Chapter 14.3, Problem 13E

To determine

To calculate: The mean deviation and the standard deviation of $\left\{200,476,721,579,152,158\right\}.$

Answer to Problem 13E

The standard deviation is $223.1441$ and mean deviation is $211$ .

Explanation of Solution

Given information:

  $\left\{200,476,721,579,152,158\right\}.$

Formula used:

A measure of variability associated with the arithmetic mean is the standard deviation.

  $\sigma =\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{\left(X_i-\overline{X}\right)}^2}}$

The arithmetic mean of the absolute values of the deviations from the mean of a set of data is called the mean deviation.

  $MD=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|X_i-\overline{X}\right|}$

Formula of Mean

  $\overline{X}=\frac{{\displaystyle \sum X}}{n}$ where X represents each of the values in the data set.

Calculation:

Consider the data ,

  $\left\{152,158,200,476,579,721\right\}.$

So, $n=6$ .

The mean of the data is

  $\begin{array}{l}=\frac{152+158+200+476+579+721}{6}\\ =\frac{2286}{6}\\ =381\end{array}$

Therefore , the standard deviation is −

$X_i$	$\overline{X}$	$X_i-\overline{X}$	${\left(X_i-\overline{X}\right)}^2$
152	381	$\begin{array}{l}=152-381\\ =-229\end{array}$	$\begin{array}{l}={\left(-229\right)}^2\\ =52441\end{array}$
158	381	$\begin{array}{l}=158-381\\ =-223\end{array}$	$\begin{array}{l}={\left(-223\right)}^2\\ =49729\end{array}$
200	381	$\begin{array}{l}=200-381\\ =-181\end{array}$	$\begin{array}{l}={\left(-181\right)}^2\\ =32761\end{array}$
476	381	$\begin{array}{l}=476-381\\ =95\end{array}$	$\begin{array}{l}={\left(95\right)}^2\\ =9025\end{array}$
579	381	$\begin{array}{l}=579-381\\ =198\end{array}$	$\begin{array}{l}={\left(198\right)}^2\\ =39204\end{array}$
721	381	$\begin{array}{l}=721-381\\ =340\end{array}$	$\begin{array}{l}={\left(340\right)}^2\\ =115600\end{array}$
${{\displaystyle \sum _{i=1}^6\left(X_i-\overline{X}\right)}}^2$			$298760$

  $\begin{array}{l}=\sigma =\sqrt{\frac{1}{6}\left(298760\right)}\\ =\sigma =\sqrt{49793.33333}\\ =\sigma =223.1441\end{array}$

The mean deviation is −

$X_i$	$\overline{X}$	$X_i-\overline{X}$
152	381	$\begin{array}{l}=\left|152-381\right|\\ =229\end{array}$
158	381	$\begin{array}{l}=\left|158-381\right|\\ =223\end{array}$
200	381	$\begin{array}{l}=\left|200-381\right|\\ =181\end{array}$
476	381	$\begin{array}{l}=\left|476-381\right|\\ =95\end{array}$
579	381	$\begin{array}{l}=\left|579-381\right|\\ =198\end{array}$
721	381	$\begin{array}{l}=\left|721-381\right|\\ =340\end{array}$
${\displaystyle \sum _{i=1}^6\left|X_i-\overline{X}\right|}$		$1266$

  $\begin{array}{l}MD=\frac{1}{6}\left(1266\right)\\ =211\end{array}$

Hence, the standard deviation is $223.1441$ and mean deviation is $=211$ .