If both the mean and the standard deviation of 50 observations x1, x2,..., x50 are equal t…

Question

If both the mean and the standard deviation of 50 observations x1, x2,..., x50 are equal to 16, then the mean of (x1 -- 4)2
, (x2 -- 4)2
,....., (x50 -- 4)2
 is :

Accepted Answer

$$Mean(\mu ) = {{\sum {{x_i}} } \over {50}} = 16$$

$$ \therefore $$ $$\sum {{x_i}}  = 16 \times 50$$

$$S.D.\left( \sigma  \right) = \sqrt {{{\sum {{x_i}^2} } \over {50}} - {{\left( \mu  \right)}^2}}  = 16$$

$$ \Rightarrow {{\sum {{x_i}^2} } \over {50}} = 256 \times 2$$

Required mean = $${{\sum {{{\left( {{x_i} - 4} \right)}^2}} } \over {50}}$$

$$ \Rightarrow {{\sum {{x_i}^2}  + 16 \times 50 - 8\sum {{x_i}} } \over {50}}$$

$$ \Rightarrow $$ 256 $$ \times $$ 2 + 16 - 8 $$ \times $$ 16

$$ \Rightarrow $$ 400

Solutions · 1