1. **State the problem:** Calculate the variance $S^2$, standard deviation $S$, and Mean Absolute Deviation (MAD) for the data set: 8, 4, 6, 2, 5. 2. **Calculate the mean (average):** $$\text{Mean} = \bar{x} = \frac{8 + 4 + 6 + 2 + 5}{5} = \frac{25}{5} = 5$$ 3. **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: $$(8-5)^2 = 3^2 = 9$$ $$(4-5)^2 = (-1)^2 = 1$$ $$(6-5)^2 = 1^2 = 1$$ $$(2-5)^2 = (-3)^2 = 9$$ $$(5-5)^2 = 0^2 = 0$$ Sum of squared deviations: $$9 + 1 + 1 + 9 + 0 = 20$$ Divide by $n-1 = 4$: $$S^2 = \frac{20}{4} = 5$$ 4. **Calculate the standard deviation $S$:** Standard deviation is the square root of variance: $$S = \sqrt{S^2} = \sqrt{5} \approx 2.236$$ 5. **Calculate the Mean Absolute Deviation (MAD):** MAD formula: $$\text{MAD} = \frac{1}{n} \sum_{i=1}^n |x_i - \bar{x}|$$ Calculate each absolute deviation: $$|8-5| = 3$$ $$|4-5| = 1$$ $$|6-5| = 1$$ $$|2-5| = 3$$ $$|5-5| = 0$$ Sum of absolute deviations: $$3 + 1 + 1 + 3 + 0 = 8$$ Divide by $n=5$: $$\text{MAD} = \frac{8}{5} = 1.6$$ **Final answers:** - Variance $S^2 = 5$ - Standard deviation $S \approx 2.236$ - Mean Absolute Deviation (MAD) = 1.6