Chapter 3.1, Problem 39E

Heavy football players: Following are the weights, in pounds, for offensive and defensive linemen on the New York Giants National Football League team at the beginning of a recent year.

1. Find the mean and median weight for the offensive linemen.
2. Find the mean and median weight for the defensive linemen.
3. Do offensive or defensive linemen tend to be heavier, or are they about the same?

To determine

To Find: the mean and median weight for offensive linemen.

Answer to Problem 39E

Weight for offensive linemen, Mean weight $=296$ and Median weight $=302.5$

Explanation of Solution

Given:

  $252,253,255,299,300,302,303,310,315,317,319,327$

Formula Used:

  $\begin{array}{l}\text{Mean}(\overline{x})=\frac{{\displaystyle \sum x_i}}{n}\\ \\ \text{Median}=\left\{\begin{array}{l}\frac{{\left(n+1\right)}^{\text{th }}\text{term}}{2},\text{When}\text{n}\text{is}\text{odd}\\ \frac{{\left(\frac{n}{2}+1\right)}^{\text{th }}\text{term + }{\left(\frac{n}{2}\right)}^{\text{th}}\text{term}}{2},\text{When n is even}\end{array}\right\}\end{array}$

Calculation:

Mean and Median weight for offensive linemen,

  $\begin{array}{c}\overline{x}=\frac{\sum x_i}{n}\\ =\frac{315+253+255+319+252+300+327+303+299+310+317+302}{12}\\ =\frac{3552}{12}\\ =296\end{array}$

For Median, first arrange data in sequential order, $252,253,255,299,300,302,303,310,315,317,319,327$

  $\begin{array}{c}\text{Median=}\left(\frac{{\left(\frac{\text{n}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{n}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\frac{\text{12}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{12}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\text{6}\right)}^{\text{th}}\text{term+}{\left(\text{7}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{\text{302+303}}{\text{2}}\right)\\ \text{=}\frac{\text{605}}{\text{2}}\\ \text{=302}\text{.5}\end{array}$

Therefore, for offensive linemen, Mean weight $=296$ and Median weight $=302.5$

To determine

To Find: the mean and median weight for defensive linemen.

Answer to Problem 39E

Weight for defensive linemen, Mean weight $=285.5$ and Median weight $=285$

Explanation of Solution

Calculation-

Mean and Median weight for defensive linemen,

  $\begin{array}{c}\overline{x}=\frac{\sum x_i}{n}\\ =\frac{304+309+306+305+265+270+264+317+255+274+261+296}{12}\\ =\frac{3426}{12}=285.5\end{array}$

For Median, first arrange data in sequential order, $255,261,264,265,270,274,296,304,305,306,309,317$

  $\begin{array}{c}\text{Median = }\left(\frac{{\left(\frac{\text{n}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{n}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\frac{\text{12}}{\text{2}}\right)}^{\text{th}}\text{term+}{\left(\frac{\text{12}}{\text{2}}\text{+1}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{{\left(\text{6}\right)}^{\text{th}}\text{term+}{\left(\text{7}\right)}^{\text{th}}\text{term}}{\text{2}}\right)\\ \text{=}\left(\frac{\text{274+296}}{\text{2}}\right)\\ \text{=}\frac{\text{570}}{\text{2}}\\ \text{=285}\end{array}$

Therefore, for defensive linemen, Mean weight $=285.5$ and Median weight $=285$

To determine

To Check: Whether defensive or offensive linemen tend to be heavier.

Answer to Problem 39E

Offensive Linemen tend to be heavier

Explanation of Solution

Offensive Linemen tend to be heavier because their median weight is $302.5$ but their mean weight is $296$ .

Defensive Linemen are same because their median weight is 285 but their mean weight is 285.5.