Question

In any discrete series (when all values are not same) the relationship between M.D. about mean and S.D. is :

• A. M.D.$=$S.D.
• B. M.D.$\ge$S.D.
• C. M.D.
• D. M.D.$\le$S.D.

Answer 1

The correct option is D M.D.$\le$S.D.

Let $\frac{x_i}{f_i}$;$i=1,2,...,n$
be the frequency distribution
Then, $S.D.=\sqrt{\frac{1}{N}{\sum }_{i=1}^n{f}_i{(x_i-\overline{X})}^2}$
and $M.D.=\frac{1}{N}\sum _{i=1}^n{f}_i|x_i-\overline{X}|$
Let $|x_i-\overline{X}|=2i$;$i=1,2,...,n$
Then ${(S.D.)}^2-{(M.D.)}^2=\frac{1}{N}\sum _{i=1}^n{f}_i{z_i}^2-{\left[\frac{1}{N}\sum _{i=1}^n{f}_i{z}_i\right]}^2$
$={{\sigma }_z}^2\ge 0\Rightarrow S.D.\ge M.D.$