Chapter 4 of 8

All 8 chapters
  1. 1Samples and bias, readunderstood
  2. 2Averages from frequency tables, readunderstood
  3. 3Cumulative frequency, readunderstood
  4. 4Box plots, readunderstood
  5. 5Histograms with unequal widths, readunderstood
  6. 6Scatter graphs and correlation, readunderstood
  7. 7Data over time, readunderstood
  8. 8Telling the truth with data, readunderstood

M4.7.4 Chapter 4

Box plots

Two groups of plants have the same median height. How can I show in one small picture that one group is much more spread out?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Draw a box plot from five key values
  • Compare two groups using box plots

Amina grows 11 bean plants in the sun. After two weeks their heights in centimetres, in order, are 12, 15, 17, 18, 20, 22, 23, 25, 27, 30, 31.

Five values sum up the list: the smallest, the lower quartile Q1, the median, the upper quartile Q3 and the largest. A box plot draws these five values on one scale.

For a short list, we find them like this. This is not the n ÷ 4 reading of the last chapter. A graph reads a running total. A short list has separate values, so we count them. The median is the middle value. Split the ordered list into a lower half and an upper half. If the list has an odd number of values, leave the middle value out of both halves. Q1 is the median of the lower half. Q3 is the median of the upper half.

Worked example

Find the five key values of Amina's 11 heights: 12, 15, 17, 18, 20, 22, 23, 25, 27, 30, 31.

  1. There are 11 values, so the median is the (11 + 1) ÷ 2 = 6th value, which is 22.
  2. Leave 22 out. The lower half is 12, 15, 17, 18, 20. Its middle value is the 3rd, so Q1 = 17.
  3. The upper half is 23, 25, 27, 30, 31. Its middle value is the 3rd, so Q3 = 27.
  4. The smallest value is 12 and the largest is 31.

Answer: Smallest 12, lower quartile 17, median 22, upper quartile 27, largest 31

Draw it first

Draw the box plot

  1. Three steps on one scale, height (cm), from 10 to 35. The plot is named Sun. Step 1 marks five dots at 12, 17, 22, 27 and 31. Step 2 draws a line from 12 to 31. Step 3 draws a box from 17 to 27 with a thick line at 22.1222311727Sun101520253035height (cm)

    1 Draw a scale and mark the five key values: 12, 17, 22, 27 and 31.

  2. Three steps on one scale, height (cm), from 10 to 35. The plot is named Sun. Step 1 marks five dots at 12, 17, 22, 27 and 31. Step 2 draws a line from 12 to 31. Step 3 draws a box from 17 to 27 with a thick line at 22.1222311727Sun101520253035height (cm)

    2 Draw a line from the smallest value to the largest.

  3. Three steps on one scale, height (cm), from 10 to 35. The plot is named Sun. Step 1 marks five dots at 12, 17, 22, 27 and 31. Step 2 draws a line from 12 to 31. Step 3 draws a box from 17 to 27 with a thick line at 22.1222311727Sun101520253035height (cm)

    3 Draw a box from Q1 to Q3. Draw a thick line across the box at the median. The parts of the line outside the box are the whiskers.

The box plot named Sun on a scale height (cm) from 10 to 35. The numbers 12, 22 and 31 are under the line, and 17 and 27 are lower down. A bracket over the box is labelled IQR 10 cm.IQR 10 cm1222311727Sun101520253035height (cm)
The box is the interquartile range: 27 − 17 = 10 cm. The whole line is the range: 31 − 12 = 19 cm.

A box plot splits the data into four parts. These are the two whiskers and the two halves of the box. About a quarter of the values lie in each part. The box holds about half of them.

Worked example

Find the five key values of 12 journey times (min): 8, 9, 11, 12, 14, 15, 17, 18, 19, 21, 24, 30.

  1. There are 12 values, so the median is halfway between the 6th and 7th values, 15 and 17. (15 + 17) ÷ 2 = 16.
  2. The lower half is the first 6 values: 8, 9, 11, 12, 14, 15. Its median is halfway between 11 and 12, which is 11.5.
  3. The upper half is the last 6 values: 17, 18, 19, 21, 24, 30. Its median is halfway between 19 and 21, which is 20.
  4. The smallest is 8 and the largest is 30.

