Steply

Box plots

Statistics · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Put them in order
  2. Find the median
  3. Find Q1 and Q3
  4. Five-number summary
  5. Draw the box plot
  6. Check

The steps for Box plot of 3, 7, 8, 5, 12, 14, 21, 13, 18. Every solution names its steps, so you always know what comes next.

Worked examples

Box plot of 3, 7, 8, 5, 12, 14, 21, 13, 18 Easy

The question

Question: draw a box plot of 3, 7, 8, 5, 12, 14, 21, 13 and 18.

3, 7, 8, 5, 12, 14, 21, 13, 18

A box plot shows how the numbers spread out.

The box holds the middle half.

box plot: the middle half is the box

1 Put them in order

First we put them in order, smallest first: 3, 5, 7, 8, 12, 13, 14, 18 and 21.

= 3, 5, 7, 8, 12, 13, 14, 18, 21

2 Find the median

9 numbers: 4 on each side of the middle one.

The median is 12.

3 Find Q1 and Q3

The median cuts the list in two halves, 4 numbers each.

The median itself is in neither half.

Careful!

The median is not counted in the halves.

Each half has 4 numbers.

Careful: the median is in neither half

The lower half has two in the middle: 5 and 7.

Q1 is halfway between them: we add them and divide by 2.

5 plus 7 is 12.

Divided by 2 is 6.

Q1 = 5 + 72 = 122 = 6

The upper half has two in the middle: 14 and 18.

Q3 is halfway between them: we add them and divide by 2.

14 plus 18 is 32.

Divided by 2 is 16.

Q3 = 14 + 182 = 322 = 16

4 Five-number summary

The five-number summary: the smallest 3, Q1 6, the median 12, Q3 16, and the biggest 21.

= min 3, Q1 6, median 12, Q3 16, max 21

5 Draw the box plot

A number line from 2 to 22, counting in 2s.

The box goes from Q1, 6, to Q3, 16.

A line across the box at the median, 12.

The whiskers reach out to the smallest, 3, and the biggest, 21.

6 Check

Let's check: the median has 4 numbers below it and 4 above it.

Each half of the plot holds the same count.

So it is right!

below the median: 4, above: 4

Let's look back.

First: put them in order.

Then: find the median.

Then: find Q1 and Q3.

Then: five-number summary.

Then: draw the box plot.

Last: check.

Answer
= min 3, Q1 6, median 12, Q3 16, max 21

▶ Watch it solved

Box plot of 21, 35, 28, 40, 33, 26, 38, 30, 45, 24 Medium

The question

Question: draw a box plot of 21, 35, 28, 40, 33, 26, 38, 30, 45 and 24.

21, 35, 28, 40, 33, 26, 38, 30, 45, 24

A box plot shows how the numbers spread out.

The box holds the middle half.

box plot: the middle half is the box

1 Put them in order

First we put them in order, smallest first: 21, 24, 26, 28, 30, 33, 35, 38, 40 and 45.

= 21, 24, 26, 28, 30, 33, 35, 38, 40, 45

2 Find the median

10 numbers: two in the middle, 30 and 33.

The median goes between them.

It cuts the list in two halves, 5 numbers each.

The median is halfway between them: we add them and divide by 2.

30 plus 33 is 63.

Divided by 2 is 31.5.

median = 30 + 332 = 632 = 31.5

3 Find Q1 and Q3

Q1 is the middle of the lower half: 26.

Q3 is the middle of the upper half: 38.

4 Five-number summary

The five-number summary: the smallest 21, Q1 26, the median 31.5, Q3 38, and the biggest 45.

= min 21, Q1 26, median 31.5, Q3 38, max 45

5 Draw the box plot

A number line from 20 to 46, counting in 2s.

The box goes from Q1, 26, to Q3, 38.

A line across the box at the median, 31.5.

The whiskers reach out to the smallest, 21, and the biggest, 45.

6 Check

Let's check: the median has 5 numbers below it and 5 above it.

Each half of the plot holds the same count.

So it is right!

below the median: 5, above: 5

Let's look back.

First: put them in order.

Then: find the median.

Then: find Q1 and Q3.

Then: five-number summary.

Then: draw the box plot.

Last: check.

Answer
= min 21, Q1 26, median 31.5, Q3 38, max 45

▶ Watch it solved

Box plot of 12, 45, 23, 38, 17, 29, 51, 34, 26, 40, 19 Harder

The question

Question: draw a box plot of 12, 45, 23, 38, 17, 29, 51, 34, 26, 40 and 19.

12, 45, 23, 38, 17, 29, 51, 34, 26, 40, 19

A box plot shows how the numbers spread out.

The box holds the middle half.

box plot: the middle half is the box

1 Put them in order

First we put them in order, smallest first: 12, 17, 19, 23, 26, 29, 34, 38, 40, 45 and 51.

