Calculate the range, population variance, and population standard deviation for the following data set. If necessary, round to one more decimal place than the largest number of decimal places given in the data. \[ 14,12,20,19,11,13,10,8,18,15 \] AnswerHow to enter your answer (opens in new window) 2 Points Population Standard Deviation:

To calculate the range, population variance, and population standard deviation for the data set \( 14, 12, 20, 19, 11, 13, 10, 8, 18, 15 \): 1. **Range**: The range is found by subtracting the smallest number from the largest number in the data set. Here, the largest number is 20, and the smallest is 8. So, the range is \( 20 - 8 = 12 \). 2. **Population Variance**: The population variance is calculated as the average of the squared differences from the mean. First, find the mean: \[ \text{Mean} = \frac{14 + 12 + 20 + 19 + 11 + 13 + 10 + 8 + 18 + 15}{10} = \frac{ 20 }{10} = 15. \] Next, calculate each squared difference from the mean, sum them, and divide by the number of data points (10): \[ \text{Variance} = \frac{(14-15)^2 + (12-15)^2 + (20-15)^2 + (19-15)^2 + (11-15)^2 + (13-15)^2 + (10-15)^2 + (8-15)^2 + (18-15)^2 + (15-15)^2}{10}. \] Solving the above gives you a variance of \( 9.2 \). 3. **Population Standard Deviation**: The population standard deviation is the square root of the variance: \[ \sigma = \sqrt{9.2} \approx 3.0. \] So, the answers are: - **Range**: 12 - **Population Variance**: 9.2 - **Population Standard Deviation**: 3.0