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Question

# 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 (a) 8.25 (b) 6.5 (c) 3.87 (d) 2.87

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Solution

## The first 10 positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Multiplying each number by −1, we get −1, −2, −3, −4, −5, −6, −7, −8, −9, −10 Adding 1 to each of these numbers, we get 0, −1, −2, −3, −4, −5, −6, −7, −8, −9 Now, $\sum _{}{x}_{i}=0+\left(-1\right)+\left(-2\right)+\left(-3\right)+\left(-4\right)+\left(-5\right)+\left(-6\right)+\left(-7\right)+\left(-8\right)+\left(-9\right)=-45$ $\sum _{}{x}_{i}^{2}=0+1+4+9+16+25+36+49+64+81=285$ ∴ Variance of the obtained numbers $=\frac{\sum _{}{x}_{i}^{2}}{10}-{\left(\frac{\sum _{}{x}_{i}}{10}\right)}^{2}\phantom{\rule{0ex}{0ex}}=\frac{285}{10}-{\left(\frac{-45}{10}\right)}^{2}\phantom{\rule{0ex}{0ex}}=28.5-20.25\phantom{\rule{0ex}{0ex}}=8.25$ Hence, the correct answer is option (a).

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