Question

139. Suppose for 30 observations, the
variance is 40. If all the observations
are increased by 10, then the
variance will be: step by step explain​

Answer 1

.

$\large\underline{\sf{Solution-}}$

$\sf \: Let \: x_1,x_2,x_3,---,x_{30} \: be \: 30 \: observations.$

$\sf \: Let \: \overline{x} \: be \: the \: mean \: and \: v \: be \: the \: variance \: of \: 30 \: observations.$

We know,

Mean is given by

$\rm :\longmapsto\: \overline{x} \: = \dfrac{1}{30}\sum_{i=1}^{30} \: x_i$

and

Variance is given by

$\rm :\longmapsto\:v = \dfrac{1}{n}\sum_{i=1}^{30} { \: (x_i \: - \: \overline{x} \: )}^{2} - - - - (1)$

Now,

Each observation is increased by 10.

$\rm :\longmapsto\:x_1 + 10,x_2 + 10,x_3 + 10,---,x_{30} + 10 \: be \: 30 \: observations$

$\bf\implies \:X_i = x_i \: + \: 10 \: \: where \: i \: = \: 1 \: to \: 30$

Now mean,

$\rm :\longmapsto\: \overline{X} \: = \dfrac{1}{30}\sum_{i=1}^{30} \:( x_i + 10)$

$\rm :\longmapsto\: \overline{X} \: = \dfrac{1}{30}\sum_{i=1}^{30} \: x_i \: + \: 10$

$\bf\implies \: \overline{X} = \overline{x} + 10$

Now,

Variance, V is given by

$\rm :\longmapsto\:V = \dfrac{1}{n}\sum_{i=1}^{30} { \: (X_i \: - \: \overline{X} \: )}^{2}$

$\rm :\longmapsto\:V = \dfrac{1}{n}\sum_{i=1}^{30} { \bigg( \: (x_i + 10) \: - \: (\overline{x} + 10\: ) \bigg)}^{2}$

$\rm :\longmapsto\:V = \dfrac{1}{n}\sum_{i=1}^{30} { \bigg( \: x_i + 10 \: - \: \overline{x} - 10\: \bigg)}^{2}$

$\rm :\longmapsto\:V = \dfrac{1}{n}\sum_{i=1}^{30} { \bigg( \: x_i \: - \: \overline{x}\: \bigg)}^{2}$

$\bf\implies \:V = v$

$\bf\implies \:Variance \: remains \: unchanged$