Amanda Boleyn, an entrepreneur who recently sold her start-up for a multi-million-dollar sum, is looking for alternate investments for her newfound fortune. She is considering an investment in wine, similar to how some people invest in rare coins and fine art. To educate herself on the properties of fine wine, she has collected data on 13 different characteristics of 178 wines. Amanda has applied k-means clustering to this data for k52, 3, and 4 and provided the summaries for each set of resulting clusters. Which value of k is the most appropriate to categorize these wines? Justify your choice with calculations.

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Inter-Cluster Distances Cluster 1 Cluster 2 Cluster 1 3.829 Cluster 2 3.829 Within-Cluster Summary Average Distance Size Cluster 1 94 3.080 Cluster 2 84 2.746 Total 178 2.922 Inter-Cluster Distances Cluster 1 Cluster 2 Cluster 3 Cluster 1 5.005 3.576 Cluster 2 5.005 3.951 Cluster 3 3.576 3.951 Within-Cluster Summary Size Average Distance Cluster 1 63 2.357 Cluster 2 51 2.438 Cluster 3 64 2.765 Total 178 2.527