Question

If mean and standard deviation of $5$ observations $x_1,x_2,x_3,x_4,x_5$ are $10$ and $3$, respectively, then the variance of $6$ observations $x_1,x_2,\cdots ,x_5$ and $-50$ is equal to :

• A. $507.5$
• B. $582.5$
• C. $509.5$
• D. $586.5$

Answer 1

The correct option is A $507.5$

$\sum x_i=50$
S.D.$=3$
$\Rightarrow {\sigma }^2=\frac{1}{5}[\sum x_i^2-\frac{1}{5}(\sum x_i{)}^2]=3^2$
$\Rightarrow [\sum x_i^2-\frac{1}{5}(50{)}^2]=45$
$\Rightarrow \sum x_i^2=545$
Now, the new variance is-
${\sigma }_N^2=\frac{1}{6}[\sum x_i^2+(-50{)}^2-\frac{1}{6}(\sum x_i-50{)}^2]$
$=\frac{1}{6}[545+(-50{)}^2-\frac{1}{6}(50-50{)}^2]$
$=507.5$