Answer: Smallest 8, lower quartile 11.5, median 16, upper quartile 20, largest 30

A box plot named Times on a scale time (min) from 5 to 35. The five values are written under it: 8, 16 and 30 on the first line, and 11.5 and 20 on the second line.8163011.520Times5101520253035time (min)
The median 16 is not in the middle of the box. The box runs from 11.5 to 20, so its middle is 15.75.

Worked example

Read the box plot of journey times below. Find the median, the interquartile range and the range.

  1. The whiskers run from 14 to 38. The thick line in the box is at 26, so the median is 26.
  2. The box runs from 20 to 30.
  3. IQR = 30 − 20 = 10 minutes.
  4. Range = 38 − 14 = 24 minutes.

Answer: The median is 26 minutes, the IQR is 10 minutes and the range is 24 minutes

A box plot named Journey on a scale time (min) from 10 to 40, with a mark every 2 minutes. The whiskers run from 14 to 38. The box runs from 20 to 30. The thick line in the box is at 26.Journey10203040time (min)
Read each end of the box and each whisker against the scale.

To compare two groups, put their box plots on the same scale. Compare the medians to compare the middle. Compare the IQRs to compare the spread. A bigger IQR means more spread. A group with a smaller spread is more consistent.

See it three ways

With things

Amina also grows 11 bean plants in the shade. Their heights in order are 8, 10, 12, 13, 15, 16, 17, 18, 19, 20, 22.

As a picture

Two box plots on one scale, height (cm), from 5 to 35. The plot named Sun has its box from 17 to 27 and its median at 22. The plot named Shade, drawn with hatching, has its box from 12 to 19 and its median at 16.SunShade5101520253035height (cm)

In symbols

Sun: median 22, IQR 27 − 17 = 10. Shade: median 16, IQR 19 − 12 = 7.

The five key values of the plants in the sun and in the shade. The head is value, Sun, Shade. Row 1: smallest, 12, 8. Row 2: lower quartile, 17, 12. Row 3: median, 22, 16. Row 4: upper quartile, 27, 19. Row 5: largest, 31, 22.valueSunShadesmallest128lower quartile1712median2216upper quartile2719largest3122
The same numbers as the box plots.

Worked example

Compare the heights of the plants in the sun and in the shade, using the median and the IQR.

  1. The medians are 22 cm in the sun and 16 cm in the shade.
  2. The IQRs are 10 cm in the sun and 7 cm in the shade.
  3. So the plants in the sun are taller in the middle, with a median 6 cm higher.
  4. The sun plants are also more spread out, because their IQR is 3 cm bigger.

Answer: The sun plants have the higher median, 22 cm against 16 cm, and the larger IQR, 10 cm against 7 cm

Try it

Draw a box plot of 11 hand spans

You need: 11 people to measure, a ruler, squared paper.

  1. Measure each hand span, from the tip of the thumb to the tip of the little finger, with the hand spread flat.
  2. Write the 11 spans in order. Find the smallest, Q1, the median, Q3 and the largest.
  3. Draw a scale on squared paper. Draw the line, the box and the thick median line.

What you will see: The box holds about the middle half of the spans. Compare your box plot with a friend's. Which is more spread out?

If you cannot try it, this is what happens.
A box plot named Spans on a scale hand span (cm) from 16 to 24. The five values are written under it: 17, 20 and 23 on the first line, and 19 and 22 on the second line.1720231922Spans1618202224hand span (cm)

A wrong idea many people have

The longer whisker holds more of the values.

Why it feels right: A longer line looks like more of something.

The test that breaks it: In Amina's sun list the lower whisker is 5 cm long and the upper whisker is 4 cm long. But each holds two values beyond the box: 12 and 15 below it, 30 and 31 above it.

What is true: Each of the four parts holds about a quarter of the values, however long it is. A long part means the values in it are more spread out.

A wrong idea many people have

The median is always in the middle of the box.

Why it feels right: The median is the middle value, and the box looks like the middle.

