Chapter 10.3, Problem 30E

Solution

To calculate : The mean number of games won by NFL teams during the 2016−2017 season from the given data and state the better average is mean or median for the given data.

The mean number of games won by NFL teams during the 2016−2017 season is $7.94$ . Either mean or median should be preferred.

Given information :

National Football League teams play $16$ games during the regular season. In $2016-2017$ , the Patriots let the league with $14$ wins while the Browns won just once. The frequency table below summarizes the numbers of games won by all $32$ teams that season.

Wins	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$
Frequency	$1$	$1$	$2$	$1$	$2$	$2$	$4$	$4$	$5$	$3$	$3$	$2$	$1$	$1$

Formula used :

The mean of a list of numbers $\left\{x_1,x_2,x_3,\mathrm{......},x_n\right\}$ with frequency $\left\{n_1,n_2,n_3,\mathrm{......},n_n\right\}$ is given by,

  $\overline{X}=\frac{x_1\cdot n_1+x_2\cdot n_2+x_3\cdot n_3+\mathrm{......}+x_n\cdot n_n}{n_1+n_2+n_3+\mathrm{....}+n_n}$

Calculation :

Consider the given data.

Make a new table with a column for product of frequency and wins.

Wins $\left(x\right)$	Frequency $\left(f\right)$	$fx$
$\begin{array}{l}1\\ 2\\ 3\\ 4\\ 5\\ 6\\ 7\\ 8\\ 9\\ 10\\ 11\\ 12\\ 13\\ 14\end{array}$	$\begin{array}{l}1\\ 1\\ 2\\ 1\\ 2\\ 2\\ 4\\ 4\\ 5\\ 3\\ 3\\ 2\\ 1\\ 1\end{array}$	$\begin{array}{l}1\\ 2\\ 6\\ 4\\ 10\\ 12\\ 28\\ 32\\ 45\\ 30\\ 33\\ 24\\ 13\\ 14\end{array}$
	$\sum f=32$	$\sum fx=254$

Find the mean number of games won by NFL teams during the 2016−2017 season.

  $\begin{array}{c}\overline{X}=\frac{254}{32}\\ \approx 7.94\end{array}$

From the given frequency distribution, it is clear that the distribution is normal. Thus, mean or median should be used.