Question

which data set has an outlier?
6, 13, 13, 15, 15, 18, 18, 22
4, 4, 4, 8, 9, 9, 11, 18
2, 3, 5, 7, 8, 8, 9, 10, 12, 17
3, 6, 7, 7, 8, 8, 9, 9, 9, 10

Explanation:

Step1: Find median, lower quartile ($Q_1$), upper quartile ($Q_3$) and inter - quartile range ($IQR = Q_3 - Q_1$) for each data set

For data set \(6,13,13,15,15,18,18,22\)

• Number of data points \(n = 8\)
• Median (\(Q_2\)): \(\frac{15 + 15}{2}=15\)
• \(Q_1=\frac{13 + 13}{2}=13\), \(Q_3=\frac{18+18}{2}=18\)
• \(IQR=18 - 13 = 5\)
• Lower fence \(=Q_1-1.5IQR=13-1.5\times5=13 - 7.5 = 5.5\)
• Upper fence \(=Q_3 + 1.5IQR=18+1.5\times5=18 + 7.5=25.5\)
• All data points \(6,13,13,15,15,18,18,22\) are within \([5.5,25.5]\)

For data set \(4,4,4,8,9,9,11,18\)

• \(n = 8\)
• Median (\(Q_2\)): \(\frac{8 + 9}{2}=8.5\)
• \(Q_1=\frac{4 + 4}{2}=4\), \(Q_3=\frac{9+11}{2}=10\)
• \(IQR=10 - 4=6\)
• Lower fence \(=Q_1-1.5IQR=4-1.5\times6=4 - 9=-5\)
• Upper fence \(=Q_3 + 1.5IQR=10+1.5\times6=10 + 9 = 19\)
• Data point \(18\) is within \([-5,19]\)

For data set \(2,3,5,7,8,8,9,10,12,17\)

• \(n = 10\)
• Median (\(Q_2\)): \(\frac{8 + 8}{2}=8\)
• \(Q_1=\frac{5+7}{2}=6\), \(Q_3=\frac{12 + 17}{2}=14.5\)
• \(IQR=14.5-6 = 8.5\)
• Lower fence \(=Q_1-1.5IQR=6-1.5\times8.5=6-12.75=-6.75\)
• Upper fence \(=Q_3 + 1.5IQR=14.5+1.5\times8.5=14.5 + 12.75=27.25\)
• Data point \(2\) is within \([-6.75,27.25]\)

For data set \(3,6,7,7,8,8,9,9,9,10\)

• \(n = 10\)
• Median (\(Q_2\)): \(\frac{8 + 8}{2}=8\)
• \(Q_1=\frac{7+7}{2}=7\), \(Q_3=\frac{9+9}{2}=9\)
• \(IQR=9 - 7=2\)
• Lower fence \(=Q_1-1.5IQR=7-1.5\times2=7 - 3 = 4\)
• Upper fence \(=Q_3 + 1.5IQR=9+1.5\times2=9 + 3=12\)
• Data point \(3\) is less than \(4\) (lower fence)

Answer:

\(3,6,7,7,8,8,9,9,9,10\)