Question

Let $x_i(1\le i\le 10)$ be ten observations of a random variable $X$. If $\sum _{i=1}^{10}(x_i-p)=3$ and $\sum _{i=1}^{10}{(x_i-p)}^2=9$ where $0\ne p\in R$, then the standard deviation of these observations is:

• A. $\frac{7}{10}$
• B. $\frac{9}{10}$
• C. $\sqrt{\frac{3}{5}}$
• D. $\frac{4}{5}$

Answer 1

The correct option is B $\frac{9}{10}$

$S.D.=\sqrt{\frac{\sum _{i=1}^n(x_i{)}^2}{n}-{\left(\frac{\sum _{i=1}^n\left(x_i\right)}{n}\right)}^2}$
When a constant is subtracted or added from the observations, $S.D.$ remains unchanged.
$\therefore S.D.=\sqrt{\frac{\sum _{i=1}^{10}(x_i-p{)}^2}{10}-{\left(\frac{\sum _{i=1}^{10}(x_i-p)}{10}\right)}^2}$
$S.D.=\sqrt{\frac{9}{10}-{\left(\frac{3}{10}\right)}^2}$
$S.D.=\frac{9}{10}$