Question

The data set shows the fuel efficiencies, in miles per gallon, of the vehicles featured in a car-buying guide.

$293140281923353843462830$
(a) What is the five-number summary of the data set?
(b) What is the range of the data set? Show your work.
Highest number $46$ minus lowest number $19$ equals $27$ range.
(c) What is the interquartile range (IQR) of the data set? Show your work.

Answer 1

Step 1: Find the five-number summary of the data set.

The given data set is:

$293140281923353843462830$.

Arrange this data set in ascending order (from lowest value to highest value) as follows :

$192328282930313538404346$.

Firstly find the median of the data set. The median of a data set is defined as the middle of a data set. There are two terms in the middle :

$Q_2=\frac{30+31}{2}=30.5$.

The median is defined as the second quartile. That is,

$Q_2=30.5$.

Now divide the data set into two parts. The median of the first part is the first quartile and the median of the second part is the third quartile.

So,

$\begin{array}{rcl}{Q}_1& =& \frac{28+28}{2}\\ & =& 28\end{array}$

and

$\begin{array}{rcl}{Q}_3& =& \frac{38+40}{2}\\ & =& 39\end{array}$.

The minimum value of the data set is $19$ and the maximum value of the data set is $46$.

The five-number summary of the data set is a collection of five values the smallest value of the data set, the largest value of the data set, and the first, second and third quartile.

Therefore, the five-number summary of the given data set is:

$\begin{array}{rcl}\mathit{M}\mathit{i}\mathit{n}\mathit{i}\mathit{m}\mathit{u}\mathit{m}\mathbf{}& \mathbf{=}& \mathbf{}\mathbf{19}\\ {\mathit{Q}}_{\mathbf{1}}& \mathbf{=}& \mathbf{28}\\ {\mathit{Q}}_{\mathbf{2}}& \mathbf{=}& \mathbf{30}\mathbf{.}\mathbf{5}\\ {\mathit{Q}}_{\mathbf{3}}& \mathbf{=}& \mathbf{39}\\ \mathit{M}\mathit{a}\mathit{x}\mathit{i}\mathit{m}\mathit{u}\mathit{m}\mathbf{}& \mathbf{=}& \mathbf{}\mathbf{46}\end{array}$

Step 2: Find the range of the data set.

The range of the data set is the difference between the maximum and minimum values of the data set.

So,

$\begin{array}{rcl}\mathrm{Range}& =& 46-19\\ & =& 27\end{array}$

Therefore, the range of the data set is $\mathbf{27}$.

Step 3: Find the interquartile range of the data set.

The interquartile range (IQR) of the data set is the difference between the first and third quartile.

Then,

$\begin{array}{rcl}IQR& =& 39-28\\ & =& 11\end{array}$

Therefore, the IQR of the data set is $\mathbf{11}$.