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

$\text{ M.D. }=\frac{1}{n}\sum _{i=1}^{n-1} \left|x_i-\overline{x}\right|$and $S.D. =\sqrt{\frac{1}{n}\sum _{i=1}^n {\left(x_i-\overline{x}\right)}^2}$

Let ${\left.a_i=\left|x_i-\overline{x}\right|\text{ so (M.D.) }\right)}^2=$${\left(\frac{1}{n}\sum _{i=1}^n a_i\right)}^2$and $(S.D{.) }^2=$

$\frac{1}{n}\sum _{i=1}^n a_i^2$. By Cauchy Schwarz inequality

$\begin{array}{l}{\left(\sum _{i=1}^n a_i\right)}^2<n\sum _{i=1}^n a_i^2\\ \Rightarrow {\left(\frac{1}{n}\sum a_i\right)}^2<\frac{1}{n}\sum a_i^2\Rightarrow \text{ M.D. <S.D. }\end{array}$