Variance of first n natural numbers is qquad | vedclass

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Question

(D) The variance $\sigma^2$ of the first $n$ natural numbers is given by the formula $\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} x_i^2 - (\bar{x})^2$.Here,

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

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(D) The variance $\sigma^2$ of the first $n$ natural numbers is given by the formula $\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} x_i^2 - (\bar{x})^2$.Here,$\sum x_i^2 = \frac{n(n+1)(2n+1)}{6}$ and $\bar{x} = \frac{n+1}{2}$.Substituting these values:$\sigma^2 = \frac{1}{n} \left( \frac{n(n+1)(2n+1)}{6} \right) - \left( \frac{n+1}{2} \right)^2$$\sigma^2 = \frac{(n+1)(2n+1)}{6} - \frac{(n+1)^2}{4}$Taking $\frac{n+1}{2}$ as a common factor:$\sigma^2 = \frac{n+1}{2} \left( \frac{2n+1}{3} - \frac{n+1}{2} \right)$$\sigma^2 = \frac{n+1}{2} \left( \frac{4n+2 - 3n-3}{6} \right)$$\sigma^2 = \frac{n+1}{2} \left( \frac{n-1}{6} \right)$$\sigma^2 = \frac{n^2-1}{12}$

Class	$0-10$	$10-20$	$20-30$	$30-40$
Frequency	$1$	$3$	$4$	$2$

$x_i$	$60$	$61$	$62$	$63$	$64$	$65$	$66$	$67$	$68$
$f_i$	$2$	$1$	$12$	$29$	$25$	$12$	$10$	$4$	$5$

Variance of first $n$ natural numbers is $\qquad$ .

Similar Questions

The $S.D.$ of $15$ items is $6$. If each item is decreased or increased by $1$,then the standard deviation will be:

What is the standard deviation of the following series?

Class $0-10$ $10-20$ $20-30$ $30-40$

Frequency $1$ $3$ $4$ $2$

The data is obtained in tabular form as follows:
$x_i$ $60$ $61$ $62$ $63$ $64$ $65$ $66$ $67$ $68$
$f_i$ $2$ $1$ $12$ $29$ $25$ $12$ $10$ $4$ $5$

Find the standard deviation of the given data. (in $.69$)

The coefficient of variation of the first $n$ natural numbers is

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