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.