Box Plots

What you'll learn

How to find the median and quartiles from an ordered list of data.
How to calculate the interquartile range and use it to test for outliers.
How to draw a box plot, including separate outlier crosses.
How to compare two distributions using medians, spreads and context.

Why box plots are useful

A box plot is a compact diagram for showing the spread of a data set. It does not show every individual value, but it quickly shows the “middle” of the data, the spread of the central values, and any unusual values.

Definition

Box plot

A box plot represents a data set using its minimum, lower quartile, median, upper quartile and maximum. If there are outliers, the whiskers stop at the smallest and largest non-outlier values, and the outliers are shown separately, usually using crosses.

This labelled diagram shows the main parts of a box plot.

A labelled box plot showing the minimum, quartiles, median, maximum, whiskers and outliers on a number line.

Annotated horizontal box plot showing quartiles, whiskers, and outliers

Key Idea

The big picture

A box plot is built from five important positions on a number line: the smallest value, $Q_1$ , the median, $Q_3$ , and the largest value — adjusted if there are outliers.

Ordering data, medians and quartiles

Before drawing a box plot, the data must be in ascending order, meaning smallest to largest.

Definition

Median and quartiles

The median is the middle value of an ordered data set. The lower quartile, written $Q_1$ , is the median of the lower half of the data. The upper quartile, written $Q_3$ , is the median of the upper half of the data.

For an odd number of data values, the median is the middle value and is not included in either half when finding $Q_1$ and $Q_3$ .

The 11 ordered values split into a lower half, the median, and an upper half, with quartile positions marked.

Example

Finding the median and quartiles

The weekly call times, in minutes, for 11 students are:

18, 22, 34, 37, 50, 52, 53, 56, 61, 78, 112

Find the median and quartiles.

Check the data are already in ascending order.
There are 11 values, so the median is the 6th value.
$median=52\text{median}=52$
The lower half is the five values below the median:
$5018,\ 22,\ 34,\ 37,\ 50$
The median of the lower half is the 3rd value, so:
$Q_1=34$
The upper half is the five values above the median:
$11253,\ 56,\ 61,\ 78,\ 112$
The median of the upper half is the 3rd value, so:
$Q_3=61$

Common Mistake

Including the median twice

When there is an odd number of values, do not include the overall median when finding the lower and upper quartiles.

Interquartile range

The full range uses the smallest and largest values, so it can be affected a lot by one extreme value. Box plots often focus on the middle half of the data instead.

Definition

Interquartile range

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

IQR=Q3−Q1\text{IQR}=Q_3-Q_1

Example

Finding the IQR

For the data from the previous example, $Q_1=34$ and $Q_3=61$ . Find the interquartile range.

Write down the formula.
$IQR=Q3−Q1\text{IQR}=Q_3-Q_1$
Substitute the quartiles.
$IQR=61−34\text{IQR}=61-34$
Calculate the answer.
$IQR=27\text{IQR}=27$

Tip

What the IQR tells you

A smaller IQR means the middle 50% of the data are more tightly clustered. A larger IQR means the middle 50% are more spread out.

Outliers

An outlier is a value that is unusually small or unusually large compared with the rest of the data.

Definition

Outlier rule

At AS Level, a common rule is:

fence=Q1−1.5×IQR\text{lower fence}=Q_1-1.5\times \text{IQR}

fence=Q3+1.5×IQR\text{upper fence}=Q_3+1.5\times \text{IQR}

A value is an outlier if it is smaller than the lower fence or greater than the upper fence.

Notice the words “smaller than” and “greater than”. A value exactly on a fence is not counted as an outlier using this rule.

A number line illustrating lower and upper outlier fences and which regions count as outliers.

Example

Showing that a value is an outlier

Using the call-time data:

18, 22, 34, 37, 50, 52, 53, 56, 61, 78, 112

Show whether there are any outliers.

From earlier, the quartiles are $Q_1=34$ and $Q_3=61$ .
Calculate the interquartile range.
$IQR=61−34=27\text{IQR}=61-34=27$
Find 1.5 times the IQR.
$1.5×27=40.51.5\times 27=40.5$
Find the lower fence.
$Q1−1.5×IQR=34−40.5=−6.5Q_1-1.5\times \text{IQR}=34-40.5=-6.5$
Find the upper fence.
$Q3+1.5×IQR=61+40.5=101.5Q_3+1.5\times \text{IQR}=61+40.5=101.5$
Compare the data values with the fences: no value is below -6.5, but 112 is greater than 101.5.
Therefore, 112 is the only outlier.

The call-time data placed against the outlier fences, showing 112 beyond the upper fence.

