1. **State the problem:** Calculate the range, variance, and standard deviation for the given data set: $$\{10, 20, 5, 13, 14, 8, 10, 17, 15, 9, 12, 19, 16, 18\}$$ 2. **Calculate the range:** The range is the difference between the maximum and minimum values. $$\text{Range} = \max(x) - \min(x)$$ From the data, $\max(x) = 20$ and $\min(x) = 5$. $$\text{Range} = 20 - 5 = 15$$ 3. **Calculate the mean ($\bar{x}$):** $$\bar{x} = \frac{\sum x_i}{n} = \frac{10 + 20 + 5 + 13 + 14 + 8 + 10 + 17 + 15 + 9 + 12 + 19 + 16 + 18}{14}$$ Calculate the sum: $$10 + 20 + 5 + 13 + 14 + 8 + 10 + 17 + 15 + 9 + 12 + 19 + 16 + 18 = 186$$ So, $$\bar{x} = \frac{186}{14} = 13.2857$$ 4. **Calculate the variance ($s^2$):** Variance formula for a sample: $$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$ Calculate each squared deviation: $(10 - 13.2857)^2 = 10.78$ $(20 - 13.2857)^2 = 45.02$ $(5 - 13.2857)^2 = 68.37$ $(13 - 13.2857)^2 = 0.08$ $(14 - 13.2857)^2 = 0.51$ $(8 - 13.2857)^2 = 27.95$ $(10 - 13.2857)^2 = 10.78$ $(17 - 13.2857)^2 = 13.80$ $(15 - 13.2857)^2 = 2.94$ $(9 - 13.2857)^2 = 18.37$ $(12 - 13.2857)^2 = 1.65$ $(19 - 13.2857)^2 = 32.66$ $(16 - 13.2857)^2 = 7.37$ $(18 - 13.2857)^2 = 22.29$ Sum of squared deviations: $$10.78 + 45.02 + 68.37 + 0.08 + 0.51 + 27.95 + 10.78 + 13.80 + 2.94 + 18.37 + 1.65 + 32.66 + 7.37 + 22.29 = 262.57$$ Divide by $n-1 = 13$: $$s^2 = \frac{262.57}{13} = 20.20$$ 5. **Calculate the standard deviation ($s$):** $$s = \sqrt{s^2} = \sqrt{20.20} = 4.49$$ 6. **Interpretation:** - The range of 15 shows the spread between the smallest and largest values. - The variance of 20.20 indicates the average squared deviation from the mean. - The standard deviation of 4.49 tells us that on average, data points deviate from the mean by about 4.49 units. This helps understand the variability in the data set practically.