The test that breaks it: The shade plants have Q1 = 12 and Q3 = 19. The middle of the box is 15.5. But the median is 16.

What is true: The thick line is where the median is. It can lie anywhere inside the box. If it is nearer one end, the values are more bunched up on that side.

Key words

box plot
a picture of the smallest value, Q1, the median, Q3 and the largest value on one scale.
whisker
the line from the box to the smallest or the largest value.
consistent
not spread out: the values stay close together.

Back to your guess

The answer is A: Draw a box plot of each group, on the same scale.

A box plot shows the middle, the spread of the middle half, and the smallest and largest values. Two box plots on one scale can be compared at a glance. A bar chart of medians hides the spread, and a total depends on how many plants there are.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Find the lower quartile of this list: 3, 5, 8, 9, 12, 14, 15.

    Tap your answer.

    1. Not this one. You kept the middle value, 9, in the lower half, so the lower half was 3, 5, 8, 9. With 7 values the middle value is left out of both halves. Go to M4.7.4. Try another.

    2. Not this one. 3 is the smallest value. The lower quartile is the median of the lower half. Go to M4.7.4. Try another.

    3. Yes, that is right. The median is the 4th value, 9. Leave it out. The lower half is 3, 5, 8. Its middle value is 5.

    4. Not this one. You took the 8, the value just below the median. The lower quartile is the middle of the lower half: the lower half is 3, 5, 8, so it is 5. Go to M4.7.4. Try another.

    See the answer

    5

    The median is the 4th value, 9. Leave it out. The lower half is 3, 5, 8. Its middle value is 5.

    • If you chose A: You kept the middle value, 9, in the lower half, so the lower half was 3, 5, 8, 9. With 7 values the middle value is left out of both halves. Go to M4.7.4.
    • If you chose B: 3 is the smallest value. The lower quartile is the median of the lower half. Go to M4.7.4.
    • If you chose D: You took the 8, the value just below the median. The lower quartile is the middle of the lower half: the lower half is 3, 5, 8, so it is 5. Go to M4.7.4.

  2. What is the median shown by this box plot?

    A box plot named Times on a scale time (min) from 0 to 60, with a mark every 5 minutes. The whiskers run from 10 to 55. The box runs from 30 to 50. A thick line in the box is at 35.Times0102030405060time (min)
    Tap your answer.

    1. Not this one. 30 is where the box starts, the lower quartile. The median is the thick line. Go to M4.7.4. Try another.

    2. Yes, that is right. The thick line marks the median. It is at 35.

    3. Not this one. 40 is the middle of the box. The median is the thick line, and it need not be in the middle of the box. Go to M4.7.4. Try another.

    4. Not this one. 50 is where the box ends, the upper quartile. The median is the thick line. Go to M4.7.4. Try another.

    See the answer

    35 minutes

    The thick line marks the median. It is at 35.

    • If you chose A: 30 is where the box starts, the lower quartile. The median is the thick line. Go to M4.7.4.
    • If you chose C: 40 is the middle of the box. The median is the thick line, and it need not be in the middle of the box. Go to M4.7.4.
    • If you chose D: 50 is where the box ends, the upper quartile. The median is the thick line. Go to M4.7.4.

  3. What is the interquartile range of the same box plot?

    A box plot named Times on a scale time (min) from 0 to 60, with a mark every 5 minutes. The whiskers run from 10 to 55. The box runs from 30 to 50. A thick line in the box is at 35.Times0102030405060time (min)
    Tap your answer.

    1. Not this one. 45 is the range, from the end of one whisker to the other. The IQR is the length of the box. Go to M4.7.4. Try another.

    2. Not this one. 35 is the median, a place on the scale. The IQR is a length: Q3 − Q1. Go to M4.7.4. Try another.

    3. Not this one. 50 is Q3, a place on the scale. The IQR is how far Q3 is above Q1. Go to M4.7.4. Try another.

    4. Yes, that is right. The box runs from 30 to 50. IQR = Q3 − Q1 = 50 − 30 = 20.

    See the answer

    20 minutes

    The box runs from 30 to 50. IQR = Q3 − Q1 = 50 − 30 = 20.

