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Maths · Further statistics

Box plots

Find the five-number summary, draw box plots, and compare two distributions using the median and interquartile range.

  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    What is the median of 3, 5, 9, 12, 20?

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    \(9\)

  2. 2

    What is the range of 3, 5, 9, 12, 20?

    Show answerHide answer

    \(17\)

  3. 3

    What is the interquartile range?

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    Upper quartile minus lower quartile

  4. 4

    What is a quartile?

    Show answerHide answer

    A value a quarter or three quarters of the way through the data

  5. 5

    Which measure of average is the middle value?

    Show answerHide answer

    Median

Learning Objectives

  1. 1Find the minimum, lower quartile, median, upper quartile and maximum.
  2. 2Draw a box plot.
  3. 3Read a box plot.
  4. 4Compare two data sets using a box plot.

BOX PLOT

A box plot shows the five-number summary: minimum, lower quartile, median, upper quartile, maximum.

The box shows the middle half of the data. The whiskers go out to the smallest and largest values.

Five-Number Summary

Find the five-number summary of 5, 8, 10, 12, 13, 15, 19, 22, 26.

Show the solutionHide the solution
  1. 1 Minimum and maximum \(5\) and \(26\)
  2. 2 Median \(13\) (the 5th of 9 values)
  3. 3 Lower quartile Middle of \(5, 8, 10, 12\): \(9\)
  4. 4 Upper quartile Middle of \(15, 19, 22, 26\): \(20.5\)

AnswerMin 5, LQ 9, median 13, UQ 20.5, max 26.

Positions in Ordered Data

For \(n\) values in order.

  • Median

    Position: \(\tfrac{n + 1}{2}\)th value

  • Lower quartile

    Position: \(\tfrac{n + 1}{4}\)th value

  • Upper quartile

    Position: \(\tfrac{3(n + 1)}{4}\)th value

  • Interquartile range

    Position: UQ \(-\) LQ

Comparing Two Box Plots

Class A: min 10, LQ 22, median 30, UQ 40, max 55. Class B: min 15, LQ 25, median 32, UQ 38, max 48. Compare the two classes.

Show the solutionHide the solution
  1. 1 Averages Median of B (32) is higher than A (30)
  2. 2 Spread: IQR A: \(40 - 22 = 18\); B: \(38 - 25 = 13\)
  3. 3 Range A: \(45\); B: \(33\)
  4. 4 Conclusion B has a slightly higher median and is more consistent

AnswerOn average class B did slightly better (median 32 against 30), and B's marks were more consistent (IQR 13 against 18).

Writing a Comparison

Always compare one average and one spread, in context.

  • An average

    Compare the medians.

  • A spread

    Compare the IQRs (or the ranges).

  • Use the context

    "Class B scored higher on average" rather than "B is higher".

  • Use numbers

    Quote the values you compared.

Read the Box

A box plot has minimum 4, LQ 10, median 15, UQ 22 and maximum 30. (a) Write down the range. (b) Write down the IQR. (c) The data set has 40 values. Roughly how many are between 10 and 22?

1. Read the five values.

2. Half of the data lie in the box.

A good answer shows: (a) \(30 - 4 = 26\). (b) \(22 - 10 = 12\). (c) About 20, because half of the data are in the box.

Can I...?

  1. 1Order the data.
  2. 2Find the median.
  3. 3Find the quartiles.
  4. 4Write the five-number summary.
  5. 5Draw a box plot.
  6. 6Find the range and IQR.
  7. 7Compare averages.
  8. 8Compare spreads.

Summary & Exam Focus

  • Five values: min, LQ, median, UQ, max.
  • The box holds the middle 50%.
  • Compare medians (average) and IQRs (spread).
  • Give the numbers you use.

