1. **Problem Statement:** We are given five data samples with coordinates $(X, Y)$ and need to apply divisive clustering using Manhattan distance to measure similarity/dissimilarity. 2. **Manhattan Distance Formula:** The Manhattan distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$d = |x_1 - x_2| + |y_1 - y_2|$$ This distance measures how far apart points are in a grid-like path (sum of absolute differences). 3. **Calculate Manhattan distances between each pair:** - Between sample 1 $(4,4)$ and sample 2 $(8,4)$: $$|4-8| + |4-4| = 4 + 0 = 4$$ - Between sample 1 and sample 3 $(15,8)$: $$|4-15| + |4-8| = 11 + 4 = 15$$ - Between sample 1 and sample 4 $(24,4)$: $$|4-24| + |4-4| = 20 + 0 = 20$$ - Between sample 1 and sample 5 $(24,12)$: $$|4-24| + |4-12| = 20 + 8 = 28$$ - Between sample 2 $(8,4)$ and sample 3 $(15,8)$: $$|8-15| + |4-8| = 7 + 4 = 11$$ - Between sample 2 and sample 4 $(24,4)$: $$|8-24| + |4-4| = 16 + 0 = 16$$ - Between sample 2 and sample 5 $(24,12)$: $$|8-24| + |4-12| = 16 + 8 = 24$$ - Between sample 3 $(15,8)$ and sample 4 $(24,4)$: $$|15-24| + |8-4| = 9 + 4 = 13$$ - Between sample 3 and sample 5 $(24,12)$: $$|15-24| + |8-12| = 9 + 4 = 13$$ - Between sample 4 $(24,4)$ and sample 5 $(24,12)$: $$|24-24| + |4-12| = 0 + 8 = 8$$ 4. **Summary of Manhattan distances:** | Pair | Distance | |-------|----------| | 1-2 | 4 | | 1-3 | 15 | | 1-4 | 20 | | 1-5 | 28 | | 2-3 | 11 | | 2-4 | 16 | | 2-5 | 24 | | 3-4 | 13 | | 3-5 | 13 | | 4-5 | 8 | 5. **Divisive Clustering:** Start with all samples in one cluster and iteratively split the cluster with the largest dissimilarity (largest Manhattan distance) until desired clusters are formed. The largest distance is 28 (between samples 1 and 5), so the first split separates these or clusters around these points. **Final answer:** The Manhattan distances between each pair of samples are calculated as above and can be used to perform divisive clustering based on these dissimilarities.