1. **State the problem:** Find the standard deviation of the data set: 1, 4, 4, 7, 7, 7, 11, 11, 13, 16, 16. 2. **Formula:** The standard deviation $\sigma$ for a sample is given by: $$\sigma = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}$$ where $n$ is the number of data points, $x_i$ are the data points, and $\bar{x}$ is the mean. 3. **Calculate the mean $\bar{x}$:** $$\bar{x} = \frac{1 + 4 + 4 + 7 + 7 + 7 + 11 + 11 + 13 + 16 + 16}{11} = \frac{97}{11} \approx 8.818$$ 4. **Calculate each squared deviation $(x_i - \bar{x})^2$:** - $(1 - 8.818)^2 = 60.53$ - $(4 - 8.818)^2 = 23.23$ - $(4 - 8.818)^2 = 23.23$ - $(7 - 8.818)^2 = 3.31$ - $(7 - 8.818)^2 = 3.31$ - $(7 - 8.818)^2 = 3.31$ - $(11 - 8.818)^2 = 4.77$ - $(11 - 8.818)^2 = 4.77$ - $(13 - 8.818)^2 = 17.53$ - $(16 - 8.818)^2 = 51.58$ - $(16 - 8.818)^2 = 51.58$ 5. **Sum the squared deviations:** $$\sum (x_i - \bar{x})^2 = 60.53 + 23.23 + 23.23 + 3.31 + 3.31 + 3.31 + 4.77 + 4.77 + 17.53 + 51.58 + 51.58 = 247.86$$ 6. **Calculate the variance:** $$s^2 = \frac{247.86}{11 - 1} = \frac{247.86}{10} = 24.786$$ 7. **Calculate the standard deviation:** $$\sigma = \sqrt{24.786} \approx 4.978$$ **Final answer:** The standard deviation of the data set is approximately $4.978$.