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Q.
Consider the first $10$ positive integers. If we multiply each number by $(-1)$ and then add $1$ to each number, the variance of the numbers so obtained is
Statistics
Solution:
First ten positive integers are $1,2,3,4,5,6,7,8,9$ and 10.
Sum of these numbers $\left(\Sigma x_j\right)=1+2+\ldots+10=55$
Sum of squares of these numbers $\left(\Sigma x_i^2\right)=1^2+2^2+\ldots+10^2$ $=385$
Standard deviation $(\sigma)=\sqrt{\frac{\sum x_i^2}{n}-\left(\frac{1}{n} \Sigma x_i\right)^2}$
$= \sqrt{\frac{385}{10}-(5.5)^2}=\sqrt{38.5-30.25}=\sqrt{8.25}$
$\therefore $ Variance $\left(\sigma^2\right)=8.25$