Box Plots

What you'll learn

How to understand the five key values used in a box plot.
How to sketch and label a box plot from a table or from raw data.
How to compare two box plots using medians and spread.
How to create a box plot from a cumulative frequency graph.

What a box plot shows

A distribution means the way a set of data values is spread out. A box plot is a compact diagram that shows the main features of a distribution on a number line.

Definition

The five-number summary

A box plot uses five values:

Minimum: the smallest value.
Lower quartile, written $Q_1$ : about 25% of the data is below this value.
Median, written $Q_2$ : the middle value.
Upper quartile, written $Q_3$ : about 75% of the data is below this value.
Maximum: the largest value.

The whiskers are the lines from the box to the minimum and maximum.

A generic box plot showing the five-number summary and the whiskers.

Key Idea

Box plots summarise, not detail

A box plot does not show every data value. It shows the centre and spread of the data using the five-number summary.

Example

Reading values from a box plot

A box plot has whiskers at 8 and 52, a box from 18 to 39, and a vertical line inside the box at 27.

Box plot with minimum 8, lower quartile 18, median 27, upper quartile 39, and maximum 52.

The whisker ends give the minimum and maximum, so the minimum is 8 and the maximum is 52.
The left edge of the box is the lower quartile, so $Q_1$ is 18.
The vertical line inside the box is the median, so the median is 27.
The right edge of the box is the upper quartile, so $Q_3$ is 39.
The range is 52 - 8 = 44, and the interquartile range is 39 - 18 = 21.

Range and interquartile range

The range measures the full spread of the data.

The interquartile range, often called the IQR, measures the spread of the middle 50% of the data.

Definition

Spread formulas

The range is found using $range=maximum−minimum\text{range}=\text{maximum}-\text{minimum}$ .
The interquartile range is found using $IQR=Q3−Q1\text{IQR}=Q_3-Q_1$ .

The IQR is often more useful than the range because it ignores the most extreme values.

Sketching a box plot from a table

When the five-number summary is given, your job is to place the five values accurately on a scale.

To draw a box plot:

Draw a horizontal number line with a sensible scale.
Mark the minimum, $Q_1$ , median, $Q_3$ , and maximum.
Draw a box from $Q_1$ to $Q_3$ .
Draw a vertical line inside the box at the median.
Draw whiskers from the box to the minimum and maximum.

Example

Sketching from five values

A set of plant heights has minimum 10 cm, $Q_1$ 26 cm, median 34 cm, $Q_3$ 41 cm, and maximum 54 cm.

Sketch of the plant height box plot using the five given values on a 0 cm to 60 cm scale.

Draw a horizontal scale from 0 cm to 60 cm.
Mark the five values on the scale: 10, 26, 34, 41, and 54.
Draw the box from 26 cm to 41 cm.
Draw a vertical line inside the box at 34 cm.
Draw whiskers from 26 cm back to 10 cm, and from 41 cm up to 54 cm.

Tip

Keep the order sensible

The five values must go in order: minimum, $Q_1$ , median, $Q_3$ , maximum. If your values are not in that order, check your calculations.

When one value is missing

Sometimes a table gives the range or IQR instead of one of the five values.

Use the formula backwards.

Example

Finding missing values before drawing

A table for potato weights gives range 96 g, $Q_1$ 109 g, median 132 g, $Q_3$ 160 g, and maximum 183 g.

Potato weight box plot after using the range to find the missing minimum of 87 g.

Use the range to find the missing minimum: 183 - 96 = 87.
The five values are now minimum 87 g, $Q_1$ 109 g, median 132 g, $Q_3$ 160 g, and maximum 183 g.
Draw the box from 109 g to 160 g.
Draw the median line at 132 g.
Draw whiskers to 87 g and 183 g.

Example

Using the IQR to find a quartile

For some puzzle times, the IQR is 8 minutes, the minimum is 12 minutes, the median is 18 minutes, $Q_3$ is 23 minutes, and the maximum is 29 minutes.

Puzzle time box plot after using the IQR to find the lower quartile of 15 minutes.

Use $IQR=Q3−Q1\text{IQR}=Q_3-Q_1$ .
Since 23 - 15 = 8, the lower quartile is 15 minutes.
The five values are 12, 15, 18, 23, and 29 minutes.
Draw the box from 15 to 23, put the median line at 18, and draw whiskers to 12 and 29.

Common Mistake

Forgetting to find the missing value

Do not start drawing if the table gives range or IQR instead of one of the five values. Work out the missing value first.

