Chapter 10.3, Problem 12E

Solution

To check: If there any outlier exists. Also, provide explanation for the same.

  $7$ is an outlier.

Given information :

The five-number summary for the number of miles (in thousands) on a company’s fleet of cars is $\left\{7,49,66,72,98\right\}$ .

Formula Used:

The five-number summary of a data set is the collection $\left\{\text{minimum value, }{Q}_1\text{, median, }{Q}_3\text{, maximum value}\right\}$ .

A number in a data set, which falls below $Q_1-1.5\left(\text{IQR}\right)$ or above $Q_3+1.5\left(\text{IQR}\right)$ are the outliers.

First quartile is $Q_1$ , third quartile is $Q_3$ , and IQR is $Q_3-Q_1$ .

Explanation :

Consider the given five-number summary: $\left\{7,49,66,72,98\right\}$ .

Here,

  $\begin{array}{c}\text{Minimum value}=7\\ Q_1=49\\ \text{Median}=66\\ Q_3=72\\ \text{Maximum value}=98\end{array}$

So, the Inter-Quartile Range is,

  $\begin{array}{c}IQR=72-49\\ =23\end{array}$

Now, the upper fence is,

  $\begin{array}{c}{Q}_3+1.5IQR=72+1.5\left(23\right)\\ =72+34.5\\ =106.5\end{array}$

And, the lower fence is,

  $\begin{array}{c}{Q}_1-1.5IQR=49-1.5\left(23\right)\\ =49-34.5\\ =14.5\end{array}$

Here, no value is above the upper fence but 7 is smaller than the lower fence.

Therefore, $7$ is an outlier.