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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
Similar Questions
Let the mean and variance of the frequency distribution
$\mathrm{x}$ | $\mathrm{x}_{1}=2$ | $\mathrm{x}_{2}=6$ | $\mathrm{x}_{3}=8$ | $\mathrm{x}_{4}=9$ |
$\mathrm{f}$ | $4$ | $4$ | $\alpha$ | $\beta$ |
be $6$ and $6.8$ respectively. If $x_{3}$ is changed from $8$ to $7 ,$ then the mean for the new data will be: