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If all the natural numbers between 1 and 20 are multiplied by 3 , then what is the variance of the resulting series?
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The correct answer is:
$299.25$
The series is $1,2,3, \ldots \ldots .20$ Variance $(\sigma)=\frac{\Sigma x^{2}}{n}-\Sigma(\bar{x})^{2}$
$=\frac{n(n+1)(2 n+1)}{6 n}-\left(\frac{n(n+1)}{2 n}\right)^{2}$
$=\frac{(n+1)}{12}(n-1)$
$=\frac{n^{2}-1}{12}=\frac{(20)^{2}-1}{12}=\frac{399}{12}=\frac{133}{4}=33.25$
- Numbers are multiplied by 3,
wiance $(\sigma)=33.25 \times 9=299,25$
$=\frac{n(n+1)(2 n+1)}{6 n}-\left(\frac{n(n+1)}{2 n}\right)^{2}$
$=\frac{(n+1)}{12}(n-1)$
$=\frac{n^{2}-1}{12}=\frac{(20)^{2}-1}{12}=\frac{399}{12}=\frac{133}{4}=33.25$
- Numbers are multiplied by 3,
wiance $(\sigma)=33.25 \times 9=299,25$
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