    • If you chose A: 45 is the range, from the end of one whisker to the other. The IQR is the length of the box. Go to M4.7.4.
    • If you chose B: 35 is the median, a place on the scale. The IQR is a length: Q3 − Q1. Go to M4.7.4.
    • If you chose C: 50 is Q3, a place on the scale. The IQR is how far Q3 is above Q1. Go to M4.7.4.

  4. The box plots show the journey times on two bus routes. Which statement is true?

    Two box plots on one scale, time (min), from 0 to 50. Route A has whiskers from 10 to 45 and a box from 18 to 32. Route B, drawn with hatching, has whiskers from 15 to 40 and a box from 22 to 28. Both have the median at 25.Route ARoute B01020304050time (min)
    Tap your answer.

    1. Yes, that is right. The medians are both 25 minutes. Route B has the smaller IQR, so its journey times are closer together.

    2. Not this one. The medians are equal, but the boxes are different lengths. The two routes have different spreads. Go to M4.7.4. Try another.

    3. Not this one. A longer box means more spread, not quicker. Both routes have the same median time. Go to M4.7.4. Try another.

    See the answer

    Route B is more consistent, because its box is shorter.

    The medians are both 25 minutes. Route B has the smaller IQR, so its journey times are closer together.

    • If you chose B: The medians are equal, but the boxes are different lengths. The two routes have different spreads. Go to M4.7.4.
    • If you chose C: A longer box means more spread, not quicker. Both routes have the same median time. Go to M4.7.4.

  5. About what share of the values lies inside the box of a box plot?

    Tap your answer.

    1. Not this one. A quarter of the values lie in each whisker, and in each half of the box. The two halves of the box add up to more. Go to M4.7.4. Try another.

    2. Yes, that is right. A quarter of the values lie below Q1 and a quarter above Q3. That leaves about half between them, in the box.

    3. Not this one. The box is only from Q1 to Q3. A quarter of the values lie below Q1 and a quarter above Q3. Go to M4.7.4. Try another.

    See the answer

    About half

    A quarter of the values lie below Q1 and a quarter above Q3. That leaves about half between them, in the box.

    • If you chose A: A quarter of the values lie in each whisker, and in each half of the box. The two halves of the box add up to more. Go to M4.7.4.
    • If you chose C: The box is only from Q1 to Q3. A quarter of the values lie below Q1 and a quarter above Q3. Go to M4.7.4.

  6. The upper whisker of a box plot is much longer than the lower whisker. What does this tell us?

    Tap your answer.

    1. Not this one. Each whisker holds about a quarter of the values, whatever its length. Go to M4.7.4. Try another.

    2. Yes, that is right. The same quarter of the values lies along a longer stretch of the scale, so those values are further apart.

    3. Not this one. Half the values lie above the median and half below, whatever the whiskers look like. A long whisker shows spread, not more values. Go to M4.7.4. Try another.

    See the answer

    The top quarter of the values is more spread out than the bottom quarter.

    The same quarter of the values lies along a longer stretch of the scale, so those values are further apart.

    • If you chose A: Each whisker holds about a quarter of the values, whatever its length. Go to M4.7.4.
    • If you chose C: Half the values lie above the median and half below, whatever the whiskers look like. A long whisker shows spread, not more values. Go to M4.7.4.

  7. From an earlier chapter

    A cumulative frequency graph shows 120 values. At what cumulative frequency do you read the lower quartile?

    See the answer

    30

    The lower quartile is a quarter of the way up: 120 ÷ 4 = 30.

  8. Find the interquartile range of these 11 values: 4, 6, 7, 10, 12, 13, 15, 18, 19, 22, 25.

    See the answer

    12

    The median is the 6th value, 13. The lower half is 4, 6, 7, 10, 12, so Q1 = 7. The upper half is 15, 18, 19, 22, 25, so Q3 = 19. IQR = 19 − 7 = 12.

Keep it

Copy this into your exercise book.

Box plot: smallest, Q1, median, Q3, largest on one scale. The box is about half the data. Compare the medians for the middle, the IQRs for the spread.

Draw: Two box plots on one scale, one above the other, with the five values marked under the first.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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