1. **State the problem:** We are given the net worth (in billions) of a sample of the richest people: 59, 52, 28, 26, 19, 19, 18, 17, 17, 17. We need to find the range, variance, and standard deviation of this data. 2. **Formulas and important rules:** - Range = Maximum value - Minimum value - Variance $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2$ where $\bar{x}$ is the mean - Standard deviation $\sigma = \sqrt{\sigma^2}$ 3. **Calculate the range:** - Maximum value = 59 - Minimum value = 17 - Range = $59 - 17 = 42$ 4. **Calculate the mean $\bar{x}$:** $$\bar{x} = \frac{59 + 52 + 28 + 26 + 19 + 19 + 18 + 17 + 17 + 17}{10} = \frac{272}{10} = 27.2$$ 5. **Calculate each squared deviation $(x_i - \bar{x})^2$:** - $(59 - 27.2)^2 = 31.8^2 = 1011.24$ - $(52 - 27.2)^2 = 24.8^2 = 615.04$ - $(28 - 27.2)^2 = 0.8^2 = 0.64$ - $(26 - 27.2)^2 = (-1.2)^2 = 1.44$ - $(19 - 27.2)^2 = (-8.2)^2 = 67.24$ - $(19 - 27.2)^2 = 67.24$ - $(18 - 27.2)^2 = (-9.2)^2 = 84.64$ - $(17 - 27.2)^2 = (-10.2)^2 = 104.04$ - $(17 - 27.2)^2 = 104.04$ - $(17 - 27.2)^2 = 104.04$ 6. **Sum of squared deviations:** $$1011.24 + 615.04 + 0.64 + 1.44 + 67.24 + 67.24 + 84.64 + 104.04 + 104.04 + 104.04 = 2159.6$$ 7. **Calculate variance:** $$\sigma^2 = \frac{2159.6}{10} = 215.96$$ 8. **Calculate standard deviation:** $$\sigma = \sqrt{215.96} \approx 14.7$$ **Final answers:** - Range = 42 - Variance = 215.96 - Standard deviation $\approx$ 14.7