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
Options
8.25
6.5
3.87
2.87
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
\[= \frac{\sum_{} x_i^2}{10} - \left( \frac{\sum_{} x_i}{10} \right)^2 \]
\[ = \frac{285}{10} - \left( \frac{- 45}{10} \right)^2 \]
\[ = 28 . 5 - 20 . 25\]
\[ = 8 . 25\]
