# 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.