Let `x_1, x_2, x_3, ,` be `n` values of a variable X. If these values are changed to `x_1+a ,x+2+a , , x_n+a ,w h e r ea in R ,` show that the variance remains unchanged.

Explanation

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

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

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

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

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

Newvariance= ` frac{-1}{n}[bar{x}+a-(x_1+a)]^{2}+[x+a-(x_{2}+a]^{2}.`

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

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

...