Finding a box plot from raw data

Sometimes you are given a list of data values.

First, make sure the data is in order from smallest to largest. Then find the median and quartiles.

For an odd number of values:

The median is the middle value.
$Q_1$ is the median of the lower half.
$Q_3$ is the median of the upper half.
Do not include the main median in either half.

Example

Sketching from race times

The race times, in seconds, are already in order:

51, 53, 55, 55, 57, 58, 59, 61, 62, 63, 65, 66, 69, 71, 74

Ordered race times split into lower half, median, and upper half to identify the quartiles.

There are 15 values, so the median is the 8th value.
The 8th value is 61, so the median is 61 seconds.
The lower half is the first 7 values: 51, 53, 55, 55, 57, 58, 59. The middle of these is 55, so $Q_1$ is 55 seconds.
The upper half is the last 7 values: 62, 63, 65, 66, 69, 71, 74. The middle of these is 66, so $Q_3$ is 66 seconds.
The minimum is 51 seconds and the maximum is 74 seconds.
Draw the box plot using 51, 55, 61, 66, and 74.

Common Mistake

Including the median twice

When there is an odd number of data values, do not include the median when finding $Q_1$ and $Q_3$ .

Comparing two box plots

Comparison questions usually want two things:

A comparison of the medians, which tells you about the typical value.
A comparison of the spread, usually using the IQR or range.

Use the context. For test scores, a higher median is better. For race times or puzzle times, a lower median means faster.

Example

Comparing two classes

Class A has minimum 8, $Q_1$ 16, median 21, $Q_3$ 32, maximum 45.

Class B has minimum 12, $Q_1$ 18, median 24, $Q_3$ 30, maximum 43.

Two box plots on the same scale for comparing Class A and Class B by median and IQR.

Compare the medians: Class B has median 24, while Class A has median 21. So Class B generally scored higher.
Find the IQR for Class A: 32 - 16 = 16.
Find the IQR for Class B: 30 - 18 = 12.
Class B has the smaller IQR, so Class B’s scores were more consistent.
A good comparison is: Class B generally scored higher and had less spread in the middle 50% of scores.

Tip

Use the word ‘median’

Instead of saying “Class B did better”, say “Class B had a higher median score”. That is more precise and earns marks.

From a cumulative frequency graph to a box plot

A cumulative frequency graph shows the running total of values up to a certain point.

To make a box plot from one:

Find the total frequency, $N$ .
Use $N /4$ for $Q_1$ .
Use $N /2$ for the median.
Use $3 N /4$ for $Q_3$ .
Read across to the curve, then down to the horizontal axis.

Example

Using a cumulative frequency graph

A cumulative frequency graph shows the weights of 60 oranges. The minimum weight is 114 g and the maximum weight is 150 g. From the graph, the readings are about 123 g at cumulative frequency 15, 130 g at cumulative frequency 30, and 137 g at cumulative frequency 45.

Cumulative frequency readings for the orange weights, followed by the corresponding box plot values.

Let $N = 60$ . The quartile positions are:
$N4=15,N2=30,3N4=45\frac{N}{4}=15,\quad \frac{N}{2}=30,\quad \frac{3N}{4}=45$
Read from cumulative frequency 15 to the curve, then down to the weight axis. This gives $Q_1$ as about 123 g.
Read from cumulative frequency 30. This gives the median as about 130 g.
Read from cumulative frequency 45. This gives $Q_3$ as about 137 g.
Use the five values 114, 123, 130, 137, and 150 to sketch the box plot.

Common Mistake

Using the wrong axis

For quartiles on a cumulative frequency graph, start on the cumulative frequency axis, not the weight or time axis.

Exam technique

In the exam

Write the five values above your scale before drawing the box plot.
In comparison questions, make one point about the median and one point about spread.
For cumulative frequency graphs, use $N /4$ , $N /2$ , and $3 N /4$ , then read across to the curve and down.

Self review

Check yourself

Can you find the five-number summary from an ordered list of 11 values?
If the IQR is 8 and $Q_3$ is 23, can you find $Q_1$ ?
When comparing two box plots, which two features should you mention?

Box Plots

What you'll learn

How to understand the five key values used in a box plot.
How to sketch and label a box plot from a table or from raw data.
How to compare two box plots using medians and spread.
How to create a box plot from a cumulative frequency graph.

What a box plot shows

A distribution means the way a set of data values is spread out. A box plot is a compact diagram that shows the main features of a distribution on a number line.