Exam focus

The box plots show the marks of two classes. Compare the distributions. (4 marks) (4 marks)

Compare one average and one spread, and refer to the context.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Box plot
A diagram showing the five-number summary.
Five-number summary
Minimum, LQ, median, UQ, maximum.
Whisker
The line from the box out to the minimum or maximum.
Interquartile range
The width of the box.
Range
Maximum minus minimum.
Consistent
Less spread out.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Compare 4 marks

    The box plots show the marks of the students in Class A and in Class B in a test. Compare the distributions of the marks.

    Two box plots comparing marks in Class A and Class B.
    Show answerHide answer

    Model answer

    The median for B (32) is higher than for A (30), so B did slightly better on average. The IQR for B (13) is less than for A (18), so B's marks were more consistent.

    Mark scheme

    • Compares medians with values — M1
    • Conclusion about average in context — A1
    • Compares IQR or range with values — M1
    • Conclusion about spread in context — A1
  2. Question 2 Work out 3 marks

    Here are nine numbers: 5, 8, 10, 12, 13, 15, 19, 22, 26. Work out the median and the interquartile range.

    Show answerHide answer

    Model answer

    Median \(= 13\). LQ \(= 9\), UQ \(= 20.5\), IQR \(= 20.5 - 9 = 11.5\).

    Mark scheme

    • Median 13 — B1
    • LQ 9 and UQ 20.5 — M1
    • 11.5 — A1
  3. Question 3 Draw 3 marks

    The five-number summary of some data is: minimum 12, lower quartile 20, median 27, upper quartile 35, maximum 50. Describe how to draw the box plot.

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    Model answer

    Draw a number line. Draw a box from 20 to 35 with a line inside at 27. Draw whiskers from the box to 12 and to 50.

    Mark scheme

    • Box from 20 to 35 — B1
    • Median line at 27 — B1
    • Whiskers to 12 and 50 — B1
  4. Question 4 Work out 2 marks

    A box plot has minimum 12, lower quartile 20, median 27, upper quartile 35 and maximum 50. Work out the range and the interquartile range.

    Show answerHide answer

    Model answer

    Range \(= 50 - 12 = 38\); IQR \(= 35 - 20 = 15\).

    Mark scheme

    • 38 — B1
    • 15 — B1
  5. Question 5 Explain 2 marks

    The box plot for 60 people has a median of 40. How many of the 60 people have a value greater than 40?

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    Model answer

    30, because the median splits the data in half.

    Mark scheme

    • 30 — B1
    • Median splits the data in half — C1
  6. Question 6 Explain 2 marks

    Give one reason why the interquartile range is a better measure of spread than the range.

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    Model answer

    The range depends on the extreme values, so one unusual value changes it. The interquartile range is not affected by extreme values.

    Mark scheme

    • Range affected by extremes — M1
    • IQR uses only the middle half — A1

Quick check

  1. The box in a box plot shows the...

    1. ARange
    2. BMean
    3. CMiddle 50% of the data
    4. DHighest value
    Show answerHide answer

    C: Middle 50% of the data

    The middle 50% of the data, from LQ to UQ.

  2. How many values are in the five-number summary?

    1. A3
    2. B5
    3. C6
    4. D7
    Show answerHide answer

    B: 5

    Minimum, LQ, median, UQ and maximum.

  3. The median of 3, 5, 9, 12, 20 is...

    1. A9
    2. B5
    3. C12
    4. D49
    Show answerHide answer

    A: 9

    The middle value.

  4. A box plot has LQ 12 and UQ 30. The IQR is...

    1. A12
    2. B30
    3. C42
    4. D18
    Show answerHide answer

    D: 18

    \(30 - 12 = 18\).

  5. The smaller the IQR, the ...

    1. AHigher the average
    2. BMore consistent the data
    3. CLarger the range
    4. DFewer the data
    Show answerHide answer

    B: More consistent the data

    More consistent the data.

  6. A box plot shows 40 values. About how many are below the median?

    1. A10
    2. B15
    3. C20
    4. D40
    Show answerHide answer

    C: 20

    Half of them: 20.

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