Exercise 1 Page 715

If a value in a data set is more than $1.5$ times the interquartile range away from the lower or upper quartiles, it is considered an outlier. Therefore, to identify any outliers we first have to find these statistical measures, including any outliers.

Analyzing the Data With Any Outliers

Consider the given data set. Let's organize the information in a list using a graphing calculator. Press $\boxed{\text{STAT}}$ and choose Edit. Enter all of the data into the first list.

Next, analyze the data by pressing the $\boxed{\text{STAT}}$ button again and navigating to the CALC menu. Press $\boxed{\text{ENTER}}$ once to select the $1\text{-Var}$ Stats option and then again to select the list in which we entered the data (usually $\text{L}1).$ This will produce most of the statistical measures we are looking to find.

We can see from our calculator that the mean $\bar{x}$ is $22.8.$ To find the median and quartiles we need to scroll down.

Examining the output will give us most of the desired statistical measures. In order to find the mode, we have to identify the value that occurs most frequently. We see that $18$ occurs the most frequently, so this is the mode of the given data set.

Identifying Outliers

To identify any outliers, we start by calculating the interquartile range (IQR). This is the difference between the first and third quartiles. Next, we need to determine the maximum and minimum values for data to be considered an outlier. Outliers are more than $1.5$ times the IQR away from the upper and lower quartiles. These calculations tell us that any values lower than $12$ or greater than $28$ are outliers. Since $54>28,$ $54$ is an outlier.

Analyzing the Data Without the Outliers

Let's repeat the process, excluding $54$ from the data set. We will type the new data set into another list, $\text{L}2.$

Once we have finished adding the new data set into $\text{L}2,$ we can analyze the data the same way as before. After selecting the $1\text{-Var}$ Stats option, however, we must remember to choose $\text{L}2$ by pushing $\boxed{\text{2nd}}$ and $\boxed{\text{2}}.$

Examining the new output will give us most of the desired statistical measures. Note that $18$ is still the most frequently occurring value. Therefore, the mode of the data set remains the same.

Summary

Finally, we summarize our findings in the table below so it is easier to compare the results.

Data set	Mean	Median	Mode
With Outlier	$22.8$	$19.5$	$18$
Without Outlier	$19.\bar{3}$	$19$	$18$

We can see that removing the outlier modified the mean and the median.