Exercise 2 Page 721

A line plot is a way of illustrating a data set in which each data point is represented with a mark above a number line. Marks representing the same elements are stacked above each other. We want to identify the shape of the distribution of the given line plot. Let's do it!

Notice that the left side of the distribution is taller than the right side. This means that the distribution is non-symmetric. Now we will look for any clusters, gaps, peaks, or outliers on the plot. Let's begin by recalling the definitions of these attributes.

With these definitions in mind, we can draw some conclusions about the characteristics of our graph.

• There are no clusters.
• There is a gap from magnitude $2.9$ to $3.6.$
• There is a peak at magnitude $1.5.$
• There appears to be an outlier at magnitude $3.6.$

Extra

Extra

We want to confirm whether $3.6$ is an outlier of our data set. Let's do using another approach for finding an outlier when given numerical data. In this case, a data value is an outlier if it is farther away from the closest quartile than $1.5$ times the interquartile range (IQR). Let's find the outliers in three steps!

1. Find the IQR of the data set.
2. Calculate the lower and upper boundary for the value of the outliers.
3. Determine whether the data set has outliers.

First, let's write all the data values as a set.

$$\begin{array}{c}1.5,1.5,1.5,1.5,1.5,1.5,1.5,1.7,1.9,\\ 1.9,2.1,2.1,2.3,2.5,2.7,2.9,3.6\end{array}$$

Now, recall that the interquartile range is the difference between $Q_3$ and $Q_1,$ the upper and lower quartiles. We can use the applet below to find the IQR of the set.

The interquartile range of the data set is $0.9.$ We know that the outliers satisfy one of the following conditions.

$$\begin{array}{c}\text{Less than }{Q}_1-1.5\cdot \text{IQR}\\ \text{Greater than }{Q}_3+1.5\cdot \text{IQR}\end{array}$$

Let's calculate these values when $Q_1=1.5$ and $Q_3=2.4.$

$Q_1-1.5\cdot \text{IQR}$	$Q_3+1.5\cdot \text{IQR}$
$1.5-1.5\cdot 0.9=1.5-1.35$	$2.4+1.5\cdot 0.9=2.4+1.35$
$0.15$	$3.75$

All values of the data set are greater than $0.15$ and less than $3.75.$ This means that $3.6$ is not actually an outlier — in fact, our data set has no outliers!