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Statistics & Probability · Higher

Box plots

Box plots (box-and-whisker diagrams) display the five-number summary of a data set: minimum, lower quartile (Q1), median, upper quartile (Q3), and maximum.

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Key facts to remember

  • 1Box plots show: minimum, Q1, median, Q3, maximum.
  • 2The box spans from Q1 to Q3 (the middle 50% of the data).
  • 3The whiskers extend to the minimum and maximum.
  • 4IQR = Q3 − Q1 — measures the spread of the middle 50%.
  • 5Use box plots to compare two data sets — compare medians and IQRs.
  • 6Outliers may be shown as crosses beyond the whiskers.
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Worked examples

Example 1

Draw a box plot for: min = 12, Q1 = 18, median = 24, Q3 = 30, max = 40.

Working

  1. Draw a number line covering 12 to 40
  2. Mark Q1 = 18 and Q3 = 30, draw a box between them
  3. Mark the median = 24 inside the box
  4. Draw whiskers from the box to min = 12 and max = 40
AnswerBox from 18 to 30, median line at 24, whiskers to 12 and 40
Example 2

Class A: median 65, IQR 20. Class B: median 72, IQR 8. Compare the two classes.

Working

  1. Class B has a higher median (72 > 65) so generally scored higher
  2. Class A has a larger IQR (20 > 8) so results are more spread out / less consistent
  3. Class B was more consistent
AnswerClass B scored higher on average (median 72 vs 65); Class A had more variation (IQR 20 vs 8).
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Common mistakes

✗Confusing the whisker ends with the mean — whiskers show min and max, not mean.
✗Comparing only medians without comparing spread (IQR) when asked to "compare".
✗Drawing the box to the mean rather than the quartiles.
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Exam tips

✓When asked to "compare" box plots, always comment on both average (median) and spread (IQR).
✓Use comparative language: "Class A had a higher median than Class B" — don't just list the values.

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