# Box-and-Whisker Plot

How to make a box-and-whisker plot (2 versions: from minimum to maximum, excluding outliers): 2 examples and their solutions.

## Example 1

### Example

### Solution

A box-and-whisker plot shows

how the values of a data is distributed.

To draw a box-and-whisker plot,

first find the quartiles (Q_{1}, Q_{2}, Q_{3}).

First find Q_{2}.

Q_{2} is the median of the data:

[6 + 7]/2 = 6.5.

Next, see the values

on the left side of Q_{2}.

Q_{1} is the median of these values:

this 5.

Next, see the values

on the right side of Q_{2}.

Q_{3} is the median of these values:

this 7.

So Q_{1} = 5, Q_{2} = 6.5, Q_{3} = 7.

It says

the plot type is

from the minimum to the maximum.

The minimum value is 1.

The maximum value is 9.

Use these values

to draw a box-and-whisker plot.

The values are from 1 to 9.

So draw the horizontal axis

that covers from 0 to 10.

Point (min), Q_{1}, Q_{2}, Q_{3}, and (max).

(min) = 1

Q_{1} = 5

Q_{2} = 6.5

Q_{3} = 7

(max) = 9

Draw the box

that passes through Q_{1}, Q_{2}, and Q_{3}.

Draw a whisker between

(min) = 1 and the left side of the box.

Draw a whisker between

the right side of the box and (max) = 9.

So this is the box-and-whisker plot

of the given data.

## Example 2

### Example

### Solution

To draw a box-and-whisker plot,

first find the quartiles (Q_{1}, Q_{2}, Q_{3}).

First find Q_{2}.

Q_{2} is the median of the data:

[6 + 7]/2 = 6.5.

Next, see the values

on the left side of Q_{2}.

Q_{1} is the median of these values:

this 5.

Next, see the values

on the right side of Q_{2}.

Q_{3} is the median of these values:

this 7.

So Q_{1} = 5, Q_{2} = 6.5, Q_{3} = 7.

It says

the plot type is

excluding outliers.

Then, find the IQR:

the interquartile range.

(IQR) = Q_{3} - Q_{1}

Q_{3} = 7

Q_{1} = 5

So (IQR) = 7 - 5 = 2.

Find 1.5(IQR).

(IQR) = 2

So 1.5(IQR) = 1.5⋅2 = 3.

Let's draw the box-and-whisker plot.

The values are from 1 to 9.

So draw the horizontal axis

that covers from 0 to 10.

Point the quartiles Q_{1}, Q_{2}, and Q_{3}.

Q_{1} = 5

Q_{2} = 6.5

Q_{3} = 7

Draw the box

that passes through Q_{1}, Q_{2}, and Q_{3}.

Recall that

(IQR) = Q_{3} - Q_{1} = 2.

So the width of the box,

Q_{3} - Q_{1},

is the (IQR): 2.

Draw the regions of 1.5(IQR)

on the left side and right side of the box.

1.5(IQR) = 3

So the left 1.5(IQR) region covers

2 ~ 5.

(5 - 3 = 2)

This 2 is the lower fence.

And the right 1.5(IQR) region covers

7 ~ 10.

(7 + 3 = 10)

This 10 is the upper fence.

See the given data.

Find the value(s)

that is (are) less than the lower fence 2.

The value 1 is less than the lower fence.

Then the value 1

is not in the left 1.5(IQR) region.

Point this value 1

on the plot.

Just like this value 1,

if a value is not in the 1.5(IQR) region,

then that value is the outlier.

So the value 1 is the outlier.

There's an outlier in the left side.

Then find the least value

that is in the left 1.5(IQR) region:

2.

So point the value 2

on the plot.

And draw a whisker

between the value 2

and the left side of the box.

Next, see the given data.

Find the value(s)

that is (are) greater than the upper fence 10.

There's no value

that is greater than the upper fence 10.

Then there's no outlier in the right side.

Then point the greatest value:

9.

And draw a whisker

between the right side of the box

and the value 9.

So this is the box-and-whisker plot

of the given data.