Chapter B.2, Problem 29E

To determine

To calculate:

Standard deviation $\left(\sigma \right)$ for the given data by using alternate formula.

Answer to Problem 29E

Standard deviation of the data set using alternate formula is $1.65$ .

Explanation of Solution

Given information:

Given data: $\text{8}.\text{1},\text{6}.\text{9},\text{3}.\text{7},\text{4}.\text{2},\text{6}.\text{1}$

Calculation:

The given data is $\text{8}.\text{1},\text{6}.\text{9},\text{3}.\text{7},\text{4}.\text{2},\text{6}.\text{1}$

The alternate formula for standard deviation is given as below:

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

  $\sigma =\sqrt{\frac{x_1{}^2,x_2{}^2,\mathrm{...},x_n}{n}-{\overline{x}}^2}$

Where,

  $\overline{x}$ is the mean.

To find the mean $\overline{x}$ ,

For a data set having $n$ , the mean $\overline{x}$ is the sum of the numbers which are divided by $n$ .

  $\begin{array}{l}\overline{x}=\frac{\text{8}.\text{1+6}.\text{9+3}.\text{7+4}.\text{2+6}.\text{1}}{5}\\ =\frac{29}{5}\\ =5.8\end{array}$

So mean is $5.8$ .

Substitute the corresponding values for $x$ , $\overline{x}$ and $n$ in the equation for standard deviation.

  $\begin{array}{l}\sigma =\sqrt{\frac{{\left(8.1\right)}^2\text{+}{\left(6.9\right)}^2\text{+}{\left(3.7\right)}^2\text{+}{\left(4.2\right)}^2\text{+}{\left(6.1\right)}^2}{5}}-{5.8}^2\\ =\sqrt{\frac{181.76}{5}}-{5.8}^2\\ =\sqrt{2712}\\ =1.65\end{array}$

Thus, the standard deviation of the data set is $1.65$ .