Box plots and comparing distributions

Drawing a box plot

Mark the lower quartile, median and upper quartile as the box with a line inside. Whiskers go out to the smallest and largest values that are not outliers. Any outliers are marked with a cross. Always draw on a labelled scale.

Outliers on a box plot

The question will tell you the rule, commonly: an outlier is a value below Q1 - 1.5 × IQR or above Q3 + 1.5 × IQR.

Example: Q1 = 12, Q3 = 21, so IQR = 9. Upper limit = 21 + 13.5 = 34.5. If the largest value is 38 and the next largest is 30, then 38 is plotted as a cross and the whisker ends at 30.

Comparing distributions

Compare a measure of location (median) and a measure of spread (IQR, or range if there are no outliers), with values, in context. You can also comment on skewness.

Good answer: 'The median time for group A (17 min) is higher than for group B (14 min), so group A generally took longer. The IQR for A (9 min) is greater than for B (6 min), so A's times were more varied.'

Worked example

Q1 = 12, median = 17, Q3 = 21, smallest value 4, largest 38, next largest 30. Outliers lie beyond 1.5 × IQR from the quartiles. Where do the whiskers end?

1. IQR = 9; lower limit = 12 - 13.5 = -1.5, upper limit = 21 + 13.5 = 34.5.
2. 4 is above -1.5, so the lower whisker goes to 4.
3. 38 > 34.5 is an outlier (a cross); the upper whisker goes to 30.

Answer: Whiskers at 4 and 30; a cross at 38.

Check yourself

1. From the box plot values Q1 = 12, median = 17, Q3 = 21, find the interquartile range.
   Show answer

   9. IQR = 21 - 12 = 9.

2. With Q1 = 12 and Q3 = 21, outliers are values above Q3 + 1.5 × IQR. Find this limit.
   Show answer

   34.5. 21 + 1.5 × 9 = 34.5.

3. With Q1 = 12 and Q3 = 21, find the lower outlier limit Q1 - 1.5 × IQR.
   Show answer

   -1.5. 12 - 13.5 = -1.5.

4. The largest values in a data set are 26, 30 and 38, with Q1 = 12 and Q3 = 21. Using the 1.5 × IQR rule, where does the upper whisker end?
   Show answer

   30. The limit is 34.5. 38 is an outlier, so the whisker ends at the largest non-outlier, 30.

5. Box plots show median 17 min and IQR 9 min for group A, and median 14 min and IQR 6 min for group B. Which comparison is best?
   • Group A generally took less time, and their times were more consistent
   • Group A generally took longer, and their times were more varied
   • Group A's times were less varied because their median is higher
   • The groups cannot be compared without the means
   Show answer

   Group A generally took longer, and their times were more varied. Higher median means longer times on average; larger IQR means more spread.

6. How is an outlier shown on a box plot?
   • By extending the whisker to it
   • By shading the box
   • It is left off the diagram
   • As a cross beyond the end of the whisker
   Show answer

   As a cross beyond the end of the whisker. The whisker stops at the last non-outlier and the outlier is plotted separately as a cross.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S1

Last updated 7 November 2026. Spotted a mistake? Tell us.