1. **Problem Statement:** Given a distribution of numbers, find the five-number summary, mode, and midhinge. 2. **Five-number summary** consists of: minimum, first quartile ($Q_1$), median ($Q_2$), third quartile ($Q_3$), and maximum. 3. **Step 1: Organize the data in ascending order.** From the given data, extract all values and sort them. 4. **Step 2: Identify minimum and maximum.** Minimum is the smallest value, maximum is the largest. 5. **Step 3: Find the median ($Q_2$).** The median is the middle value when data is ordered. 6. **Step 4: Find $Q_1$ and $Q_3$.** $Q_1$ is the median of the lower half (below median), $Q_3$ is the median of the upper half (above median). 7. **Step 5: Calculate the mode.** The mode is the value(s) that appear most frequently. 8. **Step 6: Calculate the midhinge.** The midhinge is the average of $Q_1$ and $Q_3$: $$\text{Midhinge} = \frac{Q_1 + Q_3}{2}$$ --- **Applying to the data:** - Extracted values (from the given data ignoring counts and formatting): 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,26,38,45,478,3556788999,12344568889,012344445567888899,00172235555556889,011i2222344445567788,222334557888,0146 - After cleaning and sorting (ignoring non-numeric or malformed entries like '011i...'), the sorted numeric values are approximately: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,26,38,45,146,478,222334557888,3556788999,12344568889 - Minimum = 1 - Maximum = 12344568889 - Median ($Q_2$): With 27 values, median is the 14th value: 14 - Lower half (first 13 values): 1,2,3,4,5,6,7,8,9,10,11,12,13 - Upper half (last 13 values): 15,16,17,18,26,38,45,146,478,222334557888,3556788999,12344568889 - $Q_1$ = median of lower half = 7 (7th value) - $Q_3$ = median of upper half = 45 (7th value in upper half) - Mode: From the data, 5 appears multiple times, but exact counts are unclear; assuming 5 is mode. - Midhinge = $(7 + 45)/2 = 26$ --- **Final answers:** - Five-number summary: minimum = 1, $Q_1$ = 7, median = 14, $Q_3$ = 45, maximum = 12344568889 - Mode = 5 - Midhinge = 26