Chapter B.2, Problem 28E

To determine

To calculate:

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

Answer to Problem 28E

Standard deviation of the data set is $101.55$ .

Explanation of Solution

Given information:

Given data: $246,336,473,167,219,359$

Calculation:

The given data is $246,336,473,167,219,359$

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 $n$ .

  $\begin{array}{l}\overline{x}=\frac{246+336+473+167+219+359}{6}\\ =\frac{1800}{6}\\ =300\end{array}$

So mean is $300$ .

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

  $\begin{array}{l}\sigma =\sqrt{\frac{{\left(246\right)}^2\text{+}{\left(336\right)}^2\text{+}{\left(473\right)}^2\text{+}{\left(167\right)}^2\text{+}{\left(219\right)}^2\text{+}{\left(359\right)}^2}{6}}-{300}^2\\ =\sqrt{\frac{601872}{6}}-{300}^2\\ =\sqrt{10312}\\ =101.55\end{array}$

Thus, the standard deviation of the data set is $101.55$ .