Question

Find the mean deviation about the mean for the following data:
$\begin{array}{ccccccc}{x}_i& 2& 5& 6& 8& 10& 12\\ f_i& 2& 8& 10& 7& 8& 5\end{array}$

Answer 1

$\begin{array}{ccc}{x}_i& f_i& x_i{f}_i\\ 2& 2& 2.2=4\\ 5& 8& 5.8=40\\ 6& 10& 6.10=60\\ 8& 7& 8.7=56\\ 10& 8& 10.8=80\\ 12& 5& 12.5=60\\ & \sum f_i=40& \sum x_i.f_i=300\end{array}$

Mean $\overline{x}=\frac{\sum x_i.f_i}{\sum f_i}$

$\Rightarrow \overline{x}=\frac{300}{40}$

$\Rightarrow \overline{x}=7.5$

$\begin{array}{cccc}{x}_i& f_i& |x_i-\overline{x}|& f_i|x_i-\overline{x}|\\ 2& 2& 5.5& 11\\ 5& 8& 2.5& 20\\ 6& 10& 1.5& 15\\ 8& 7& 0.5& 3.5\\ 10& 8& 2.5& 20\\ 12& 5& 4.5& 22.5\\ & \sum f_i=40& & \sum f_i|x_i-\overline{x}|=92\end{array}$

$\sum f_i=40,\sum f_i|x_i-\overline{x}|=92$

Mean deviation about mean $=\frac{\sum f_i|x_i-\overline{x}|}{\sum f_i}$

$M.D.\left(\overline{x}\right)=\frac{92}{40}=2.3$