Lesson
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Notes
\(\text{A point is an Outlier if}\)
\(\text{it is below } Q1-1.5IQR\)
\(\text{ or it is above } Q3+1.5IQR\)
\(\text{Inner Quartile Range}\)
\(IQR=Q3-Q1\)
Practice Problems
Do the following data sets contain an outlier?
\(\textbf{1)}\) \( \{1,2,3,4,5,6,7,14\} \) \(14 \text{ is an outlier.}\)
\(\,\,\,\,\,\,Q1 = 2.5\)
\(\,\,\,\,\,\,Q3 = 6.5\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 6.5-2.5 = 4\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 4 = 6\)
\(\,\,\,\,\,\,Q1-1.5IQR=2.5-6=-3.5\)
\(\,\,\,\,\,\,Q3+1.5IQR=6.5+6=12.5\)
\(\,\,\,\,\,\,\text{Are any points outside of } -3.5 \text{ and } 12.5\)?
\(\,\,\,\,\,\,14 \gt 12.5\)
\(\,\,\,\,\,\,14 \text{ is an outlier.}\)
\(\,\,\,\,\,\,Q1 = 2.5\)
\(\,\,\,\,\,\,Q3 = 6.5\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 6.5-2.5 = 4\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 4 = 6\)
\(\,\,\,\,\,\,Q1-1.5IQR=2.5-6=-3.5\)
\(\,\,\,\,\,\,Q3+1.5IQR=6.5+6=12.5\)
\(\,\,\,\,\,\,\text{Are any points outside of } -3.5 \text{ and } 12.5\)?
\(\,\,\,\,\,\,14 \gt 12.5\)
\(\,\,\,\,\,\,14 \text{ is an outlier.}\)
\(\textbf{2)}\) \( \{7,3,9,17,4,6,2\} \) \(\text{No, there is not an outlier.}\)
\(\,\,\,\,\,\,Q1 = 3\)
\(\,\,\,\,\,\,Q3 = 9\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 9-3 = 6\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 6 = 9\)
\(\,\,\,\,\,\,Q1-1.5IQR=3-9=-6\)
\(\,\,\,\,\,\,Q3+1.5IQR=9+9=18\)
\(\,\,\,\,\,\,\text{All data points are in between } -6 \text{ and } 18\).
\(\,\,\,\,\,\,\text{There is not an outlier.}\)
\(\,\,\,\,\,\,Q1 = 3\)
\(\,\,\,\,\,\,Q3 = 9\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 9-3 = 6\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 6 = 9\)
\(\,\,\,\,\,\,Q1-1.5IQR=3-9=-6\)
\(\,\,\,\,\,\,Q3+1.5IQR=9+9=18\)
\(\,\,\,\,\,\,\text{All data points are in between } -6 \text{ and } 18\).
\(\,\,\,\,\,\,\text{There is not an outlier.}\)
\(\textbf{3)}\) \( \{1,2,3,4,5,6\} \) \(\text{No, there is not an outlier.}\)
\(\,\,\,\,\,\,Q1 = 2\)
\(\,\,\,\,\,\,Q3 = 5\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 5-2 = 3\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 3 = 4.5\)
\(\,\,\,\,\,\,Q1-1.5IQR=2-4.5=-2.5\)
\(\,\,\,\,\,\,Q3+1.5IQR=5+4.5=9.5\)
\(\,\,\,\,\,\,\text{All data points are in between } -2.5 \text{ and } 9.5\).
\(\,\,\,\,\,\,\text{There is not an outlier.}\)
\(\,\,\,\,\,\,Q1 = 2\)
\(\,\,\,\,\,\,Q3 = 5\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 5-2 = 3\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 3 = 4.5\)
\(\,\,\,\,\,\,Q1-1.5IQR=2-4.5=-2.5\)
\(\,\,\,\,\,\,Q3+1.5IQR=5+4.5=9.5\)
\(\,\,\,\,\,\,\text{All data points are in between } -2.5 \text{ and } 9.5\).
