Chapter 10.3, Problem 33E

Solution

To calculate : The standard deviation and variance for the given data set.

The standard deviation and variance of the given data is $9.71\text{ and }94.28$ .

Given information :

The data set is: $\left\{23,45,29,34,39,41,19,22\right\}$ .

Formula used :

The standard deviation of a population $\left\{x_1,x_2,x_3,\mathrm{......},x_n\right\}$ is given by,

  $s=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}}$ .

Variance $={\left(s\right)}^2$ .

Calculation :

Consider the given data and present it in tabular form as below:

$x$	$x-\overline{x}$	${\left(x-\overline{x}\right)}^2$
$\begin{array}{l}23\\ 45\\ 29\\ 34\\ 39\\ 41\\ 19\\ 22\end{array}$	$\begin{array}{l}-8.5\\ 13.5\\ -2.5\\ 2.5\\ 7.5\\ 9.5\\ -12.5\\ -9.5\end{array}$	$\begin{array}{l}72.25\\ 182.25\\ 6.25\\ 6.25\\ 56.25\\ 90.25\\ 156.25\\ 90.25\end{array}$
$\sum x=252$		$\sum {\left(x-\overline{x}\right)}^2=660$

Find the mean $\left(\mu \right)$ of the given data.

  $\begin{array}{c}\overline{x}=\frac{252}{8}\\ =31.5\end{array}$

Find the standard deviation $\left(s\right)$ of the given data.

  $\begin{array}{c}s=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}}\\ =\sqrt{\frac{1}{8-1}\left(660\right)}\\ \approx 9.71\end{array}$

Find the variance for the given data.

  $s^2\approx 94.28$

Therefore, the standard deviation and variance of the given data is $9.71\text{ and }94.28$ .