Chapter 3.4, Problem 16E

Size of Dams These data represent the volumes in cubic yards of the largest dams in the United States and in South America. Construct a boxplot of the data for each region and compare the distributions.

United States	South America
125,628 92,000 78,008 77,700 66,500 62,850 52,435 50,000	311,539 274,026 105,944 102,014 56,242 46,563

To determine

The boxplot of the given data and comparison of the distribution.

Answer to Problem 16E

The boxplot for the capacity of dams in the United States is shown below,

Fig (1)

The boxplot for the capacity of dams in the South America is shown below,

Fig (2)

The range and variation of the capacity of the dams in South America is larger than the capacity of the dams in United Stated.

Explanation of Solution

Given info:

The data of the capacity of dams in the United States and the South America is shown in the table below.

United States	South America
125628	311539
92000	274026
78008	105944
77700	102014
66500	56242
62850	46563
52435
50000

Calculation:

The boxplot is used to graphically depict groups of numerical data through their quantities.

Arrange the densities of dams in the United States in ascending order as shown below.

United States
50000
52435
62850
66500
77700
78008
92000
125628

The minimum capacity of the dam from the above table is 50000 and the maximum capacity of the dam is 125628.

The number of data in the table is 8.

The formula to calculate the median for even data is shown below.

$\text{Median}=\frac{{\left(\frac{n}{2}\right)}^{\text{th}}\text{term}+{\left(\frac{n}{2}+1\right)}^{\text{th}}\text{term}}{2}$

The total number of terms in the data is 8 so, substitute 8 for n in the above formula.

$\begin{array}{c}\text{Median}=\frac{{\left(\frac{8}{2}\right)}^{\text{th}}\text{term}+{\left(\frac{8}{2}+1\right)}^{\text{th}}\text{term}}{2}\\ =\frac{{\left(4\right)}^{\text{th}}\text{term}+{\left(5\right)}^{\text{th}}\text{term}}{2}\end{array}$ (1)

$4^{\text{th}}\text{term}$ from the arranged data is 66500 and the $5^{\text{th}}\text{term}$ is 77700.

Substitute 66500 for $4^{\text{th}}\text{term}$ and 77700 for $5^{\text{th}}\text{term}$ in equation (1).

$\begin{array}{c}\frac{4^{\text{th}}\text{term}+5^{\text{th}}\text{term}}{2}=\frac{66500+77700}{2}\\ =72,100\end{array}$

Thus, the median of the data is 72,100.

Software procedure:

Step-by-step procedure to construct boxplot by using Excel add-in (MegaStat).

• First select the data for which boxplot is obtained.
• Click on Add-Ins option in the top.
• Click on MegaStat option on the left side of the screen and then click on descriptive statistics.
• Select the boxplot option then in input range, select the data cells and then click ok.

Observation:

Since median falls to the right of the centre of the box thus the distribution is slightly negatively skewed.

Now, arrange the data of densities of dams in the South America in ascending order as shown below.

South America

46563

56242

102014

105944

274026

311539

The minimum capacity of the dam from the above table is 46563 and the maximum capacity of the dam is 311539.

The number of data in the table is 6.

The formula to calculate the median for even data is shown below.

$\text{Median}=\frac{{\left(\frac{n}{2}\right)}^{\text{th}}\text{term}+{\left(\frac{n}{2}+1\right)}^{\text{th}}\text{term}}{2}$

The total number of terms in the data is 6 so, substitute 6 for n in the above formula.

$\begin{array}{c}\text{Median}=\frac{{\left(\frac{6}{2}\right)}^{\text{th}}\text{term}+{\left(\frac{6}{2}+1\right)}^{\text{th}}\text{term}}{2}\\ =\frac{{\left(3\right)}^{\text{rd}}\text{term}+{\left(4\right)}^{\text{th}}\text{term}}{2}\end{array}$ (2)

$3^{\text{rd}}\text{term}$ from the arranged data is 102014 and the $4^{\text{th}}\text{term}$ is 105944.

Substitute 102014 for $3^{\text{rd}}\text{term}$ and 105944 for $4^{\text{th}}\text{term}$ in equation (2).

$\begin{array}{c}\frac{3^{\text{rd}}\text{term}+4^{\text{th}}\text{term}}{2}=\frac{102014+105944}{2}\\ =103,979\end{array}$

Thus, the median of the data is 103,979.

Software procedure:

Step-by-step procedure to construct boxplot by using Excel add-in (MegaStat).

• First select the data for which boxplot is obtained.
• Click on Add-Ins option in the top.
• Click on MegaStat option on the left side of the screen and then click on descriptive statistics.
• Select the boxplot option then in input range, select the data cells and then click ok.

Observation:

Since median falls to the left of the centre of the box thus the distribution is slightly positively skewed.

The range and variation of the capacity of the dams in South America is larger than the capacity of the dams in United Stated.