Question

Variance of first $20$ natural number is

• A. $\frac{133}{4}$
• B. $\frac{379}{12}$
• C. $\frac{133}{2}$
• D. $\frac{399}{4}$

Answer 1

The correct option is A $\frac{133}{4}$

Variance of first $n$ natural numbers is $=\frac{\sum (x_i{)}^2}{n}-(Mean{)}^2$

$=\frac{1^2+2^2+3^2.....n^2}{n}-(\frac{n+1}{2}{)}^2$

${\sigma }^2=\frac{n(n+1)(2n+1)}{6n}-\frac{(n+1{)}^2}{4}$

${\sigma }^2=\frac{n^2-1}{12}$

So variance of first $20$ natural number

$={\sigma }^2=\frac{{20}^2-1}{12}=\frac{399}{12}=\frac{133}{4}$