Question

predicting the effects of an outlier
if a data set has only one outlier, which value will always change when the outlier is excluded?
the median
the range
the interquartile range
the upper quartile

Explanation:

• Median: If the data set has an odd number of values, excluding the outlier may or may not change the median. For example, in a data set \( \{1,2,3,4,100\} \) (where \( 100 \) is the outlier), the median is \( 3 \). After excluding \( 100 \), the data set is \( \{1,2,3,4\} \) and the median is \( \frac{2 + 3}{2}=2.5 \). But if the data set is \( \{1,100,3\} \), the median is \( 3 \), and after excluding \( 100 \), the median is \( 1 \). However, if the data set size is large (e.g., \( n = 101 \) and the outlier is at one end), removing it may not change the median (e.g., if the median was the \( 51^{st} \) value in a sorted data - set of \( 101 \) values, and the outlier was the \( 101^{st} \) value).
• Range: The range is defined as \( R=\text{Max}-\text{Min} \). An outlier is either the maximum or the minimum value of the data set. If we exclude the outlier, then either the new maximum or the new minimum (or both) will change. For example, if the data set is \( \{1,2,3,4,100\} \), \( R = 100 - 1=99 \). After excluding \( 100 \), \( R=4 - 1 = 3 \). If the data set is \( \{ - 100,1,2,3,4\} \), \( R=4-(-100)=104 \), and after excluding \(-100\), \( R = 4 - 1=3 \).
• Inter - quartile range: The inter - quartile range \( IQR=Q_3 - Q_1 \). If the outlier is an extreme value (either very large or very small), and the data set is relatively large (e.g., \( n\geq5 \)), removing the outlier may not affect \( Q_1 \) and \( Q_3 \). For example, consider the data set \( \{1,2,3,4,100\} \). First, we find the quartiles. For \( n = 5 \), \( Q_1\) is the \( 2^{nd} \) value and \( Q_3 \) is the \( 4^{th} \) value. \( Q_1 = 2\), \( Q_3=4\), \( IQR = 2 \). After removing \( 100 \), \( Q_1 = 2\), \( Q_3=3\), \( IQR=1 \). But if the data set is \( \{1,2,3,4,5,6,7,8,9,100\} \), \( n = 10 \). \( Q_1=\frac{3 + 4}{2}=3.5\), \( Q_3=\frac{8+9}{2}=8.5\), \( IQR = 5 \). After removing \( 100 \), \( Q_1=\frac{3 + 4}{2}=3.5\), \( Q_3=\frac{8 + 9}{2}=8.5\), \( IQR = 5 \).
• Upper quartile (\( Q_3 \)): If the outlier is the maximum value and \( n\) is odd (e.g., \( n = 5\): \( \{1,2,3,4,100\} \), \( Q_3 = 4 \). After removing \( 100\), \( n = 4\), \( Q_3=\frac{3+4}{2}=3.5 \)). But if \( n\) is large (e.g., \( n = 101\) and the outlier is the \( 101^{st} \) value in a sorted data - set), \( Q_3\) (the \( 76^{th} \) value) may not change when the \( 101^{st} \) value is removed.

Answer:

the range