Gujarati
13.Statistics
medium

The variance of the first $n$ natural numbers is

A

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

B

$\frac{{{n^2} - 1}}{6}$

C

$\frac{{{n^2} + 1}}{6}$

D

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

Solution

(a) Variance $ = {({\rm{S}}{\rm{.D}}{\rm{.}})^2}$$ = \frac{1}{n}\Sigma {x^2} – {\left( {\frac{{\Sigma x}}{n}} \right)^2}$,$\left( {\because \;\;\bar x = \frac{{\Sigma x}}{n}} \right)$

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

$= \frac{{{n^2} – 1}}{{12}}$.

Standard 11
Mathematics

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