Question

Find the mean deviation about the mean for the following data: $6,7,10,12,13,4,8,12$

Answer 1

Given data: $6,7,10,12,13,4,8,12$

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

$\Rightarrow \overline{x}=\frac{1}{8}{\sum }_{i=1}^8{x}_i$

$\Rightarrow \overline{x}=\frac{(4+6+7+8+10+12+12+13)}{8}$

$\Rightarrow \overline{x}=\frac{72}{8}=9$

Mean, $\overline{x}=9$.

$\begin{array}{ccc}{x}_i& x_i-\overline{x}& |x_i-\overline{x}|\\ 4& 4-9=-5& 5\\ 6& 6-9=-3& 3\\ 7& 7-9=-2& 2\\ 8& 8-9=-1& 1\\ 10& 10-9=1& 1\\ 12& 12-9=3& 3\\ 12& 12-9=3& 3\\ 13& 13-9=4& 4\end{array}$

Mean deviation about the mean is given by:

$\begin{array}{cc}{x}_i& |x_i-\overline{x}|\\ 4& 5\\ 6& 3\\ 7& 2\\ 8& 1\\ 10& 1\\ 12& 3\\ 12& 3\\ 13& 4\end{array}$

$M.D.\left(\overline{x}\right)=\frac{1}{n}{\sum }_{i=1}^n|x_i-\overline{x}|$
$=\frac{1}{8}{\sum }_{i=1}^8|x_i-\overline{x}|$

$=\frac{1}{8}(5+3+2+1+1+3+3+4)$

$=\frac{1}{8}\times 22=2.75$