Chapter 13, Problem 15RE

To determine

To calculate: The standard deviation of the data set, $2,5,5,7,7,9,11$ .

Answer to Problem 15RE

The standard deviation of the data set, $2,5,5,7,7,9,11$ is $2.72$ .

Explanation of Solution

Given information:

The data set, $2,5,5,7,7,9,11$ .

Formula used:

For the data set $\left\{x_1,x_2,x_3,\dots x_n\right\}$ that consists of n data points,mean is denoted by $\overline{x}$ and mathematically expressed as, $\overline{x}=\frac{x_1+x_2+\cdots x_n}{n}$ .

For the data set $\left\{x_1,x_2,x_3,\dots x_n\right\}$ that consists of n data points, standard deviation is denoted by $\sigma$ and mathematically expressed as, $\sigma =\sqrt{\frac{x_1^2+x_2^2+\cdots +x_n^2}{n}-{\overline{x}}^2}$ , where $\overline{x}$ is the mean of the data.

Calculation:

Consider the data set, $2,5,5,7,7,9,11$ . There are seven data points.

Recall that for the data set $\left\{x_1,x_2,x_3,\dots x_n\right\}$ that consists of n data points,mean is denoted by $\overline{x}$ and mathematically expressed as, $\overline{x}=\frac{x_1+x_2+\cdots x_n}{n}$ .

For the provided data set, mean is evaluated as,

  $\begin{array}{c}\overline{x}=\frac{x_1+x_2+\cdots x_n}{n}\\ =\frac{2+5+5+7+7+9+11}{7}\\ =\frac{46}{7}\\ =6.57\end{array}$

Therefore, mean of the data is $6.57$ .

Next, recall that for the data set $\left\{x_1,x_2,x_3,\dots x_n\right\}$ that consists of n data points, standard deviation is denoted by $\sigma$ and mathematically expressed as, $\sigma =\sqrt{\frac{x_1^2+x_2^2+\cdots +x_n^2}{n}-{\overline{x}}^2}$ , where $\overline{x}$ is the mean of the data.

For the provided data set, standard deviation is evaluated as,

  $\begin{array}{c}\sigma =\sqrt{\frac{x_1^2+x_2^2+\cdots +x_n^2}{n}-{\overline{x}}^2}\\ =\sqrt{\frac{2^2+5^2+5^2+7^2+7^2+9^2+{11}^2}{7}-{\left(6.57\right)}^2}\\ =\sqrt{7.388}\\ =2.72\end{array}$

Thus, the standard deviation of the data set, $2,5,5,7,7,9,11$ is $2.72$ .