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Answer
Since, the first $10$ positive integers are $1, 2, 3, 4, 5, 6, 7, 8, 9$ and $10.$
On multiplying each number by $-1,$ we get $-1, -2, -3, -4, -5, -6, -7, -8, -9, -10$ On adding $1$ in each number.
We get $0, -1, -2, -3, -4, -5, -6, -7, -8, -9.$
$\therefore\ \sum\text{x}_\text{i}=-\frac{9\times10}{2}=-45$
and $\sum\text{x}^2_\text{i}=0^2+(-1)^2+(-2)^2+\ ....\ +(9)^2=\frac{9\times10\times19}{6}=285$
$\text{SD}=\sqrt{\frac{285}{10}-\Big(\frac{-45}{10}\Big)^2}$
$=\sqrt{\frac{285}{10}-\frac{2025}{100}}$
$=\sqrt{\frac{2850-2025}{100}}$
$=\sqrt{8.25}$
Now, $\text{variance}=(\text{SD})^2$
$=\big(\sqrt{8.25}\big)^2$
$=8.25$
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