Question

The variance of first $50$ even natural numbers is:

• A. $\frac{833}{4}$
• B. $833$
• C. $437$
• D. $\frac{437}{4}$

Answer 1

The correct option is B $833$

Variance
$=\frac{\sum x_i^2}{n}-{\left(\frac{\sum x_i}{n}\right)}^2$
${\sigma }^2=\left(\frac{2^2+4^2+6^2+...+{100}^2}{50}\right)-{\left(\frac{2+4+6+...+100}{50}\right)}^2$
$=\left(\frac{\sum _i^{50}(2n{)}^2}{50}\right)-{\left(\frac{\sum _i^{50}\left(2n\right)}{50}\right)}^2$
$=\left(\frac{\frac{4\cdot 50\cdot 51\cdot 101}{6}}{50}\right)-{\left(\frac{\frac{50\cdot 102}{2}}{50}\right)}^2$
$=3434-2601$
$\Rightarrow {\sigma }^2=833$