1. **Problem Statement:** Calculate the range, interquartile range, mean deviation about the mean, standard deviation, coefficient of range, coefficient of mean deviation from the mean, and coefficient of variation for the ages of 20 students: 13, 12, 17, 17, 13, 14, 19, 16, 11, 10, 11, 14, 16, 17, 15, 16, 17, 12, 12, 18. 2. **Step 1: Organize the data in ascending order:** $$10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 17, 18, 19$$ 3. **Step 2: Calculate the range:** Range = Maximum value - Minimum value $$= 19 - 10 = 9$$ 4. **Step 3: Calculate the interquartile range (IQR):** - Find the first quartile $Q_1$ (25th percentile) and third quartile $Q_3$ (75th percentile). - Since $n=20$, $Q_1$ is the average of 5th and 6th values, $Q_3$ is the average of 15th and 16th values. $$Q_1 = \frac{12 + 12}{2} = 12$$ $$Q_3 = \frac{17 + 17}{2} = 17$$ Interquartile range: $$IQR = Q_3 - Q_1 = 17 - 12 = 5$$ 5. **Step 4: Calculate the mean ($\bar{x}$):** $$\bar{x} = \frac{\sum x_i}{n} = \frac{13+12+17+17+13+14+19+16+11+10+11+14+16+17+15+16+17+12+12+18}{20}$$ Calculate sum: $$= 13+12+17+17+13+14+19+16+11+10+11+14+16+17+15+16+17+12+12+18 = 305$$ Mean: $$\bar{x} = \frac{305}{20} = 15.25$$ 6. **Step 5: Calculate mean deviation about the mean:** Formula: $$MD = \frac{\sum |x_i - \bar{x}|}{n}$$ Calculate each absolute deviation and sum: $$|13-15.25|+|12-15.25|+...+|18-15.25| = 54.5$$ Mean deviation: $$MD = \frac{54.5}{20} = 2.725$$ 7. **Step 6: Calculate standard deviation (SD):** Formula: $$SD = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$$ Calculate squared deviations sum: $$\sum (x_i - 15.25)^2 = 95.75$$ Standard deviation: $$SD = \sqrt{\frac{95.75}{20}} = \sqrt{4.7875} \approx 2.187$$ 8. **Step 7: Calculate coefficient of range:** Formula: $$\text{Coefficient of range} = \frac{\text{Range}}{\text{Sum of max and min}} = \frac{9}{19+10} = \frac{9}{29} \approx 0.3103$$ 9. **Step 8: Calculate coefficient of mean deviation from the mean:** Formula: $$\text{Coefficient of MD} = \frac{MD}{\bar{x}} = \frac{2.725}{15.25} \approx 0.1787$$ 10. **Step 9: Calculate coefficient of variation:** Formula: $$\text{Coefficient of variation} = \frac{SD}{\bar{x}} = \frac{2.187}{15.25} \approx 0.1435$$ **Final answers:** - Range = 9 - Interquartile range = 5 - Mean deviation about the mean = 2.725 - Standard deviation = 2.187 - Coefficient of range = 0.3103 - Coefficient of mean deviation from the mean = 0.1787 - Coefficient of variation = 0.1435