1. Find the range, standard deviation, and variance for the sample data: 3, 4, 6, 7, 12, 2, 1. 2. Find the standard deviation for the population data: 4, 9, 12, 16, 17, 20. --- ### Problem 1: Sample data analysis 1. **Range**: The range is the difference between the maximum and minimum values. $$\text{Range} = 12 - 1 = 11$$ 2. **Mean calculation:** $$\bar{x} = \frac{3 + 4 + 6 + 7 + 12 + 2 + 1}{7} = \frac{35}{7} = 5$$ 3. **Variance for sample:** $$s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$$ Calculations of squared deviations: $$(3-5)^2 = 4, (4-5)^2 = 1, (6-5)^2 = 1, (7-5)^2 = 4, (12-5)^2 = 49, (2-5)^2 = 9, (1-5)^2 = 16$$ Sum of squared deviations: $$4 + 1 + 1 + 4 + 49 + 9 + 16 = 84$$ Variance: $$s^2 = \frac{84}{6} = 14$$ 4. **Standard deviation:** $$s = \sqrt{14} \approx 3.7$$ --- ### Problem 2: Population standard deviation 1. **Mean calculation:** $$\mu = \frac{4 + 9 + 12 + 16 + 17 + 20}{6} = \frac{78}{6} = 13$$ 2. **Variance for population:** $$\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2$$ Squared deviations: $$(4 - 13)^2 = 81, (9 - 13)^2 = 16, (12 - 13)^2 = 1, (16 - 13)^2 = 9, (17 - 13)^2 = 16, (20 - 13)^2 = 49$$ Sum: $$81 + 16 + 1 + 9 + 16 + 49 = 172$$ Variance: $$\sigma^2 = \frac{172}{6} \approx 28.7$$ 3. **Standard deviation:** $$\sigma = \sqrt{28.7} \approx 5.4$$