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: Calculate quartiles and IQR for first dataset

For \(6,13,13,15,15,18,18,22\):

• \(n = 8\)
• \(Q_1\) (median of first half \(6,13,13,15\)): \(\frac{13 + 13}{2}=13\)
• \(Q_3\) (median of second half \(15,18,18,22\)): \(\frac{18+18}{2} = 18\)
• \(IQR=Q_3 - Q_1=18 - 13 = 5\)
• Lower fence \(=Q_1-1.5\times IQR=13-1.5\times5 = 13 - 7.5=5.5\)
• Upper fence \(=Q_3 + 1.5\times IQR=18+1.5\times5=18 + 7.5 = 25.5\)
• No data outside \([5.5,25.5]\)

Step2: Calculate quartiles and IQR for second dataset

For \(4,4,4,8,9,9,11,18\):

• \(n = 8\)
• \(Q_1\) (median of first half \(4,4,4,8\)): \(\frac{4 + 4}{2}=4\)
• \(Q_3\) (median of second half \(9,9,11,18\)): \(\frac{9+11}{2}=10\)
• \(IQR = 10 - 4=6\)
• Lower fence \(=4-1.5\times6=4 - 9=-5\)
• Upper fence \(=10+1.5\times6=10 + 9 = 19\)
• \(18<19\), no outlier

Step3: Calculate quartiles and IQR for third dataset

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

• \(n = 10\)
• \(Q_1\) (value at \(\frac{n + 1}{4}\)th position, \(\frac{10+1}{4}=2.75\)th value). \(Q_1=5+(7 - 5)\times0.75=6.5\)
• \(Q_3\) (value at \(3\times\frac{n + 1}{4}\)th position, \(3\times\frac{10 + 1}{4}=8.25\)th value). \(Q_3=10+(12 - 10)\times0.25 = 10.5\)
• \(IQR=10.5 - 6.5 = 4\)
• Lower fence \(=6.5-1.5\times4=6.5 - 6=0.5\)
• Upper fence \(=10.5+1.5\times4=10.5 + 6=16.5\)
• \(17>16.5\)

Step4: Calculate quartiles and IQR for fourth dataset

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

• \(n = 10\)
• \(Q_1\) (value at \(\frac{n + 1}{4}\)th position, \(\frac{10+1}{4}=2.75\)th value). \(Q_1=7+(7 - 7)\times0.75=7\)
• \(Q_3\) (value at \(3\times\frac{n + 1}{4}\)th position, \(3\times\frac{10+1}{4}=8.25\)th value). \(Q_3=9+(9 - 9)\times0.25=9\)
• \(IQR=9 - 7=2\)
• Lower fence \(=7-1.5\times2=7 - 3 = 4\)
• Upper fence \(=9+1.5\times2=9+3=12\)
• No data outside \([4,12]\)

Answer:

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