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 $x_1{x}_2,...,x_5$ and $-50$ is equal

Answer 1

Answer: 507.5

$\because \bar{x}=\frac{{\sum }_{i=1}^5{x}_i}{5}\Rightarrow {\sum }_{i=1}^5{x}_i=10\times 5=50\Rightarrow {\sum }_{i=1}^{6}xi=50-50=0\frac{{\sum }_{i=1}^5{x}_i^2}{5}-(10{)}^2=3^2=9$
$\Rightarrow {\sum }_{i=1}^5{x}_i^2=545$
Then, $\Rightarrow {\sum }_{i=1}^6{x}_i^2={\sum }_{i=1}^5{x}_i^2+(-50{)}^2=545+(-50{)}^2=3045$ Variance
$=\frac{{\sum }_{i=1}^6{x}_i^2}{6}-{\left(\frac{{\sum }_{i=1}^{6}xi}{6}\right)}^2=\frac{3045}{6}-0=507.5$
Variance
$=\frac{{\sum }_{i=1}^6{x}_i^2}{6}-{\left(\frac{{\sum }_{i=1}^{6}xi}{6}\right)}^2=\frac{3045}{6}-0=507.5$