1. **Problem:** Find the mean absolute deviation (MAD), standard deviation (σ), and variance (σ²) of the data set \{90, 86, 81, 62, 45, 93, 100, 75\}. 2. **Step 1: Calculate the mean (average)** The mean is given by: $$\text{Mean} = \frac{\sum x_i}{n}$$ where $x_i$ are the data points and $n$ is the number of points. Calculate the sum: $$90 + 86 + 81 + 62 + 45 + 93 + 100 + 75 = 632$$ Number of data points $n = 8$ Mean: $$\frac{632}{8} = 79$$ 3. **Step 2: Calculate the Mean Absolute Deviation (MAD)** MAD is the average of the absolute deviations from the mean: $$\text{MAD} = \frac{\sum |x_i - \text{Mean}|}{n}$$ Calculate each absolute deviation: $$|90 - 79| = 11$$ $$|86 - 79| = 7$$ $$|81 - 79| = 2$$ $$|62 - 79| = 17$$ $$|45 - 79| = 34$$ $$|93 - 79| = 14$$ $$|100 - 79| = 21$$ $$|75 - 79| = 4$$ Sum of absolute deviations: $$11 + 7 + 2 + 17 + 34 + 14 + 21 + 4 = 110$$ MAD: $$\frac{110}{8} = 13.75$$ 4. **Step 3: Calculate the Variance (σ²)** Variance is the average of the squared deviations from the mean: $$\sigma^2 = \frac{\sum (x_i - \text{Mean})^2}{n}$$ Calculate each squared deviation: $$ (90 - 79)^2 = 11^2 = 121$$ $$ (86 - 79)^2 = 7^2 = 49$$ $$ (81 - 79)^2 = 2^2 = 4$$ $$ (62 - 79)^2 = 17^2 = 289$$ $$ (45 - 79)^2 = 34^2 = 1156$$ $$ (93 - 79)^2 = 14^2 = 196$$ $$ (100 - 79)^2 = 21^2 = 441$$ $$ (75 - 79)^2 = 4^2 = 16$$ Sum of squared deviations: $$121 + 49 + 4 + 289 + 1156 + 196 + 441 + 16 = 2272$$ Variance: $$\frac{2272}{8} = 284$$ 5. **Step 4: Calculate the Standard Deviation (σ)** Standard deviation is the square root of the variance: $$\sigma = \sqrt{284} \approx 16.85$$ **Final answers:** Mean Absolute Deviation = 13.75 Variance = 284 Standard Deviation = 16.85