Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 8 / b

Every Euclidean distance obeys the triangle inequality. The three recorded entries $d_{14}=0.67$, $d_{16}=2.88$ and $d_{46}=4.37$ violate it:At least one of $0.67,2.88,4.37$ must be incorrect. Two are in the final row as required. The gap is far larger than any possible rounding of entries to two decimal places.

Complete-linkage clustering defines the dissimilarity between clusters in cluster analysis $A,B$ as $d_{\mathrm{CL}}(A,B)={\max }_{i\in A,j\in B}{d}_{ij}$. Remove point 6 and first merge 1 with 4 at height $0.67$. The distances from $\{1,4\}$ to 2, 3, 5 are respectivelyThe next merge therefore attaches 2 at $1.41$. Now the distances from $\{1,2,4\}$ to 3 and 5 are $2.95$ and $3.32$, whereas $d_{35}=2.82$. Merge 3 and 5 at $2.82$. The final distance between the two remaining clusters in cluster analysis is the largest cross-entry, $3.32$. Thus the dendrogram has merges

Cutting strictly between heights $2.82$ and $3.32$ gives the two clusters in cluster analysisThe figure is an original drawing of these computed merges.