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. $\bar{x}=\frac{2+4+6+\ldots . .+98+100}{50}$
$\Rightarrow \bar{x}=\frac{2(1+2+3+\ldots+49+50)}{50}$
$\Rightarrow \bar{x}=\frac{2 \times 50 \times 51}{2 \times 50}=51$
$\therefore \sigma^{2}=\frac{\Sigma\, x_{i}^{2}}{n}-(\bar{x})^{2}$
$=\frac{\left(2^{2}+4^{2}+6^{2}+\ldots+98^{2}+100^{2}\right)}{50}-(51)^{2}$
$=\frac{2^{2}\left(1^{2}+2^{2}+3^{2}+\ldots+49^{2}+50^{2}\right)-(51)^{2} \times 50}{50}$
$=\frac{4(50)(51)(101)-(51) \times(51) \times 50 \times 6}{6 \times 50}$
$ =\frac{50 \times 51(404-306)}{6 \times 50}=\frac{51 \times 98}{6}=833$