Question

The standard deviation of $n$ observations $x_1,x_2,.......,x_n$ is $2$. If $\sum _{i=1}^n{x}_i=20$ and $\sum _{i=1}^n{x}_i^2=100$, then $n=$

• A. $10$ or $20$
• B. $5$ or $10$
• C. $5$ or $20$
• D. $5$ or $15$
• E. $25$

Answer 1

The correct option is C

$5$ or $20$

Explanation for the correct option:

Given, standard deviation$=2$

${\left(\mathrm{standard}\mathrm{deviation}\right)}^2=4$

$\Rightarrow \mathrm{standard}\mathrm{deviation}=\sqrt{\frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{x}}_{\mathrm{i}}^2}}{\mathrm{n}}-{\left(\frac{\underset{\mathrm{i}=1}{\overset{\mathrm{n}}{\sum {\mathrm{x}}_{\mathrm{i}}}}}{\mathrm{n}}\right)}^2}$

$$\begin{gathered} \Rightarrow \frac{100}{n}-\frac{{\left(20\right)}^2}{n^2}=4 \\ \Rightarrow \frac{100n-400}{n^2}=4 \\ \Rightarrow 4n^2=100n-400 \\ \Rightarrow n^2-25n+100=0 \\ \Rightarrow n^2-20n-5n+100=0 \\ \Rightarrow n(n-20)-5(n-20)=0 \\ \Rightarrow n=20⋮n=5 \end{gathered}$$

Hence, option (C) is the correct answer.