Question

The variance of the ten integers 11, 12, 13, ..., 20 is:

• A. 7.64
• B. 7.82
• C. 8.16
• D. 8.25

Hint:

When calculating variance, first find the mean and then compute the sum of squared differences from the mean.

Answer 1

The Correct Option is D

The variance of a set of numbers is calculated using the formula: \[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n} \] Where: - \( x_i \) are the data points, - \( \bar{x} \) is the mean of the data, - \( n \) is the number of data points. The integers in question are 11, 12, 13, ..., 20. Let's calculate the mean first: \[ \bar{x} = \frac{11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20}{10} = \frac{145}{10} = 14.5 \] Now, calculate the sum of the squared differences from the mean: \[ \sum (x_i - \bar{x})^2 = (11 - 14.5)^2 + (12 - 14.5)^2 + \cdots + (20 - 14.5)^2 = 14.5 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 \] \[+ 12.25 + 20.25 + 30.25 = 100.5 \] Finally, divide by the number of data points (10): \[ \text{Variance} = \frac{100.5}{10} = 8.25 \] Thus, the correct answer is 8.25.