1. **Problem Statement:** Given the distribution data: 1 3 2 1 4 2 5 8 5 6 478 5 7 15 8 3556788999 26 9 12344568889 45 10 0123444445567888899 17 11 00122235555556889 38 12 01112222344445567788 18 13 222334557888 6 14 0146 2 15 5 1 16 1, find the five-number summary, the mode, and the median. 2. **Step 1: Organize the data** Extract all numbers and list them in ascending order. The data appears to be grouped by some keys but we consider all values as a single dataset: Numbers extracted: 1,1,1,1,2,2,2,3,4,5,5,5,5,6,6,7,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,14,14,14,15,15,15,16,16 3. **Step 2: Five-number summary** The five-number summary consists of: - Minimum: smallest value - Q1: first quartile (25th percentile) - Median: middle value (50th percentile) - Q3: third quartile (75th percentile) - Maximum: largest value Sorted data count $n=74$. - Minimum = 1 - Maximum = 16 Median position = $\frac{n+1}{2} = \frac{74+1}{2} = 37.5$th value Median = average of 37th and 38th values Q1 position = $\frac{n+1}{4} = \frac{75}{4} = 18.75$th value Q1 = interpolate between 18th and 19th values Q3 position = $3\times \frac{n+1}{4} = 3 \times 18.75 = 56.25$th value Q3 = interpolate between 56th and 57th values 4. **Step 3: Find values at these positions** Counting values: - Values 1 to 9 appear multiple times, so 18th and 19th values are 8 and 8 - 37th and 38th values are 10 and 10 - 56th and 57th values are 11 and 11 So: $$Q1 = 8$$ $$Median = 10$$ $$Q3 = 11$$ 5. **Step 4: Mode** Mode is the most frequent value. Counting frequencies, 8 appears most frequently. 6. **Final answers:** - Five-number summary: Minimum = 1, Q1 = 8, Median = 10, Q3 = 11, Maximum = 16 - Mode = 8 - Median = 10