# 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.

Q1 = 5
Q2 = 6.5
Q3 = 7

Quartiles

Draw three points at the quartiles:
5, 6.5, 7.

Draw a box
that passes through Q1 = 5 and Q3 = 7.

And draw a vertical line
that passes through Q2 = 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 Q1 = 5.

Draw a point at the maximum value 9.
Draw a line that connects
Q3 = 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:
Q1 = 5, Q2 = 6.5, Q3 = 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.

Q1 = 5
Q2 = 6.5
Q3 = 7

Quartiles

Before drawing the box,
find the IQR.

The IQR is the interquartile range.
It means the width of the box.

So (IQR) = Q3 - Q1
= 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 Q1 = 5 and Q3 = 7.

And draw a vertical line
that passes through Q2 = 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 Q1 = 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
Q3 = 7 and the maximum point 9.

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

The box shows the quartiles:
Q1 = 5, Q2 = 6.5, Q3 = 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'.