Mean Square Deviation Formula and Example

Here you will learn mean square deviation formula and relation between mean square deviation and variance with example.

Let’s begin –

Mean Square Deviation Formula

The mean square deviation of a distribution is the mean of the square of deviations of variate from assumed mean. It is denoted by $S^2$.

Hence $S^2$ = $\sum{x_i – a}^2\over n$ = $\sum{d_i}^2\over n$      (for ungrouped dist.)

$S^2$ = $\sum{x_i – a}^2\over N$ = $\sum{f_id_i}^2\over N$    (for frequency dist.),    where $d_i$ = $x_i – a$

Relation between variance and mean square deviation

$\because$    ${\sigma}^2$ = $\sum{f_id_i}^2\over N$ – $({\sum f_i{d_i}\over N})^2$

$\implies$    ${\sigma}^2$ = $s^2$ – $d^2$,    where d = $\bar{x} – a$ = ${\sum f_i{d_i}\over N}$

$\implies$    $s^2$ = ${\sigma}^2$ + $d^2$,    $\implies$ $s^2$ $\geq$ ${\sigma}^2$

Hence the variance is the minimum value of mean square deviation of a distribution.

Example : Find the variance of the following freq. dist.

class	0 – 2	2 – 4	4 – 6	6 – 8	8 – 10	10 – 12
$f_i$	2	7	12	19	9	1

Solution : Let a = 7 and h = 2

class	$x_i$	$f_i$	$u_i$ = $x_i – a\over h$	$f_iu_i$	$f_iu_i^2$
0 – 2	1	2	-3	-6	18
2 – 4	3	7	-2	-14	28
4 – 6	5	12	-1	-12	12
6 – 8	7	19	0	0	0
8 – 10	9	9	1	9	9
10 – 12	11	1	2	2	4
		N = 50		$\sum{f_iu_i}$ = -21	$\sum{f_iu_i^2}$ = 71

$\because$     ${\sigma^2}$ = $h^2$[$\sum f_i{u_i}^2\over n$ – $({\sum f_i{u_i}\over n})^2$]

= 4[$71\over 50$ – (${-21\over 50}^2$)]

= 4[1.42 – 0.1764] = 4.97

Mathematical Properties of Variance

(i) Var.$(x_i + p$) = Var.($x_i$)

(ii) Var.$(px_i$) = $p^2$Var.($x_i$)

(iii) Var$(ax_i + b$) = $a^2$.Var($x_i$)

where p, a, b are constants.