Question

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

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

Answer 1

Answer: (B)

Mean of first $50$ even natural numbers
i.e. $\bar{x}=\frac{2+4+6+\dots ..+98+100}{50}$
$\Rightarrow \bar{x}=\frac{2(1+2+3+\dots +49+50)}{50}$
$\Rightarrow \bar{x}=\frac{2\times 50\times 51}{2\times 50}=51$
$\therefore {\sigma }^2=\frac{\Sigma x_i^2}{n}-(\bar{x}{)}^2$
$=\frac{\left(2^2+4^2+6^2+\dots +98^2+100^2\right)}{50}-(51{)}^2$
$=\frac{2^2\left(1^2+2^2+3^2+\dots +49^2+50^2\right)-(51{)}^2\times 50}{50}$
$=\frac{4(50)(51)(101)-(51)\times (51)\times 50\times 6}{6\times 50}$
$=\frac{50\times 51(404-306)}{6\times 50}=\frac{51\times 98}{6}=833$