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

Answer 1

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

$$\begin{gathered} =\frac{\sum _{}{x}_i^2}{10}-{\left(\frac{\sum _{}{x}_i}{10}\right)}^2 \\ =\frac{285}{10}-{\left(\frac{-45}{10}\right)}^2 \\ =28.5-20.25 \\ =8.25 \end{gathered}$$

Hence, the correct answer is option (a).