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

First 10 positive integers are 1, 2, 3,4 ,5 ,6, 7, 8, 9, 10

on multiplying each number by – 1, we get

- 1, -2, -3, -4, -5, -6, -7, -8, -9, -10

on adding 1 to each of the number, we get

0, - 1, -2, -3, -4, -5, -6, -7, -8, -9

∴∑x_i = 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 = - 45

and

∑ x_i²= 0² + (- 1)² + (-2)² + (-3)² + (-4)² + … + (-9)²

But we know , so the above equation on applying this formula when n=9, we get

Now we know,

Substituting the corresponding values, we get

Now for variance we will square on both sides, we get

σ²=8.25

Hence the variance of the numbers so obtained is 8.25