Chapter 10.3, Problem 37E

Solution

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

The standard deviation and variance of the given data is $\$7.04\text{ thousands and \$}49.56\text{ thousands}$ respectively.

Given information :

The salary in thousands of dollar is:

  $33.5,35.3,33.8,29.3,36.7,32.8,31.7,37.3,33.5,28.2,34.8,33.5,29.7,38.5,32.7,34.8,34.2,31.6,35.4$

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:

$x$	$x-\overline{x}$	${\left(x-\overline{x}\right)}^2$
$\begin{array}{l}33.5\\ 35.3\\ 33.8\\ 29.3\\ 36.7\\ 32.8\\ 31.7\\ 37.3\\ 33.5\\ 28.2\\ 34.8\\ 33.5\\ 29.7\\ 38.5\\ 32.7\\ 34.8\\ 34.2\\ 31.6\\ 35.4\end{array}$	$\begin{array}{l}0\\ 1.8\\ 0.3\\ -4.2\\ 3.2\\ -0.7\\ -1.8\\ 3.8\\ 0\\ -5.3\\ 1.3\\ 0\\ -3.8\\ 5\\ -0.8\\ 1.3\\ 0.7\\ -1.9\\ 1.9\end{array}$	$\begin{array}{l}0\\ 3.24\\ 0.09\\ 17.64\\ 10.24\\ 0.49\\ 1.26\\ 14.44\\ 0\\ 28.09\\ 1.69\\ 0\\ 14.44\\ 25\\ 0.64\\ 1.69\\ 0.49\\ 3.61\\ 3.61\end{array}$
$\sum x=637.3$		$\sum {\left(x-\overline{x}\right)}^2=126.66$

Find the mean of the given data.

  $\begin{array}{c}\overline{x}=\frac{637.3}{19}\\ \approx 33.5\end{array}$

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

  $\begin{array}{c}s=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}}\\ =\sqrt{\frac{1}{19-1}\left(126.66\right)}\\ \approx 7.04\end{array}$

Find the variance for the given data.

  $s^2\approx 49.56$

Therefore, the standard deviation and variance of the given data is $\$7.04\text{ thousands and \$}49.56\text{ thousands}$ respectively.