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Outliers and Modified Boxplots

Outliers and Modified Boxplots

Author: Sophia Tutorial
Description:

Calculate the value of an outlier by using the 1.5XIQR rule.

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Tutorial
what's covered
This tutorial will cover the topic of outliers and modified boxplots. Our discussion breaks down as follows:

  1. Outliers
  2. The 1.5xIQR Rule
  3. Boxplots


1. Outliers

You may recall that outliers are values that are far outside the pattern established by the rest of the data. They're either very high or very low in comparison to the rest of the data set.

Boxplots, introduced in another tutorial, are a way to graphically display the five number summary for a data set. This tutorial will present a modified version of boxplots so that it is easier to observe outliers in them.

EXAMPLE

Here is a set of test scores.

90, 98, 89, 88, 46, 90, 91, 84, 94

Almost everyone scored in the 80's or 90's, except for one student, who scored a 46. That student is an outlier.
terms to know
Outlier
A point that is so large or small as to be unusual, given the rest of the data points.
Modified Boxplot
A graphical display showing a modified version of the five number summary. If a distribution has outliers, then the "whiskers" only extend to the highest and lowest points that are not outliers.


2. The 1.5xIQR Rule

To make it easier to find outliers, there is a mathematical rule for determining whether a point is an outlier or not. This is called the “1.5xIQR rule.” IQR stands for Interquartile Range.

So, how do you use the 1.5xIQR method?

step by step

Step 1: Find the quartiles of the data set.
Step 2: Find the interquartile range (IQR).
Step 3: If we have a point that is 1.5 IQR's below the first quartile or 1.5 IQR's or more above the third quartile, then it is an outlier.

EXAMPLE

Consider the data set of test scores from above.

90 98 89 88 46 90 91 84 94


Step 1: First, find the quartiles of the data set. To do this, order the data from least to greatest. find the median, and find the medians within each of the low and high data sets.

46 84 88 89 90 90 91 94 98

Q1=86

Median

Q3=92.5


The median of this data set is 90. The median of the first quartile is actually between 84 and 88, or at 86, and the median of the third quartile is between 91 and 94, which is at 92.5.

Step 2: Next, find the interquartile range, or IQR. The interquartile range is the distance between the first and third quartiles. The difference between 92.5 and 86 is 6.5.

table attributes columnalign left end attributes row cell Q subscript 1 equals 86 end cell row cell Q subscript 2 equals 92.5 end cell row cell I Q R equals space 92.5 space minus space 86 space equals 6.5 end cell end table

Step 3: Any point that is 1.5 IQR's below the first quartile or 1.5 IQR's or more above the third quartile will be considered an outlier.

1.5 IQR below the first quartile:
Q 1 minus 1.5 I Q R equals 86 minus open parentheses 1.5 close parentheses open parentheses 6.5 close parentheses equals 86 minus 9.75 equals 76.25

1.5 IQR above the third quartile:
Q 3 plus 1.5 I Q R equals 92.5 plus open parentheses 1.5 close parentheses open parentheses 6.5 close parentheses equals 92.5 plus 9.75 equals 102.25

This indicates that any test score higher than 102.25 points would be considered an outlier on the high side. Anything below 76.25 will be considered an outlier on the low side.

Of the test scores, only 46 falls outside this range, so this test score would be an outlier.

try it
Suppose that you have the data set of all house prices for homes purchased in Albuquerque, New Mexico, from February to April in 1993. These are in thousands of dollars.

The first and third quartiles have been calculated:

Home Prices in Albuquerque, New Mexico
From February - April, 1993
205 72 93.9 99.5 87.5 105
208 72 82 97.5 88.9 104.5
215 74.9 78 97.5 85.5 105
215 73.1 77 90 83.5 102
199.9 72.5 70 96 81 100
190 67 62 86 80.5 103
180 215 54 169.5 79.9 97.5
156 159.9 107 155.3 75 95
145 135 210 125 75.9 94
144.9 129.9 72.5 130 75.5 92
137.5 125 66 102 75 94.5
127 123.9 60 102 73 87.4
125 120 58 92.2 72.9 87.2
123.5 112.5 184.4 92.5 71 87
117 110 158 89.9 77.3 86.9
118 108 69.9 85 69 76.6
115.5 105 133 87.6 67 73.9
111 104.9 116 89 61.9
113.9 95.5 110.9 87 129.5
99.5 93.4 112.9 70 97.5
Q1 = 78, Q3 = 120


What is the range for outliers? What is the lower fence for outliers and the upper fence for outliers?
IQR
I Q R equals Q 1 minus Q 3 equals 120 minus 78 equals 42
1.5 IQR below the first quartile:
Q 1 minus 1.5 I Q R equals 78 minus open parentheses 1.5 close parentheses open parentheses 42 close parentheses equals 78 minus 63 equals 15
1.5 IQR above the third quartile:
Q 3 plus 1.5 I Q R equals 120 plus open parentheses 1.5 close parentheses open parentheses 42 close parentheses equals 120 plus 63 equals 183
The interquartile range is 42, so any point below 78 minus 1.5xIQR, which is 15, or above 120 plus 1.5xIQR, which is 183, will be an outlier.
Notice that there's nothing in the list below 15, but there are seven above 183. This means that there are seven outliers in this data set, which, by the way, is a completely legitimate and legal occurrence in this situation.
term to know

1.5xIQR Rule
If a point is larger than Q3 + 1.5xIQR, or smaller than Q1 - 1.5xIQR, then it is an outlier.

3. Boxplots

You can use this new information to create a new version of an already existing plot that you have. You’ve made boxplots in another tutorial; now you can modify them to show outliers.

Generally, you would make the whiskers on the box-and-whisker plot extend all the way out to the maximum and minimum. If the minimum or maximum (or both) are outliers, that will make the whiskers really long. For a modified boxplot, instead of going all the way out to those outliers, you can extend them only to the highest and lowest values that aren't outliers and notate the outliers separately.

EXAMPLE

Refer back to the student data set from the section above. Here are the values from least to greatest.

46 84 88 89 90 90 91 94 98

Q1=86

Q3=92.5


Mark the same values that you would have if you were making a regular box-and-whisker plot. However, don’t go all the way down to 46 for your minimum--even though 46 is the actual minimum. 46 is an outlier, so instead go to the next lowest number that isn't an outlier---84--and make your line there. Then you can make your box and whiskers.

File:4260-outlier2.png

You still have to show the 46 as part of this data set somehow, so you will mark it with a dot. This is a modified boxplot.

File:4261-outlier3.png

In the home value data set, there were seven high outliers. This is a modified plot for that data set:

File:4262-outlier4.png


summary
You can determine in some measurable way if a point within a data set is an outlier using the 1.5xIQR rule. Data sets might have no outliers, or they might have one or more outliers on the low side, one or more outliers on the high side, or both. There's no rule for how many outliers are allowed in a data set. Whatever outliers exist, you can use a modified boxplot to visually display them.

Good luck!

Source: Adapted from Sophia tutorial by Jonathan Osters.

Terms to Know
1.5xIQR Rule

If a point is larger than Q3 + 1.5xIQR, or smaller than Q1 - 1.5xIQR, then it is an outlier.

Modified Boxplot

A graphical display showing a modified version of the five number summary. If a distribution has outliers, then the "whiskers" only extend to the highest and lowest points that are not outliers.

Outlier

A point that is so large or small as to be unusual, given the rest of the data points.