### Chapter 1. Descriptive Statistics: Measures of Variability

### Interquartile Range Rule for Identifying Outliers

A common method for identifying outliers is the *Interquartile Range Rule*.

Interquartile Range Rule

According to the **Interquartile Range Rule**, a score #X# is considered an outlier if:

- The score lies more than #1.5\cdot IQR\,# below the first quartile: #X < (Q_1 - 1.5\cdot IQR)#
- The score lies more than #1.5\cdot IQR\,# above the third quartile: #X > (Q_3 + 1.5\cdot IQR)#

\[50,\,\,\,55,\,\,\,51,\,\,\,53,\,\,\,54,\,\,\,53,\,\,\,53,\,\,\,70,\,\,\,60,\,\,\,46,\,\,\,60,\,\,\,44,\,\,\,47\]

Based on the

*Interquartile Range Rule*, how many outliers are there in the sample?

To calculate the interquartile range, first sort the values in ascending order:

\[44,\,\,\,46,\,\,\,47,\,\,\,50,\,\,\,51,\,\,\,53,\,\,\,53,\,\,\,53,\,\,\,54,\,\,\,55,\,\,\,60,\,\,\,60,\,\,\,70\]

Next, calculate the first quartile. To find the index #i_1# of the first quartile (#Q=1#), use the following formula:

\[\begin{array}{rcl}

i_1 &=& \cfrac{Q}{4}(n-1)+1\\

&=& \cfrac{1}{4}(13 - 1) + 1=4

\end{array}\]

Since #i_1=4# is an integer, the first quartile is the score located at the #4^{th}# position of the ordered data:

\[X_{4} = 50\]

Next, calculate the third quartile. To find the index #i_3# of the third quartile (#Q=3#), use the following formula:

\[\begin{array}{rcl}

i_3 &=& \cfrac{Q}{4}(n-1)+1\\

&=& \cfrac{3}{4}(13 - 1) + 1=10

\end{array}\]

Since #i_3=10# is an integer, the third quartile is the score located at the #10^{th}# position of the ordered data:

\[X_{10} = 55\]

Calculate the *interquartile range*:

\[\text{IQR}=Q_3-Q_1=55-50=5\]

According to the *Interquartile Range Rule*, a score #X# is considered an outlier if:

- The score lies more than #1.5\cdot IQR\,# below the first quartile: #X < (Q_1 - 1.5\cdot IQR)#

\[Q_1 - 1.5\cdot IQR = 50 - 1.5 \cdot 5 = 42.5\] - The score lies more than #1.5\cdot IQR\,# above the third quartile: #X > (Q_3 + 1.5\cdot IQR)#

\[Q_3 + 1.5\cdot IQR = 55 + 1.5 \cdot 5 = 62.5\]

This means that any score #X<42.5# or #X>62.5# should be considered an outlier, of which there is #1# in the sample, namely: #70#.

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