1. **Problem Statement:** Given a distribution, find the five-number summary, mode, midhinge, and mean using the empirical relation among mode, median, and mean. 2. **Five-number summary:** - Minimum: smallest value in the data - Q1: first quartile (25th percentile) - Median (Q2): middle value (50th percentile) - Q3: third quartile (75th percentile) - Maximum: largest value in the data 3. **Given values:** - Minimum = 11 - Maximum = 4785 - Q1 = 88 - Median (Q2) = 115 - Q3 = 133 4. **Mode and Midhinge:** - Mode is the most frequent value; given as 5 (appears 5 times) - Midhinge is the average of Q1 and Q3: $$\text{Midhinge} = \frac{Q1 + Q3}{2} = \frac{88 + 133}{2} = \frac{221}{2} = 110.5$$ 5. **Mean calculation using empirical relation:** Empirical formula: $$\text{mode} = 3 \times \text{median} - 2 \times \text{mean}$$ Rearranged to solve for mean: $$\text{mean} = \frac{3 \times \text{median} - \text{mode}}{2}$$ Substitute values: $$\text{mean} = \frac{3 \times 115 - 5}{2} = \frac{345 - 5}{2} = \frac{340}{2} = 170$$ **Final answers:** - Five-number summary: minimum = 11, Q1 = 88, median = 115, Q3 = 133, maximum = 4785 - Mode = 5 - Midhinge = 110.5 - Mean = 170