Question

which data sets have outliers? check all that apply.
14, 21, 24, 25, 27, 32, 35
15, 30, 35, 41, 44, 50, 78
16, 32, 38, 39, 41, 42, 58
17, 23, 28, 31, 39, 45, 75
18, 30, 34, 38, 43, 45, 68

Explanation:

Step1: Recall outlier - finding method

We can use the inter - quartile range (IQR) method. First, we need to find the first quartile ($Q_1$), the third quartile ($Q_3$), and then calculate the IQR = $Q_3 - Q_1$. Outliers are values less than $Q_1-1.5\times IQR$ or greater than $Q_3 + 1.5\times IQR$.

Step2: For the data set 14, 21, 24, 25, 27, 32, 35

Arrange in ascending order. There are $n = 7$ data points.
The median (second - quartile $Q_2$) is the 4th value, so $Q_2=25$.
The lower half of the data is 14, 21, 24. The median of the lower half ($Q_1$) is 21.
The upper half of the data is 27, 32, 35. The median of the upper half ($Q_3$) is 32.
$IQR=Q_3 - Q_1=32 - 21 = 11$.
$Q_1-1.5\times IQR=21-1.5\times11=21 - 16.5 = 4.5$ and $Q_3 + 1.5\times IQR=32+1.5\times11=32 + 16.5 = 48.5$. There are no outliers.

Step3: For the data set 15, 30, 35, 41, 44, 50, 78

Arrange in ascending order. $n = 7$ data points.
$Q_2$ is the 4th value, so $Q_2 = 41$.
The lower half is 15, 30, 35, $Q_1 = 30$.
The upper half is 44, 50, 78, $Q_3 = 50$.
$IQR=Q_3 - Q_1=50 - 30 = 20$.
$Q_1-1.5\times IQR=30-1.5\times20=30 - 30 = 0$ and $Q_3 + 1.5\times IQR=50+1.5\times20=50 + 30 = 80$. There are no outliers.

Step4: For the data set 16, 32, 38, 39, 41, 42, 58

Arrange in ascending order. $n = 7$ data points.
$Q_2$ is the 4th value, so $Q_2 = 39$.
The lower half is 16, 32, 38, $Q_1 = 32$.
The upper half is 41, 42, 58, $Q_3 = 42$.
$IQR=Q_3 - Q_1=42 - 32 = 10$.
$Q_1-1.5\times IQR=32-1.5\times10=32 - 15 = 17$ and $Q_3 + 1.5\times IQR=42+1.5\times10=42 + 15 = 57$. 58 is an outlier.

Step5: For the data set 17, 23, 28, 31, 39, 45, 75

Arrange in ascending order. $n = 7$ data points.
$Q_2$ is the 4th value, so $Q_2 = 31$.
The lower half is 17, 23, 28, $Q_1 = 23$.
The upper half is 39, 45, 75, $Q_3 = 45$.
$IQR=Q_3 - Q_1=45 - 23 = 22$.
$Q_1-1.5\times IQR=23-1.5\times22=23 - 33=-10$ and $Q_3 + 1.5\times IQR=45+1.5\times22=45 + 33 = 78$. There are no outliers.

Step6: For the data set 18, 30, 34, 38, 43, 45, 68

Arrange in ascending order. $n = 7$ data points.
$Q_2$ is the 4th value, so $Q_2 = 38$.
The lower half is 18, 30, 34, $Q_1 = 30$.
The upper half is 43, 45, 68, $Q_3 = 45$.
$IQR=Q_3 - Q_1=45 - 30 = 15$.
$Q_1-1.5\times IQR=30-1.5\times15=30 - 22.5 = 7.5$ and $Q_3 + 1.5\times IQR=45+1.5\times15=45 + 22.5 = 67.5$. 68 is an outlier.

Answer:

16, 32, 38, 39, 41, 42, 58; 18, 30, 34, 38, 43, 45, 68