# Write the Variance of First N Natural Numbers. - Mathematics

Write the variance of first n natural numbers.

#### Solution

​Sum of first n natural numbers

$= \frac{n\left( n + 1 \right)}{2}$
Mean, $\bar{X} = \frac{\text{ Sum of all the observations} }{\text{ Total number of observations } }$
$= \frac{\frac{n\left( n + 1 \right)}{2}}{n} = \frac{n + 1}{2}$

$\therefore \sigma^2 = \frac{\sum \left( x_i - \bar{X} \right)^2}{n} = \frac{\sum \left( x_i - \frac{n + 1}{2} \right)^2}{n}$

$= \frac{1}{n}\sum \left[ {x_i}^2 - x_i \left( n + 1 \right) + \left( \frac{n + 1}{2} \right)^2 \right]$

$= \frac{n\left( n + 1 \right)\left( 2n + 1 \right)}{6n} - \left[ \frac{n\left( n + 1 \right)}{2} \right]\left( \frac{n + 1}{n} \right) + \frac{\left( n + 1 \right)^2}{4n} \times n$

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

$= \frac{\left( n + 1 \right)\left( n - 1 \right)}{12} = \frac{\left( n^2 - 1 \right)}{12}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 32 Statistics
Q 1 | Page 49