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Box and Whisker Plots

```
Date: 02/13/2000 at 21:39:15
From: Ramiro Martinez
Subject: Box and Whisker Plots

I don't understand box and whisker plots. All I know is that a box and
whisker plot is used to display data. I can't find information on this
anywhere else.

Sincerely,
Ramiro Martinez
```

```
Date: 02/14/2000 at 12:03:38
From: Doctor TWE
Subject: Re: Box and Whisker Plots

Hi Ramiro - thanks for writing to Dr. Math.

A box-and-whisker plot (often simply called a box plot) is a graphical
way of showing data. It is useful for quickly finding outliers - data
points out of line with the rest of the data set.

Suppose we want to construct a box plot of the following test scores:

50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95, 100

If they're not already in numerical order, it's best to arrange them
in ascending order.

First, we need to construct the "box." To do so, we must find the
upper and lower quartiles and the median. The median is the number in
the middle of our set (when arranged in numerical order). The upper
and lower quartiles are the values 1/4 of the way from the top or
bottom of our set. In our example:

50, 60, 73, 77, 80, 81, 82, 83, 84, 84, 84, 85, 88, 95, 100
^               ^               ^
L.Q.           Median           U.Q.

To draw the box, we'll put a scale on the x-axis and draw a box from
the lower quartile to the upper quartile. We'll add a vertical line to
mark the median, like so:

LQ     M UQ
+-------+
|     | |
+-------+
^.........^.........^.........^.........^.........^.........^
50       60        70        80        90        100       110

where LQ = Lower Quartile, M = Median, UQ = Upper Quartile.

Now we add "fences." First, we compute the inner quartile range (IQR).
The IQR = UQ - LQ. So in our example IQR = 85 - 77 = 8. The inner
fences are 1.5*IQR below the L.Q. and 1.5*IQR above the U.Q. For our
example, the inner fences are at:

77 - 1.5*8 = 77 - 12 = 65
and at   85 + 1.5*8 = 85 + 12 = 97

We'll mark these with a dotted line (I'll use colons ":"). Sometimes
the fences are not drawn on the box plot, but we'll put them in so we
can see where they are:

LIF         LQ     M UQ         UIF
:           +-------+           :
:           |     | |           :
:           +-------+           :
^.........^.........^.........^.........^.........^.........^
50       60        70        80        90        100       110

where LIF = Lower Inner Fence, UIF = Upper Inner Fence.

There is also a set of outer fences. These are 3*IQR below the L.Q.
and 3*IQR above the U.Q. For our example, the outer fences are at:

77 - 3*8 = 77 - 24 = 53
and at   85 + 3*8 = 85 + 24 = 109

We'll mark these with another dotted line. These are always twice as
far out as the inner fences. Here's what we have so far:

LOF         LIF         LQ     M UQ         UIF         UOF
:           :           +-------+           :           :
:           :           |     | |           :           :
:           :           +-------+           :           :
^.........^.........^.........^.........^.........^.........^
50       60        70        80        90        100       110

where LOF = Lower Outer Fence, UOF = Upper Outer Fence.

Now we add the "whiskers." Find the first value above (to the right
of) the Lower Inner Fence. Mark it with an X and draw a line
connecting it to the box. Similarly, find the first value below (to
the left of) the Upper Inner Fence. Mark it with an X and draw a line
connecting it to the box as well. In our example, the end values for
our whiskers are at 73 (the first value above 65) and 95 (the first
value below 97.) Our plot now looks like this:

LOF         LIF         LQ     M UQ         UIF         UOF
:           :           +-------+           :           :
:           :       X---|     | |---------X :           :
:           :           +-------+           :           :
^.........^.........^.........^.........^.........^.........^
50       60        70        80        90        100       110

Finally, we have to mark the outliers. Values between the inner and
outer fences are called "suspect outliers." We mark them with an
asterisk "*".

Values outside the outer fences are called "highly suspect outliers."
We mark them with an "o". In our example, we have two suspect
outliers: the 60 and the 100. We also have one highly suspect outlier:
the 50. Once we mark these on our plot, we're finished:

LOF         LIF         LQ     M UQ         UIF         UOF
:           :           +-------+           :           :
o  :      *    :       X---|     | |---------X :  *        :
:           :           +-------+           :           :
^.........^.........^.........^.........^.........^.........^
50       60        70        80        90        100       110

We could "erase" the fences and labels, but I'd probably leave them in
so that the person looking at the graph can see where they are. If we
erase them, we'll have:

+-------+
o         *            X---|     | |---------X    *
+-------+
^.........^.........^.........^.........^.........^.........^
50       60        70        80        90        100       110

As you can see, this plot quickly gives an idea of what our data look
like. Half the numbers are between 77 and 85, the middle of the data
set is at 83, the "reasonable" range of the data goes from 73 to 95,
and we have three suspect data values at 50, 60, and 100.

A nice feature of this kind of plot is that all the computations are
subtract, and multiply by 1.5 and 3.

I hope this helps! If you have any more questions, write back.

- Doctor TWE, The Math Forum
http://mathforum.org/dr.math/
```

```
For more on the meanings of "quartile" and mathematicians'

Defining Quartiles
http://mathforum.org/library/drmath/view/60969.html

- Doctor Melissa, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Statistics
High School Statistics
Middle School Statistics

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