= 12, 17, 19, 23, 26, 29, 34, 38, 40, 45, 51

2 Find the median

11 numbers: 5 on each side of the middle one.

The median is 29.

3 Find Q1 and Q3

The median cuts the list in two halves, 5 numbers each.

The median itself is in neither half.

Careful!

The median is not counted in the halves.

Each half has 5 numbers.

Careful: the median is in neither half

Q1 is the middle of the lower half: 19.

Q3 is the middle of the upper half: 40.

4 Five-number summary

The five-number summary: the smallest 12, Q1 19, the median 29, Q3 40, and the biggest 51.

= min 12, Q1 19, median 29, Q3 40, max 51

5 Draw the box plot

A number line from 10 to 55, counting in 5s.

The box goes from Q1, 19, to Q3, 40.

A line across the box at the median, 29.

The whiskers reach out to the smallest, 12, and the biggest, 51.

6 Check

Let's check: the median has 5 numbers below it and 5 above it.

Each half of the plot holds the same count.

So it is right!

below the median: 5, above: 5

Let's look back.

First: put them in order.

Then: find the median.

Then: find Q1 and Q3.

Then: five-number summary.

Then: draw the box plot.

Last: check.

Answer
= min 12, Q1 19, median 29, Q3 40, max 51

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. box plot of 3, 7, 8, 5, 12, 14, 21, 13, 18
    Answer
    = min 3, Q1 6, median 12, Q3 16, max 21
  2. box plot of 4, 9, 6, 2, 11, 7, 8, 5
    Answer
    = min 2, Q1 4.5, median 6.5, Q3 8.5, max 11
  3. box plot of 1, 3, 5, 7, 9, 11, 13
    Answer
    = min 1, Q1 3, median 7, Q3 11, max 13
  4. box plot of 10, 12, 15, 18, 20, 22, 25
    Answer
    = min 10, Q1 12, median 18, Q3 22, max 25
  5. box plot of 5, 8, 6, 9, 7, 10, 4
    Answer
    = min 4, Q1 5, median 7, Q3 9, max 10
  6. box plot of 20, 30, 25, 35, 40, 45, 50, 28
    Answer
    = min 20, Q1 26.5, median 32.5, Q3 42.5, max 50

Medium

  1. box plot of 21, 35, 28, 40, 33, 26, 38, 30, 45, 24
    Answer
    = min 21, Q1 26, median 31.5, Q3 38, max 45
  2. box plot of 6, 2, 9, 4, 12, 7, 15, 3, 10, 8, 5
    Answer
    = min 2, Q1 4, median 7, Q3 10, max 15
  3. box plot of 55, 61, 48, 70, 66, 52, 59, 63, 74
    Answer
    = min 48, Q1 53.5, median 61, Q3 68, max 74
  4. box plot of 15, 22, 18, 30, 25, 20, 27, 17, 35, 23, 19, 28
    Answer
    = min 15, Q1 18.5, median 22.5, Q3 27.5, max 35
  5. box plot of 0, 2, 5, 3, 8, 6, 4
    Answer
    = min 0, Q1 2, median 4, Q3 6, max 8
  6. box plot of 31, 27, 40, 22, 35, 29, 44, 38, 25, 33, 36
    Answer
    = min 22, Q1 27, median 33, Q3 38, max 44
  7. box plot of 9, 14, 11, 7, 16, 12, 10, 18, 13, 8
    Answer
    = min 7, Q1 9, median 11.5, Q3 14, max 18
  8. box plot of 62, 58, 71, 65, 54, 68, 75, 60
    Answer
    = min 54, Q1 59, median 63.5, Q3 69.5, max 75

Harder

  1. box plot of 12, 45, 23, 38, 17, 29, 51, 34, 26, 40, 19
    Answer
    = min 12, Q1 19, median 29, Q3 40, max 51
  2. box plot of 2, 4, 4, 5, 7, 8, 8, 9, 12, 15
    Answer
    = min 2, Q1 4, median 7.5, Q3 9, max 15
  3. five-number summary of 81, 75, 92, 68, 88, 79, 95, 84, 72, 90
    Answer
    = min 68, Q1 75, median 82.5, Q3 90, max 95
  4. box plot of 1, 1, 2, 3, 3, 4, 5, 6, 6, 8, 9, 12
    Answer
    = min 1, Q1 2.5, median 4.5, Q3 7, max 12
  5. box plot of 105, 98, 112, 120, 95, 108, 115, 101, 110
    Answer
    = min 95, Q1 99.5, median 108, Q3 113.5, max 120
  6. five-number summary of 3, 9, 4, 7, 15, 6, 8, 5, 11
    Answer
    = min 3, Q1 4.5, median 7, Q3 10, max 15