Definition

The five-number summary

A box plot uses five values:

Minimum: the smallest value.
Lower quartile, written $Q_1$ : about 25% of the data is below this value.
Median, written $Q_2$ : the middle value.
Upper quartile, written $Q_3$ : about 75% of the data is below this value.
Maximum: the largest value.

The whiskers are the lines from the box to the minimum and maximum.

A generic box plot showing the five-number summary and the whiskers.

Key Idea

Box plots summarise, not detail

A box plot does not show every data value. It shows the centre and spread of the data using the five-number summary.

Example

Reading values from a box plot

A box plot has whiskers at 8 and 52, a box from 18 to 39, and a vertical line inside the box at 27.

Box plot with minimum 8, lower quartile 18, median 27, upper quartile 39, and maximum 52.

The whisker ends give the minimum and maximum, so the minimum is 8 and the maximum is 52.
The left edge of the box is the lower quartile, so $Q_1$ is 18.
The vertical line inside the box is the median, so the median is 27.
The right edge of the box is the upper quartile, so $Q_3$ is 39.
The range is 52 - 8 = 44, and the interquartile range is 39 - 18 = 21.

Range and interquartile range

The range measures the full spread of the data.

The interquartile range, often called the IQR, measures the spread of the middle 50% of the data.

Definition

Spread formulas

The range is found using $range=maximum−minimum\text{range}=\text{maximum}-\text{minimum}$ .
The interquartile range is found using $IQR=Q3−Q1\text{IQR}=Q_3-Q_1$ .

The IQR is often more useful than the range because it ignores the most extreme values.

Sketching a box plot from a table

When the five-number summary is given, your job is to place the five values accurately on a scale.

To draw a box plot:

Draw a horizontal number line with a sensible scale.
Mark the minimum, $Q_1$ , median, $Q_3$ , and maximum.
Draw a box from $Q_1$ to $Q_3$ .
Draw a vertical line inside the box at the median.
Draw whiskers from the box to the minimum and maximum.

Example

Sketching from five values

A set of plant heights has minimum 10 cm, $Q_1$ 26 cm, median 34 cm, $Q_3$ 41 cm, and maximum 54 cm.

Sketch of the plant height box plot using the five given values on a 0 cm to 60 cm scale.

Draw a horizontal scale from 0 cm to 60 cm.
Mark the five values on the scale: 10, 26, 34, 41, and 54.
Draw the box from 26 cm to 41 cm.
Draw a vertical line inside the box at 34 cm.
Draw whiskers from 26 cm back to 10 cm, and from 41 cm up to 54 cm.

Tip

Keep the order sensible

The five values must go in order: minimum, $Q_1$ , median, $Q_3$ , maximum. If your values are not in that order, check your calculations.

When one value is missing

Sometimes a table gives the range or IQR instead of one of the five values.

Use the formula backwards.

Example

Finding missing values before drawing

A table for potato weights gives range 96 g, $Q_1$ 109 g, median 132 g, $Q_3$ 160 g, and maximum 183 g.

Potato weight box plot after using the range to find the missing minimum of 87 g.

Use the range to find the missing minimum: 183 - 96 = 87.
The five values are now minimum 87 g, $Q_1$ 109 g, median 132 g, $Q_3$ 160 g, and maximum 183 g.
Draw the box from 109 g to 160 g.
Draw the median line at 132 g.
Draw whiskers to 87 g and 183 g.

Example

Using the IQR to find a quartile

For some puzzle times, the IQR is 8 minutes, the minimum is 12 minutes, the median is 18 minutes, $Q_3$ is 23 minutes, and the maximum is 29 minutes.

Puzzle time box plot after using the IQR to find the lower quartile of 15 minutes.

Use $IQR=Q3−Q1\text{IQR}=Q_3-Q_1$ .
Since 23 - 15 = 8, the lower quartile is 15 minutes.
The five values are 12, 15, 18, 23, and 29 minutes.
Draw the box from 15 to 23, put the median line at 18, and draw whiskers to 12 and 29.

Common Mistake

Forgetting to find the missing value

Do not start drawing if the table gives range or IQR instead of one of the five values. Work out the missing value first.

Finding a box plot from raw data

Sometimes you are given a list of data values.

First, make sure the data is in order from smallest to largest. Then find the median and quartiles.

For an odd number of values:

The median is the middle value.
$Q_1$ is the median of the lower half.
$Q_3$ is the median of the upper half.
Do not include the main median in either half.

