Question

Find the mean deviation about the mean for the following data: $12,3,18,17,4,9,17,19,20,15,8,17,2,3,16,11,3,1,0,5$

Answer 1

Given data: $12,3,18,17,4,9,17,19,20,15,8,17,2,3,16,11,3,1,0,5$

Mean $\overline{x}=\frac{1}{n}{\sum }_{i=1}^n{x}_i$

$\Rightarrow \overline{x}=\frac{1}{20}{\sum }_{i=1}^{20}{x}_i$

$\Rightarrow \overline{x}=\frac{\left(\begin{array}{c}12+3+18+17+4+9+17+19+20+15\\ +8+17+2+3+16+11+3+1+0+5\end{array}\right)}{20}$

$=\frac{200}{20}$

$\Rightarrow \overline{x}=10$

Finding deviations of the respective observations from the mean $\overline{x}$.

$\begin{array}{cccccccc}{x}_i& 12& 3& 18& 17& 4& 9& 17\\ x_i-\overline{x}& 2& -7& 8& 7& -6& -1& 7\end{array}$

$\begin{array}{cccccccc}{x}_i& 19& 20& 15& 8& 17& 2& 3\\ x_i-\overline{x}& 9& 10& 5& -2& 7& -8& -7\end{array}$

$\begin{array}{ccccccc}{x}_i& 16& 11& 3& 1& 0& 5\\ x_i-\overline{x}& 6& 1& -7& -9& -10& -5\end{array}$


Now,
$|x_i-\overline{x}|=2,7,8,7,6,1,7,9,10,5,2,7,8,7,6,1,7,9,10,5$

Mean deviation about the mean is given by:

M.D. $\left(\overline{x}\right)=\frac{1}{n}{\sum }_{i=1}^n|x_i-\overline{x}|$

$=\frac{\left(\begin{array}{c}2+7+8+7+6+1+7+9+10+5\\ +2+7+8+7+6+1+7+9+10+5\end{array}\right)}{20}$

$=\frac{1}{20}\times 124$

$=6.2$