1. **Problem statement:** We have a dataset with 10 values: 4.8, 5.2, 5.5, 6.1, 4.9, 5.0, 6.2, 5.8, 5.1, 5.7. **a) Calculate the mean and standard deviation.** 2. **Calculate the mean:** The mean is the sum of all values divided by the number of values. $$\text{Mean} = \frac{4.8 + 5.2 + 5.5 + 6.1 + 4.9 + 5.0 + 6.2 + 5.8 + 5.1 + 5.7}{10}$$ 3. Calculate the sum: $$4.8 + 5.2 + 5.5 + 6.1 + 4.9 + 5.0 + 6.2 + 5.8 + 5.1 + 5.7 = 54.3$$ 4. Calculate the mean: $$\text{Mean} = \frac{54.3}{10} = 5.43$$ 5. **Calculate the standard deviation:** First, find the squared differences from the mean for each value: $$ (4.8 - 5.43)^2 = 0.3969 $$ $$ (5.2 - 5.43)^2 = 0.0529 $$ $$ (5.5 - 5.43)^2 = 0.0049 $$ $$ (6.1 - 5.43)^2 = 0.4489 $$ $$ (4.9 - 5.43)^2 = 0.2809 $$ $$ (5.0 - 5.43)^2 = 0.1849 $$ $$ (6.2 - 5.43)^2 = 0.5929 $$ $$ (5.8 - 5.43)^2 = 0.1369 $$ $$ (5.1 - 5.43)^2 = 0.1089 $$ $$ (5.7 - 5.43)^2 = 0.0729 $$ 6. Sum the squared differences: $$0.3969 + 0.0529 + 0.0049 + 0.4489 + 0.2809 + 0.1849 + 0.5929 + 0.1369 + 0.1089 + 0.0729 = 2.281\approx 2.28$$ 7. Calculate the variance (using sample standard deviation, divide by $n-1=9$): $$\text{Variance} = \frac{2.28}{9} = 0.2533$$ 8. Calculate the standard deviation: $$\text{Standard deviation} = \sqrt{0.2533} \approx 0.503$$ **b) Calculate the range, quartiles, and interquartile range (IQR).** 9. Sort the data: $$4.8, 4.9, 5.0, 5.1, 5.2, 5.5, 5.7, 5.8, 6.1, 6.2$$ 10. Calculate the range: $$\text{Range} = 6.2 - 4.8 = 1.4$$ 11. Calculate the quartiles: - First quartile ($Q_1$) is the median of the lower half (first 5 values): 4.8, 4.9, 5.0, 5.1, 5.2 $$Q_1 = 5.0$$ - Second quartile ($Q_2$) is the median of the entire dataset (10 values): average of 5th and 6th values $$Q_2 = \frac{5.2 + 5.5}{2} = 5.35$$ - Third quartile ($Q_3$) is the median of the upper half (last 5 values): 5.5, 5.7, 5.8, 6.1, 6.2 $$Q_3 = 5.8$$ 12. Calculate the interquartile range (IQR): $$\text{IQR} = Q_3 - Q_1 = 5.8 - 5.0 = 0.8$$ **Final answers:** - Mean = 5.43 - Standard deviation $\approx$ 0.503 - Range = 1.4 - $Q_1 = 5.0$ - $Q_2 = 5.35$ - $Q_3 = 5.8$ - IQR = 0.8