Question

Using the same prediction errors listed in Exercise 1, construct a boxplot and include the values of the 5-number summary.

Short answer

The five-number summary is given below.

Minimum	–7
First quartile	–4.5
Median	0.5
Third quartile	2.5
Maximum	22

And the boxplot is given below.

Step-by-step solution

Given information

The errors in terms of minutes are listed below.

4 –7 0 1 –1 1 –4 –7 22 7 –5 1

Obtain the five-number summary

A five-number summary is a group of five measures from the quantitative data.The measures form a pattern of box-plots.

The five measures are:

• Minimum value
• First quartile
• Median
• Third quartile
• Maximum value

Compute the median

In the increasing order arrangement of the observations, the middle observation is the median value for odd counts of data, and the average of the middle observations is the median value for even counts of data, respectively.

Arrange the data in ascending order.

–7
–7
–5
–4
–1
0
1
1
1
4
7
22

The count of observations is 12.

The two middle observations are 0 and 1.

The median is computed as shown below.

$$\begin{gathered} M=\frac{0+1}{2} \\ =0.5 \end{gathered}$$

Thus, the median is 0.5 minutes.

Compute the first and third quartiles

Themedian values for the lower and upper halves of the data set give the first and third quartile measures, respectively.

The lower half of the dataset is shown below.

–7

–7

–5

–4

–1

0

The upper half of the dataset is shown below.

1

1

1

4

7

22

The median of the lower half of the data set is computed below.

$$\begin{gathered} Q_1=\frac{-5+\left(-4\right)}{2} \\ =-4.5 \end{gathered}$$

Thus, the first quartile is –4.5 minutes.

The median of the upper half of the dataset is computed below.

$$\begin{gathered} Q_3=\frac{1+4}{2} \\ =2.5 \end{gathered}$$

Thus, the third quartile is 2.5 minutes.

Maximum and minimum values

The maximum value of the data set is 22.

The minimum value of the data set is –7.

Thus, the five-number summary is obtained as follows:

Minimum	–7
First quartile	–4.5
Median	0.5
Third quartile	2.5
Maximum	22

Boxplot of the data

The steps to construct a box plot are given below.

• Draw a horizontal number line.
• Mark the points of the five-number summary on the line.
• Draw a closed rectangular curve extending from the first (–4.5) to the third quartile (2.5), with a line dividing the curve at 0.5.
• Extend two lines—left of –4.5 up to –7 and right of 2.5 up to 22.

The boxplot is sketched as follows.