Common Mistake

Using the overall range instead of the IQR

The outlier test uses $Q_3-Q_1$ , not maximum minus minimum. Always find the IQR first.

Drawing a box plot

To draw a box plot, you need:

the lower quartile, $Q_1$
the median
the upper quartile, $Q_3$
the smallest non-outlier
the largest non-outlier
any outliers, marked separately

The box goes from $Q_1$ to $Q_3$ . The median is drawn as a vertical line inside the box. The whiskers go to the smallest and largest non-outliers.

Example

Drawing a box plot with an outlier

Draw a box plot for the data:

18, 22, 34, 37, 50, 52, 53, 56, 61, 78, 112

Use the quartiles already found.
$Q1=34,median=52,Q3=61Q_1=34,\quad \text{median}=52,\quad Q_3=61$
Use the outlier test already completed. The value 112 is an outlier.
The smallest value is 18, and it is not an outlier, so the lower whisker starts at 18.
The largest non-outlier is 78, so the upper whisker ends at 78.
Draw a horizontal number line with a suitable scale, for example from 0 to 120.
Draw the box from 34 to 61, and put a vertical median line at 52.
Draw whiskers from 34 down to 18 and from 61 up to 78.
Mark the outlier 112 with a cross, not as the end of the whisker.

The completed box plot for the call-time data, with the outlier shown separately from the whisker.

Tip

Choosing a scale

Pick a scale that includes all important values, including outliers. Make sure the spacing is consistent: if one square represents 10 minutes, every square must represent 10 minutes.

Common Mistake

Whiskering to an outlier

If a value is an outlier, do not use it as the end of the whisker. Mark it separately with a cross.

Reading a box plot

You may be asked to read values directly from a box plot.

The phrase “the time by which 75% had finished” means the upper quartile, $Q_3$ . This is because 75% of the data are at or below $Q_3$ .

Example

Interpreting a box plot in context

A box plot shows times taken to complete a puzzle. The lower quartile is 10 minutes, the median is 13 minutes, the upper quartile is 18 minutes, and crosses are plotted at 28 and 29 minutes.

To find the time by which 75% of children had completed the puzzle, identify $Q_3$ .
The upper quartile is 18 minutes.
Therefore, 75% of the children had completed the puzzle by 18 minutes.
The crosses at 28 and 29 minutes represent outliers.
In context, these are children who took unusually long times to complete the puzzle.

Key Idea

Percentages in a box plot

Roughly 25% of the data lie in each section: below $Q_1$ , between $Q_1$ and the median, between the median and $Q_3$ , and above $Q_3$ .

A box plot divided into four sections, each representing about 25% of the data.

Comparing two box plots

When comparing distributions, do not just list values. You should compare them in context.

Useful comparisons include:

Average location: compare the medians.
Spread of the middle 50%: compare the IQRs.
Overall spread: compare the ranges, but be careful with outliers.
Outliers: mention unusually high or low values and what they mean in context.

Example

Comparing two distributions

Two groups of students completed the same puzzle.

Two box plots on the same scale for comparing medians, IQRs, ranges and outliers between Group A and Group B.

Group A has median 13 minutes, $Q_1=10$ , $Q_3=18$ , smallest non-outlier 6, largest non-outlier 24, and outliers at 28 and 29 minutes.

Group B has median 15 minutes, $Q_1=12$ , $Q_3=17$ , minimum 7 and maximum 22, with no outliers.

Compare the two distributions.

Compare the medians. Group A has the lower median, 13 minutes compared with 15 minutes, so a typical student in Group A completed the puzzle faster.
Compare the IQRs.
$A=18−10=8\text{IQR for Group A}=18-10=8$ $B=17−12=5\text{IQR for Group B}=17-12=5$
Group B has the smaller IQR, so the middle 50% of Group B’s times were more consistent.
Compare the overall spread. Ignoring outliers, Group A’s non-outlier range is from 6 to 24, while Group B’s range is from 7 to 22.
Mention the outliers in context. Group A had two unusually slow times, 28 and 29 minutes, while Group B had no outliers.

Exam technique

In the exam

Put raw data in order first, then find the median, $Q_1$ and $Q_3$ carefully.
For outliers, write the IQR and both fences clearly before making your conclusion.
When comparing box plots, write sentences in context: compare medians for typical value, IQRs for consistency, and mention outliers.

Self review

Check yourself

If a data set has 15 values, which position is the median, and which values form the lower half?
What are the two outlier fences when $Q_1=20$ and $Q_3=44$ ?
In a box plot, why might the largest data value not be the end of the upper whisker?

Box Plots

What you'll learn

How to find the median and quartiles from an ordered list of data.
How to calculate the interquartile range and use it to test for outliers.
How to draw a box plot, including separate outlier crosses.
How to compare two distributions using medians, spreads and context.