Example

Sketching from race times

The race times, in seconds, are already in order:

51, 53, 55, 55, 57, 58, 59, 61, 62, 63, 65, 66, 69, 71, 74

Ordered race times split into lower half, median, and upper half to identify the quartiles.

There are 15 values, so the median is the 8th value.
The 8th value is 61, so the median is 61 seconds.
The lower half is the first 7 values: 51, 53, 55, 55, 57, 58, 59. The middle of these is 55, so $Q_1$ is 55 seconds.
The upper half is the last 7 values: 62, 63, 65, 66, 69, 71, 74. The middle of these is 66, so $Q_3$ is 66 seconds.
The minimum is 51 seconds and the maximum is 74 seconds.
Draw the box plot using 51, 55, 61, 66, and 74.

Common Mistake

Including the median twice

When there is an odd number of data values, do not include the median when finding $Q_1$ and $Q_3$ .

Comparing two box plots

Comparison questions usually want two things:

A comparison of the medians, which tells you about the typical value.
A comparison of the spread, usually using the IQR or range.

Use the context. For test scores, a higher median is better. For race times or puzzle times, a lower median means faster.

Example

Comparing two classes

Class A has minimum 8, $Q_1$ 16, median 21, $Q_3$ 32, maximum 45.

Class B has minimum 12, $Q_1$ 18, median 24, $Q_3$ 30, maximum 43.

Two box plots on the same scale for comparing Class A and Class B by median and IQR.

Compare the medians: Class B has median 24, while Class A has median 21. So Class B generally scored higher.
Find the IQR for Class A: 32 - 16 = 16.
Find the IQR for Class B: 30 - 18 = 12.
Class B has the smaller IQR, so Class B’s scores were more consistent.
A good comparison is: Class B generally scored higher and had less spread in the middle 50% of scores.

Tip

Use the word ‘median’

Instead of saying “Class B did better”, say “Class B had a higher median score”. That is more precise and earns marks.

From a cumulative frequency graph to a box plot

A cumulative frequency graph shows the running total of values up to a certain point.

To make a box plot from one:

Find the total frequency, $N$ .
Use $N /4$ for $Q_1$ .
Use $N /2$ for the median.
Use $3 N /4$ for $Q_3$ .
Read across to the curve, then down to the horizontal axis.

Example

Using a cumulative frequency graph

Cumulative frequency readings for the orange weights, followed by the corresponding box plot values.

Let $N = 60$ . The quartile positions are:
$N4=15,N2=30,3N4=45\frac{N}{4}=15,\quad \frac{N}{2}=30,\quad \frac{3N}{4}=45$
Read from cumulative frequency 15 to the curve, then down to the weight axis. This gives $Q_1$ as about 123 g.
Read from cumulative frequency 30. This gives the median as about 130 g.
Read from cumulative frequency 45. This gives $Q_3$ as about 137 g.
Use the five values 114, 123, 130, 137, and 150 to sketch the box plot.

Common Mistake

Using the wrong axis

For quartiles on a cumulative frequency graph, start on the cumulative frequency axis, not the weight or time axis.

Exam technique

In the exam

Write the five values above your scale before drawing the box plot.
In comparison questions, make one point about the median and one point about spread.
For cumulative frequency graphs, use $N /4$ , $N /2$ , and $3 N /4$ , then read across to the curve and down.

Self review

Check yourself

Can you find the five-number summary from an ordered list of 11 values?
If the IQR is 8 and $Q_3$ is 23, can you find $Q_1$ ?
When comparing two box plots, which two features should you mention?

Box Plots

What you'll learn

What a box plot shows

Range and interquartile range

Sketching a box plot from a table

When one value is missing

Finding a box plot from raw data

Comparing two box plots

From a cumulative frequency graph to a box plot

Recap questions

You've reached the end

Box Plots

What you'll learn

What a box plot shows

Range and interquartile range

Sketching a box plot from a table

When one value is missing

Finding a box plot from raw data

Comparing two box plots

From a cumulative frequency graph to a box plot

Recap questions

You've reached the end

Box Plots

What you'll learn

What a box plot shows

Range and interquartile range

Sketching a box plot from a table

When one value is missing

Finding a box plot from raw data

Comparing two box plots

From a cumulative frequency graph to a box plot

Recap questions

You've reached the end

Box Plots Revision Guide

Box Plots

What you'll learn

What a box plot shows

Range and interquartile range

Sketching a box plot from a table

When one value is missing

Finding a box plot from raw data

Comparing two box plots

From a cumulative frequency graph to a box plot

Recap questions

You've reached the end

Box Plots Revision Guide