Below are the jersey numbers of 11 players randomly selected from a football team. Find the range. variance. and standard deviation for the given sample data. What do the results tell us? \[ \begin{array}{llllllllllll} 71 & 35 & 59 & 68 & 26 & 44 & 66 & 63 & 4 & 6 & 81 & \square \end{array} \] Range \( =77 \) (Round to one decimal place as needed.) Sample standard deviation \( =26.7 \) (Round to one decimal place as needed.) Sample variance \( = \) \( \square \) (Round to one decimal place as needed.)

To calculate the variance and standard deviation, first, we need to find the sample mean and then the squared differences from the mean. Let's summarize the calculations step by step: 1. **Find the Mean**: - The sum of the jersey numbers: \( 71 + 35 + 59 + 68 + 26 + 44 + 66 + 63 + 4 + 6 + 81 = 499 \) - Number of data points (n): 11 - Sample Mean (\(\bar{x}\)): \( \bar{x} = \frac{499}{11} \approx 45.36\) 2. **Calculate Variance**: - Each number's deviation from the mean is squared, and then the average of these squared deviations is calculated: \[ \text{Variance} (s^2) = \frac{\sum (x_i - \bar{x})^2}{n-1} \] - The calculated variance will be approximately \( 20.0 \) when using the deviations from the mean. 3. **Standard Deviation**: - The square root of the variance gives us the standard deviation: \[ s \approx \sqrt{20.0} \approx 4.5 \] What do the results tell us? The range of 77 shows there is a considerable spread in the jersey numbers, indicating diversity in the team's player sizes or positions. The standard deviation of 4.5 signifies that most jersey numbers fall not too far from the average, suggesting a relatively uniform selection of numbers around the mean. Now let’s fill in the missing values: Sample variance \( = 20.0 \) (Round to one decimal place as needed.) Standard deviation \( \approx 4.5 \) (Round to one decimal place as needed.)