1. Descriptive Statistics: Measures of Variability
Interquartile Range Rule for Identifying Outliers
A common method for identifying outliers is the Interquartile Range Rule.
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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)#
\[59\,\,\,\,\,\,81\,\,\,\,\,\,63\,\,\,\,\,\,72\,\,\,\,\,\,93\,\,\,\,\,\,52\,\,\,\,\,\,70\,\,\,\,\,\,71\,\,\,\,\,\,66\,\,\,\,\,\,68\,\,\,\,\,\,68\,\,\,\,\,\,72\,\,\,\,\,\,65\,\,\,\,\,\,\]
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:
\[52\,\,\,\,\,\,59\,\,\,\,\,\,63\,\,\,\,\,\,65\,\,\,\,\,\,66\,\,\,\,\,\,68\,\,\,\,\,\,68\,\,\,\,\,\,70\,\,\,\,\,\,71\,\,\,\,\,\,72\,\,\,\,\,\,72\,\,\,\,\,\,81\,\,\,\,\,\,93\,\,\,\,\,\,\]
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:
\[X_{4} = 65\]
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:
\[X_{10} = 72\]
Calculate the interquartile range:
\[\text{IQR}=Q_3-Q_1=72-65=7\]
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)#
\[Q_1 - 1.5\cdot IQR = 65 - 1.5 \cdot 7 = 54.5\] - The score lies more than #1.5\cdot IQR\,# above the third quartile: #X > (Q_3 + 1.5\cdot IQR)#
\[Q_3 + 1.5\cdot IQR = 72 + 1.5 \cdot 7 = 82.5\]
This means that any score #X<54.5# or #X>82.5# should be considered an outlier, of which there are #2# in the sample, namely: #52# and #93#.