Why box plots are useful

Definition

Box plot

This labelled diagram shows the main parts of a box plot.

A labelled box plot showing the minimum, quartiles, median, maximum, whiskers and outliers on a number line.

Annotated horizontal box plot showing quartiles, whiskers, and outliers

Key Idea

The big picture

A box plot is built from five important positions on a number line: the smallest value, $Q_1$ , the median, $Q_3$ , and the largest value — adjusted if there are outliers.

Ordering data, medians and quartiles

Before drawing a box plot, the data must be in ascending order, meaning smallest to largest.

Definition

Median and quartiles

For an odd number of data values, the median is the middle value and is not included in either half when finding $Q_1$ and $Q_3$ .

The 11 ordered values split into a lower half, the median, and an upper half, with quartile positions marked.

Example

Finding the median and quartiles

The weekly call times, in minutes, for 11 students are:

18, 22, 34, 37, 50, 52, 53, 56, 61, 78, 112

Find the median and quartiles.

Check the data are already in ascending order.
There are 11 values, so the median is the 6th value.
$median=52\text{median}=52$
The lower half is the five values below the median:
$5018,\ 22,\ 34,\ 37,\ 50$
The median of the lower half is the 3rd value, so:
$Q_1=34$
The upper half is the five values above the median:
$11253,\ 56,\ 61,\ 78,\ 112$
The median of the upper half is the 3rd value, so:
$Q_3=61$

Common Mistake

Including the median twice

When there is an odd number of values, do not include the overall median when finding the lower and upper quartiles.

Interquartile range

The full range uses the smallest and largest values, so it can be affected a lot by one extreme value. Box plots often focus on the middle half of the data instead.

Definition

Interquartile range

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

IQR=Q3−Q1\text{IQR}=Q_3-Q_1

Example

Finding the IQR

For the data from the previous example, $Q_1=34$ and $Q_3=61$ . Find the interquartile range.

Write down the formula.
$IQR=Q3−Q1\text{IQR}=Q_3-Q_1$
Substitute the quartiles.
$IQR=61−34\text{IQR}=61-34$
Calculate the answer.
$IQR=27\text{IQR}=27$

Tip

What the IQR tells you

A smaller IQR means the middle 50% of the data are more tightly clustered. A larger IQR means the middle 50% are more spread out.

Outliers

An outlier is a value that is unusually small or unusually large compared with the rest of the data.

Definition

Outlier rule

At AS Level, a common rule is:

fence=Q1−1.5×IQR\text{lower fence}=Q_1-1.5\times \text{IQR}

fence=Q3+1.5×IQR\text{upper fence}=Q_3+1.5\times \text{IQR}

A value is an outlier if it is smaller than the lower fence or greater than the upper fence.

Notice the words “smaller than” and “greater than”. A value exactly on a fence is not counted as an outlier using this rule.

A number line illustrating lower and upper outlier fences and which regions count as outliers.

Example

Showing that a value is an outlier

Using the call-time data:

18, 22, 34, 37, 50, 52, 53, 56, 61, 78, 112

Show whether there are any outliers.

From earlier, the quartiles are $Q_1=34$ and $Q_3=61$ .
Calculate the interquartile range.
$IQR=61−34=27\text{IQR}=61-34=27$
Find 1.5 times the IQR.
$1.5×27=40.51.5\times 27=40.5$
Find the lower fence.
$Q1−1.5×IQR=34−40.5=−6.5Q_1-1.5\times \text{IQR}=34-40.5=-6.5$
Find the upper fence.
$Q3+1.5×IQR=61+40.5=101.5Q_3+1.5\times \text{IQR}=61+40.5=101.5$
Compare the data values with the fences: no value is below -6.5, but 112 is greater than 101.5.
Therefore, 112 is the only outlier.

The call-time data placed against the outlier fences, showing 112 beyond the upper fence.

Common Mistake

Using the overall range instead of the IQR

The outlier test uses $Q_3-Q_1$ , not maximum minus minimum. Always find the IQR first.

Drawing a box plot

To draw a box plot, you need:

the lower quartile, $Q_1$
the median
the upper quartile, $Q_3$
the smallest non-outlier
the largest non-outlier
any outliers, marked separately

The box goes from $Q_1$ to $Q_3$ . The median is drawn as a vertical line inside the box. The whiskers go to the smallest and largest non-outliers.

