Finding Outliers

Date: 09/28/2000 at 23:25:07
From: Caitlin 
Subject: 1.5xIQR

How does the 1.5*IQR criterion work? How is it used to find outliers 
in statistics?

Thank you.

Date: 10/02/2000 at 12:53:21
From: Doctor TWE
Subject: Re: 1.5xIQR

Hi Caitlin - thanks for writing to Dr. Math.

In statistics, an outlier is an observation (data point) that is 
unusually large or small relative to the other values in the data set. 
For example, suppose we record the height of all the 7th graders. 
Suppose all of them are in the range of 48 inches to 66 inches except 
one, Mikey, who is 80 inches tall. Mikey's height would be an outlier. 
(It would also be an outlier if Mikey were, say, 32 inches tall.)

Outliers can be the result of an error in measurement of the value 
(perhaps Mikey was measured in centimeters instead of inches), a value 
from a different population (perhaps Mikey is really a 12th grader), 
or simply a rare chance event (perhaps Mikey really is a 6'8" 7th 
grader).

To determine that a value is an outlier for a given population, we use 
the 1.5*IQR criterion.

Suppose we want to find any outliers in the following set of 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 must find the upper and lower quartiles. 
They 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.                            U.Q.

Next, we compute the inner quartile range (IQR). The IQR = UQ - LQ. So 
in our example IQR = 85 - 77 = 8. Suspect outliers are any data points 
that are 1.5*IQR below the L.Q. or 1.5*IQR above the U.Q. For our 
example:

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

so suspect outliers would be values below 65 or above 97. Sometimes we 
also use the criterion 3*IQR below the L.Q. or 3*IQR above the U.Q. to 
determine "highly suspect" outliers. For our example:

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

so values below 53 or above 109 are highly suspect outliers. In our 
example, we have two suspect outliers: the 60 and the 100. We also 
have one highly suspect outlier: the 50.
 
A nice feature of this criterion is that the computations are 
relatively simple. Here, we never had to do anything more than adding, 
subtracting, and multiplying 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' 
disagreements about them, see

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

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