Let `x_1,\ x_2,\....,\ x_n` values of variable `X` and let `'a'` be non zero real number. Then prove that the variance of the observations `a x_1,\ a x_2, ... ,\ a x_n` is `a^2` Var `(X)` . Also, find their standard deviation.

Explanation

variance `= σ^2`

`∴σ^2=1/n(sum​y_i​(x_i​−barx)^2) `....(1)

If each observation is multiplied by `a ` and the new observation are `y_i`​ then

`y_i​=ax_i​,i.e,x_i​=1/a​y_i​`

so`bary​=1/n​sum​y_i​`

`=1/n​sum​ax_i`

`​=a/n​sum​x_i`

`​=abarx`

...