# Box-and-Whisker Plot

How to draw a box-and-whisker plot from the given data: examples and their solution. (Two plot types: from minimum to maximum, excluding outliers)

## Example 1: from Minimum to Maximum

Find the quartiles of the given data.*Q*_{1} = 5*Q*_{2} = 6.5*Q*_{3} = 7

Quartiles

Draw three points at the quartiles:

5, 6.5, 7.

Draw a box

that passes through *Q*_{1} = 5 and *Q*_{3} = 7.

And draw a vertical line

that passes through *Q*_{2} = 6.5.

The minimum value is 1.

And the maximum value is 9.

Draw a point at the minimum value 1.

Draw a line that connects

the minimum value 1 and *Q*_{1} = 5.

Draw a point at the maximum value 9.

Draw a line that connects*Q*_{3} = 7 and the maximum value 9.

These two lines are the whiskers.

So this box-and-whisker plot is the answer.

The box shows the quartiles:*Q*_{1} = 5, *Q*_{2} = 6.5, *Q*_{3} = 7.

And the whisker shows

the minimum and the maximum values:

1, 9.

So, by reading the box-and-whisker plot,

you can easily know

how the data looks like.

## Example 2: Excluding Outliers

Find the quartiles of the given data.*Q*_{1} = 5*Q*_{2} = 6.5*Q*_{3} = 7

Quartiles

Before drawing the box,

find the IQR.

The IQR is the interquartile range.

It means the width of the box.

So (IQR) = *Q*_{3} - *Q*_{1}

= 7 - 5

= 2.

Then find 1.5⋅(IQR).

(IQR) = 2

So 1.5⋅(IQR) = 1.5⋅2

= 3.

Draw three points at the quartiles:

5, 6.5, 7.

Draw a box

that passes through *Q*_{1} = 5 and *Q*_{3} = 7.

And draw a vertical line

that passes through *Q*_{2} = 6.5.

Draw the IQR above the box.

(green double arrow)

Its width is the IQR: 2.

Draw the lower fence

at the left side of the IQR.

(purple double arrow)

Draw the upper fence

at the right side of the IQR.

(purple double arrow)

The widths of these two fences are 1.5⋅(IQR): 3.

Let's see the left half.

See if there's any number

that is not covered by the lower fence.

The width of the lower fence is 3.

So the lower fence covers

from 2 (= 5 - 3) to 5.

1 is not covered by the lower fence.

So 1 is the outlier.

So draw a point on 1.

Draw a point

at the end of the lower fence: 2.

And draw a line that connects

the endpoint of the lower fence 2 and *Q*_{1} = 5.

Let's see the right half.

See if there's any number

that is not covered by the upper fence.

The width of the upper fence is 3.

So the upper fence covers

from 7 to 10 (= 7 + 3).

The maximum value 9

is covered by the upper fence.

This means

the right half is covered by the right fence.

So there's no outlier at the right side.

So draw a point on 9.

And draw a line that connects*Q*_{3} = 7 and the maximum point 9.

So this box-and-whisker plot is the answer.

The box shows the quartiles:*Q*_{1} = 5, *Q*_{2} = 6.5, *Q*_{3} = 7.

The whisker shows

the farthest values in the fences:

2, 9.

And you can see the outlier:

1.

If you compare this plot

to the previous example's plot,

you can see that the outlier is excluded

from the whisker.

This is why this plot type is 'excluding outliers'.