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

mean `=bar{x}=frac{x_{1}+x_{2}+x_{3}+cdots+x}{n}`

Variance `=frac{(bar{x}-x_{1})^{2}+(bar{x}-x_{2})^{2}+cdots(bar{x}-x_{n})^{2}}{n}`

new mean `=frac{x_{1}+a+x_{2}+a+x_{3}+a+x_{n}+x^{n}}{n}`

`=frac{(x_{1}+x_{2}+ldots+x_{n})+(a+a+a ldots . . .)}{n}`

`=bar{x}+frac{a x}{x} Rightarrow barx+a`

newvariance `frac{1}{n}[bar{x}+a-(x,+a]^{2}+[x+a-(x_{2}+a)]^{2}.`

` ldots . . .+[bar{x}+a-(x_{n}+a)]^{2} `

`=frac{1}{n}(bar{x}-x_{1})^{2}+(bar x-x_{2})^{2}+ldots .+(bar{x}- x_{n})^{2}`

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