\(\,\,\,\,\,\,\text{There is not an outlier.}\)
\(\textbf{4)}\) \( \{4,2,6,7,1,16,8,8\} \) \(16 \text{ is an outlier.}\)
\(\,\,\,\,\,\,Q1 = 3\)
\(\,\,\,\,\,\,Q3 = 8\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 8-3 = 5\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 5 = 7.5\)
\(\,\,\,\,\,\,Q1-1.5IQR=3-7.5=-4.5\)
\(\,\,\,\,\,\,Q3+1.5IQR=8+7.5=15.5\)
\(\,\,\,\,\,\,\text{Are any points outside of } -4.5 \text{ and } 15.5\)?
\(\,\,\,\,\,\,16 \gt 15.5\)
\(\,\,\,\,\,\,16 \text{ is an outlier.}\)
\(\,\,\,\,\,\,Q1 = 3\)
\(\,\,\,\,\,\,Q3 = 8\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 8-3 = 5\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 5 = 7.5\)
\(\,\,\,\,\,\,Q1-1.5IQR=3-7.5=-4.5\)
\(\,\,\,\,\,\,Q3+1.5IQR=8+7.5=15.5\)
\(\,\,\,\,\,\,\text{Are any points outside of } -4.5 \text{ and } 15.5\)?
\(\,\,\,\,\,\,16 \gt 15.5\)
\(\,\,\,\,\,\,16 \text{ is an outlier.}\)
\(\textbf{5)}\) \( \{5,5,6,2,5,4\} \) \(\text{No, there is not an outlier.}\)
\(\,\,\,\,\,\,Q1 = 4\)
\(\,\,\,\,\,\,Q3 = 5\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 5-4 = 1\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 1 = 1.5\)
\(\,\,\,\,\,\,Q1-1.5IQR=4-1.5=2.5\)
\(\,\,\,\,\,\,Q3+1.5IQR=5+1.5=6.5\)
\(\,\,\,\,\,\,\text{All data points are in between } 2.5 \text{ and } 6.5\).
\(\,\,\,\,\,\,\text{There is not an outlier.}\)
\(\,\,\,\,\,\,Q1 = 4\)
\(\,\,\,\,\,\,Q3 = 5\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 5-4 = 1\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 1 = 1.5\)
\(\,\,\,\,\,\,Q1-1.5IQR=4-1.5=2.5\)
\(\,\,\,\,\,\,Q3+1.5IQR=5+1.5=6.5\)
\(\,\,\,\,\,\,\text{All data points are in between } 2.5 \text{ and } 6.5\).
\(\,\,\,\,\,\,\text{There is not an outlier.}\)
\(\textbf{6)}\) \( \{1,2,3,4,5,6,7\} \) \(\text{No, there is not an outlier.}\)
\(\,\,\,\,\,\,Q1 = 2\)
\(\,\,\,\,\,\,Q3 = 6\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 6-2 = 4\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 4 = 6\)
\(\,\,\,\,\,\,Q1-1.5IQR=2-6=-4\)
\(\,\,\,\,\,\,Q3+1.5IQR=6+6=12\)
\(\,\,\,\,\,\,\text{All data points are in between } -4 \text{ and } 12\).
\(\,\,\,\,\,\,\text{There is not an outlier.}\)
\(\,\,\,\,\,\,Q1 = 2\)
\(\,\,\,\,\,\,Q3 = 6\)
\(\,\,\,\,\,\,IQR = Q3-Q1 = 6-2 = 4\)
\(\,\,\,\,\,\,1.5IQR = 1.5 \cdot 4 = 6\)
\(\,\,\,\,\,\,Q1-1.5IQR=2-6=-4\)
\(\,\,\,\,\,\,Q3+1.5IQR=6+6=12\)
\(\,\,\,\,\,\,\text{All data points are in between } -4 \text{ and } 12\).
\(\,\,\,\,\,\,\text{There is not an outlier.}\)
