What is the IQR?
Show answer
9
Subtract lower quartile from upper quartile:23−14=9.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- Calculate IQR when Q1=7.5 cm and Q3=12.0 cm.2 marks
- State the five-number summary shown.2 marks
- Draw a labelled box plot for the summary 2, 5, 8, 11, 15 seconds.3 marks
- A: median 20, IQR 6. B: median 24, IQR 4. Compare the typical mass and central spread.3 marks
- Can a box plot alone tell you the exact mean? Explain.2 marks
- Two plots use horizontal scales 0–20 and 0–100. Why should you redraw them before comparing box widths?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Calculate IQR when Q1=7.5 cm and Q3=12.0 cm.2 marks
12.0−7.5=4.5.
4.5 cm - State the five-number summary shown.2 marks
Read the two whisker ends, the box ends and the median line in order.
5, 10, 15, 20, 25 minutes - Draw a labelled box plot for the summary 2, 5, 8, 11, 15 seconds.3 marks
Choose a consistent scale covering 2 to 15.
Mark all five summary values; join the box and whiskers.
Whiskers at 2 and 15; box from 5 to 11; median at 8, on a seconds scale. - A: median 20, IQR 6. B: median 24, IQR 4. Compare the typical mass and central spread.3 marks
24>20 compares medians.
4<6 compares IQRs.
B has higher median mass and smaller central spread. - Can a box plot alone tell you the exact mean? Explain.2 marks
A box plot supplies five positions, not every observation.
Different data sets with that summary may have different means.
No. - Two plots use horizontal scales 0–20 and 0–100. Why should you redraw them before comparing box widths?2 marks
A centimetre on the page represents different amounts.
Use a common numerical scale for a fair visual comparison.
The different scales distort apparent spread.
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 ↗