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

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
$M . D . \geq S . D .$
C
$M . D . < S . D .$
D
$M . D . \leq S . D .$

Solution:

Let $x_{i} / f_{i}; i = 1, 2, \dots, n$ be a frequency

distribution. Then,

$= \sqrt{\frac{1}{N} \sum_{i = 1}^{n} f_{i} {(x_{i} - \bar{X})}^{2}}$ and $M.D. = \frac{1}{N} \sum_{i = 1}^{n} f_{i} |x_{i} - \bar{X}|$

let $|x_{i} - \bar{X}| = z_{i}; i = 1, 2, \dots, n .$ Then,

${(S.D.)}^{2} - (M.D.)^{2} = \frac{1}{N} \sum_{i = 1}^{n} f_{i} z_{i}^{2} - {(\frac{1}{N} \sum_{i = 1}^{n} f_{i} z_{i})}^{2} = σ_{z}^{2} \geq 0$

$\Rightarrow S.D. > M.D.$

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