Example

Drawing a box plot with an outlier

Draw a box plot for the data:

18, 22, 34, 37, 50, 52, 53, 56, 61, 78, 112

Use the quartiles already found.
$Q1=34,median=52,Q3=61Q_1=34,\quad \text{median}=52,\quad Q_3=61$
Use the outlier test already completed. The value 112 is an outlier.
The smallest value is 18, and it is not an outlier, so the lower whisker starts at 18.
The largest non-outlier is 78, so the upper whisker ends at 78.
Draw a horizontal number line with a suitable scale, for example from 0 to 120.
Draw the box from 34 to 61, and put a vertical median line at 52.
Draw whiskers from 34 down to 18 and from 61 up to 78.
Mark the outlier 112 with a cross, not as the end of the whisker.

The completed box plot for the call-time data, with the outlier shown separately from the whisker.

Tip

Choosing a scale

Pick a scale that includes all important values, including outliers. Make sure the spacing is consistent: if one square represents 10 minutes, every square must represent 10 minutes.

Common Mistake

Whiskering to an outlier

If a value is an outlier, do not use it as the end of the whisker. Mark it separately with a cross.

Reading a box plot

You may be asked to read values directly from a box plot.

The phrase “the time by which 75% had finished” means the upper quartile, $Q_3$ . This is because 75% of the data are at or below $Q_3$ .

Example

Interpreting a box plot in context

A box plot shows times taken to complete a puzzle. The lower quartile is 10 minutes, the median is 13 minutes, the upper quartile is 18 minutes, and crosses are plotted at 28 and 29 minutes.

To find the time by which 75% of children had completed the puzzle, identify $Q_3$ .
The upper quartile is 18 minutes.
Therefore, 75% of the children had completed the puzzle by 18 minutes.
The crosses at 28 and 29 minutes represent outliers.
In context, these are children who took unusually long times to complete the puzzle.

Key Idea

Percentages in a box plot

Roughly 25% of the data lie in each section: below $Q_1$ , between $Q_1$ and the median, between the median and $Q_3$ , and above $Q_3$ .

A box plot divided into four sections, each representing about 25% of the data.

Comparing two box plots

When comparing distributions, do not just list values. You should compare them in context.

Useful comparisons include:

Average location: compare the medians.
Spread of the middle 50%: compare the IQRs.
Overall spread: compare the ranges, but be careful with outliers.
Outliers: mention unusually high or low values and what they mean in context.

Example

Comparing two distributions

Two groups of students completed the same puzzle.

Two box plots on the same scale for comparing medians, IQRs, ranges and outliers between Group A and Group B.

Group A has median 13 minutes, $Q_1=10$ , $Q_3=18$ , smallest non-outlier 6, largest non-outlier 24, and outliers at 28 and 29 minutes.

Group B has median 15 minutes, $Q_1=12$ , $Q_3=17$ , minimum 7 and maximum 22, with no outliers.

Compare the two distributions.

Compare the medians. Group A has the lower median, 13 minutes compared with 15 minutes, so a typical student in Group A completed the puzzle faster.
Compare the IQRs.
$A=18−10=8\text{IQR for Group A}=18-10=8$ $B=17−12=5\text{IQR for Group B}=17-12=5$
Group B has the smaller IQR, so the middle 50% of Group B’s times were more consistent.
Compare the overall spread. Ignoring outliers, Group A’s non-outlier range is from 6 to 24, while Group B’s range is from 7 to 22.
Mention the outliers in context. Group A had two unusually slow times, 28 and 29 minutes, while Group B had no outliers.

Exam technique

In the exam

Put raw data in order first, then find the median, $Q_1$ and $Q_3$ carefully.
For outliers, write the IQR and both fences clearly before making your conclusion.
When comparing box plots, write sentences in context: compare medians for typical value, IQRs for consistency, and mention outliers.

Self review

Check yourself

If a data set has 15 values, which position is the median, and which values form the lower half?
What are the two outlier fences when $Q_1=20$ and $Q_3=44$ ?
In a box plot, why might the largest data value not be the end of the upper whisker?

Box Plots

What you'll learn

Why box plots are useful

Ordering data, medians and quartiles

Interquartile range

Outliers

Drawing a box plot

Reading a box plot

Comparing two box plots

Recap questions

You've reached the end

Box Plots

What you'll learn

Why box plots are useful

Ordering data, medians and quartiles

Interquartile range

Outliers

Drawing a box plot

Reading a box plot

Comparing two box plots

Recap questions

You've reached the end

Box Plots

What you'll learn

Why box plots are useful

Ordering data, medians and quartiles

Interquartile range

Outliers

Drawing a box plot

Reading a box plot

Comparing two box plots

Recap questions

You've reached the end

Box Plots Revision Guide

Box Plots

What you'll learn

Why box plots are useful

Ordering data, medians and quartiles

Interquartile range

Outliers

Drawing a box plot

Reading a box plot

Comparing two box plots

Recap questions

You've reached the end

Box Plots Revision Guide