The numbers of regular season wins for 10 football teams in a given season are given below. Determine the range, mean, variance, and standard deviation of the population data set. \( 2,8,15,3,14,10,13,9,4,7 \)

To calculate the range, mean, variance, and standard deviation of the given data set, we will follow these steps: ### Given Data The numbers of regular season wins are: \[ 2, 8, 15, 3, 14, 10, 13, 9, 4, 7 \] ### Step 1: Calculate the Range The range is calculated as: \[ \text{Range} = \text{Maximum} - \text{Minimum} \] - Maximum = 15 - Minimum = 2 \[ \text{Range} = 15 - 2 = 13 \] ### Step 2: Calculate the Mean The mean is calculated as: \[ \text{Mean} = \frac{\sum x_i}{N} \] where \( N \) is the number of data points. Calculating the sum: \[ \sum x_i = 2 + 8 + 15 + 3 + 14 + 10 + 13 + 9 + 4 + 7 = 91 \] Number of data points \( N = 10 \). Now, calculate the mean: \[ \text{Mean} = \frac{91}{10} = 9.1 \] ### Step 3: Calculate the Variance The variance is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \] where \( \mu \) is the mean. First, we calculate \( (x_i - \mu)^2 \) for each data point: - For \( 2 \): \( (2 - 9.1)^2 = (-7.1)^2 = 50.41 \) - For \( 8 \): \( (8 - 9.1)^2 = (-1.1)^2 = 1.21 \) - For \( 15 \): \( (15 - 9.1)^2 = (5.9)^2 = 34.81 \) - For \( 3 \): \( (3 - 9.1)^2 = (-6.1)^2 = 37.21 \) - For \( 14 \): \( (14 - 9.1)^2 = (4.9)^2 = 24.01 \) - For \( 10 \): \( (10 - 9.1)^2 = (0.9)^2 = 0.81 \) - For \( 13 \): \( (13 - 9.1)^2 = (3.9)^2 = 15.21 \) - For \( 9 \): \( (9 - 9.1)^2 = (-0.1)^2 = 0.01 \) - For \( 4 \): \( (4 - 9.1)^2 = (-5.1)^2 = 26.01 \) - For \( 7 \): \( (7 - 9.1)^2 = (-2.1)^2 = 4.41 \) Now, sum these squared differences: \[ \sum (x_i - \mu)^2 = 50.41 + 1.21 + 34.81 + 37.21 + 24.01 + 0.81 + 15.21 + 0.01 + 26.01 + 4.41 = 194.1 \] Now, calculate the variance: \[ \sigma^2 = \frac{194.1}{10} = 19.41 \] ### Step 4: Calculate the Standard Deviation The standard deviation is the square root of the variance: \[ \sigma = \sqrt{\sigma^2} = \sqrt{19.41} \approx 4.41 \] ### Summary of Results - **Range**: 13 - **Mean**: 9.1 - **Variance**: 19.41 - **Standard Deviation**: 4.41 These are the calculated statistics for the given data set.