Draw and interpret common-scale box-and-whisker plots
Place five summary values on one numerical axis and compare centre and middle-half variation without inferring hidden raw values.
Exam scope and assessment notes
Probability & Statistics 1 (Paper 5, AS & A Level) · Mathematics 9709, 2026–2027, section 5.1. Label variables, units, class boundaries and diagram keys. In these raw-data quartile exercises, use the median of each ordered half, excluding the overall median when the number of observations is odd; follow any convention supplied in an examination question. Cumulative-frequency quartiles use ranks n/4 and 3n/4, with a median rank n/2. Grouped-data interpolation and midpoint moments are estimates, not recovered raw measurements. Histogram area represents frequency, with height proportional to frequency density. Use descriptive variance with divisor n, not the sample-estimator divisor n − 1. Coded and combined totals must refer to the same units and compatible coding before addition. Retain unrounded working and distinguish variance units from standard-deviation units. Written practice is self-checked; human subject-teacher review remains pending.
Before you begin
- Calculate raw median and quartiles using a stated convention.
- Read a linear measurement axis.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which endpoints define the box itself?
What you will learn
- Draw a five-number box plot with the correct locations.
- Compare box plots and explain what they cannot determine.
Use a scale rather than equally spaced summary marks
For the eight-minute dataset 2, 3, 3, 4, 8, 9, 10, 11, the five-number summary is minimum 2, Q1 3, median 6, Q3 9.5 and maximum 11. Draw the box from 3 to 9.5 and a median line at 6; extend whiskers to 2 and 11. Distances follow the numerical time axis, so summary points are not equally spaced.
This lesson uses the basic minimum-to-maximum box plot. Some software uses an outlier convention with whiskers ending at different values. Do not automatically apply a 1.5 × IQR rule or alter endpoints unless the question specifies it. Label the unit and use a common axis when comparing groups.
Read centre and spread separately
A second group with summary 1, 4, 5, 7, 13 minutes has median 5 and IQR 7 − 4 = 3. It has a lower median and smaller middle-half spread than the first group, whose median is 6 and IQR 6.5. However, the second group’s range is 12, larger than the first group’s 9. The words ‘less variable’ need the chosen measure of variation.
The five marks do not give a unique exact mean, mode or sample size. Unequal whiskers can suggest tail asymmetry but do not justify treating lengths as counts. A box covering the middle half is a summary of ordered positions; it does not show that measurements are uniformly distributed inside it. Compare like observations in context.
Pause and explain
B has IQR 3 and range 12; A has IQR 6.5 and range 9. Which claim is justified?
Worked example
Draw box plots for A summary (2, 3, 6, 9.5, 11) and B summary (1, 4, 5, 7, 13), in minutes, and compare median and IQR.
Show the worked solution
- Plot both summaries on a single linear time axis and draw each box from Q1 to Q3.
- A median is 6 and IQR 6.5; B median is 5 and IQR 3 minutes.
- B has lower typical time and smaller middle-half spread; its larger range does not contradict that IQR comparison.
Answer B has lower median (5 versus 6) and smaller IQR (3 versus 6.5 minutes), but larger range (12 versus 9).
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.
Give the box, median line and whisker endpoints for summary (2, 3, 6, 9.5, 11) minutes under the basic min-to-max rule.
Give me a hint
Identify the role of each ordered summary entry.
Compare my reasoning
- Draw the box from Q1 = 3 to Q3 = 9.5 minutes.
- Draw the median line at 6 minutes inside the box.
- Extend whiskers to minimum 2 and maximum 11 on the same linear axis.
Box 3–9.5; median 6; whiskers 2 and 11 minutes.
Look for these in your work
- Use the actual numerical spacing.
- Keep minimum and maximum outside the box endpoints.
Compare A summary (2, 3, 6, 9.5, 11) with B summary (1, 4, 5, 7, 13) for completion times in minutes.
Give me a hint
Calculate IQR and range separately before writing a contextual comparison.
Compare my reasoning
- A median 6, IQR 6.5 and range 9; B median 5, IQR 3 and range 12.
- B has a smaller typical completion time and more concentrated middle-half times.
- B’s overall range is larger, so do not claim it has smaller spread by every measure.
B median 5 versus 6; IQR 3 versus 6.5; range 12 versus 9 minutes.
Look for these in your work
- Support each comparison with values and units.
- Specify which spread measure supports consistency.
A report claims that summary (1, 4, 5, 7, 13) determines an exact mean of 6 and that the long upper whisker contains more values. Evaluate both claims.
Give me a hint
Ask whether the five summary positions specify the sum or individual frequencies.
Compare my reasoning
- The five-number summary does not fix all raw observations or their sum.
- Therefore an exact mean of 6 cannot be recovered from it alone.
- Whisker length is a value interval, not a frequency count; a longer interval does not show more observations.
Neither claim follows: the exact mean is undetermined, and whisker length is not frequency.
Look for these in your work
- Explain the missing information for the mean.
- Separate numerical distance from observation count.
Common mistakes
- Place points according to a common numerical scale.
- Name IQR or range when describing variation.
- Do not recover an exact mean or count from the five-number summary alone.
What can you explain now?
Does a wider right whisker mean it contains more observations than the left whisker?
Compare with the explanation
No; length measures numerical spread, not a larger observation count.
Whisker lengths are not histogram areas or frequencies.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Draw five values to scale; compare median, IQR and range by name; keep hidden means and counts undetermined.
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.