\begin{tabular}{l} 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 \\ \( \begin{array}{lllllllll}71 & 35 & 59 & 68 & 26 & 44 & 66 & 63 & 4\end{array} 6 \) \\ \hline\end{tabular}

**Step 1. List the Data** The sample data (jersey numbers) are assumed to be \[ 71,\; 35,\; 59,\; 68,\; 26,\; 44,\; 66,\; 63,\; 4,\; 6. \] There are \( n = 10 \) data values. **Step 2. Find the Range** The range is the difference between the maximum and minimum values. Here, - Maximum = \(71\) - Minimum = \(4\) Thus, the range is \[ \text{Range} = 71 - 4 = 67. \] **Step 3. Compute the Sample Mean** The sample mean \(\bar{x}\) is given by \[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}. \] Calculating the sum: \[ 71 + 35 + 59 + 68 + 26 + 44 + 66 + 63 + 4 + 6 = 442. \] Thus, the mean is \[ \bar{x} = \frac{442}{10} = 44.2. \] **Step 4. Compute the Sample Variance** The sample variance \( s^2 \) is computed using \[ s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}. \] Calculate each squared deviation: 1. For \(71\): \(71 - 44.2 = 26.8\); \((26.8)^2 = 718.24\). 2. For \(35\): \(35 - 44.2 = -9.2\); \((-9.2)^2 = 84.64\). 3. For \(59\): \(59 - 44.2 = 14.8\); \((14.8)^2 = 219.04\). 4. For \(68\): \(68 - 44.2 = 23.8\); \((23.8)^2 = 566.44\). 5. For \(26\): \(26 - 44.2 = -18.2\); \((-18.2)^2 = 331.24\). 6. For \(44\): \(44 - 44.2 = -0.2\); \((-0.2)^2 = 0.04\). 7. For \(66\): \(66 - 44.2 = 21.8\); \((21.8)^2 = 475.24\). 8. For \(63\): \(63 - 44.2 = 18.8\); \((18.8)^2 = 353.44\). 9. For \(4\): \(4 - 44.2 = -40.2\); \((-40.2)^2 = 1616.04\). 10. For \(6\): \(6 - 44.2 = -38.2\); \((-38.2)^2 = 1459.24\). Sum all squared deviations: \[ 718.24 + 84.64 + 219.04 + 566.44 + 331.24 + 0.04 + 475.24 + 353.44 + 1616.04 + 1459.24 = 5823.6. \] Now, the sample variance is \[ s^2 = \frac{5823.6}{10 - 1} = \frac{5823.6}{9} \approx 647.07. \] **Step 5. Compute the Sample Standard Deviation** The standard deviation \( s \) is the square root of the variance: \[ s = \sqrt{647.07} \approx 25.45. \] **Interpretation of the Results** - **Range:** A range of \(67\) indicates that the jersey numbers span a wide interval, from \(4\) to \(71\). - **Variance and Standard Deviation:** The sample variance is approximately \(647.07\) and the standard deviation is about \(25.45\). This tells us that the jersey numbers deviate on average by roughly \(25\) from the mean \(44.2\). In practical terms, there is a considerable spread in the values among the players’ jersey numbers. These results highlight that the selected sample has a large dispersion around the mean.