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 correct option is A

8.25

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,

$\sum x_i=0+(-1)+(-2)+(-3)+(-4)+(-5)+(-6)+(-7)+(-8)+(-9)=-45$

$\sum x_i^2=0+1+4+9+16+25+36+49+64+81=225$

$\therefore$ Variance of the obtained numbers.

$=\frac{\sum x_i^2}{10}-{\left(\frac{\sum x_i}{10}\right)}^2$

$=\frac{285}{10}-{\left(\frac{-45}{10}\right)}^2$

$=28.5-20.25=8.25$

Hence, the correct answer is option (a).