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

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