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Solve this following

Question:

If mean and standard deviation of 5 observations $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \mathrm{x}_{4}, \mathrm{x}_{5}$ are 10 and 3 , respectively, then the variance of 6 observations $\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots, \mathrm{x}_{5}$ and $-50$ is equal to :

  1. $582.5$

  2. $507.5$

  3. $586.5$

  4. $509.5$


Correct Option: , 2

Solution:

$\bar{x}=10 \Rightarrow \sum_{i=1}^{5} x_{i}=50$

S.D. $=\sqrt{\frac{\sum_{i=1}^{5} \mathrm{x}_{\mathrm{i}}^{2}}{5}-(\overline{\mathrm{x}})^{2}}=8$

$\Rightarrow \sum_{i=1}^{5}\left(x_{i}\right)^{2}=109$

variance $=\frac{\sum_{i=1}^{5}\left(x_{i}\right)^{2}+(-50)^{2}}{6}-\left(\sum_{i=1}^{5} \frac{x_{i}-50}{6}\right)$

$=507.5$

Option (2)

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