STD 11 - 13. statistics — Maths STD 11 Science — Question

Then,${\rm{S}}{\rm{.D}}{\rm{.}} = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^n {{f_i}{{({x_i} - \bar x)}^2}} } $

and ${\rm{M}}{\rm{.D}}{\rm{.}} = \frac{1}{N}\sum\limits_{i = 1}^n {{f_i}|{x_i}} - \bar x|$

Let $|{x_i} - \bar x| = {z_i};i = 1,2,.....n$ .

Then,

$(S.D.)2 -(M.D.)2$ $ = \frac{1}{N}\sum\limits_{i = 1}^n {{f_i}z_i^2 - {{\left( {\frac{1}{N}\sum\limits_{i = 1}^n {{f_i}{z_i}} } \right)}^2}} $

$ = \sigma _z^2 \ge 0$==> S. D. $ \ge $ $M.D.$