Which metric is defined as the sum of squared distances between cluster centroids and the global centroid, indicating cluster separation?

Which metric is defined as the sum of squared distances between cluster centroids and the global centroid, indicating cluster separation?

• A. Within-Cluster Sum of Squares (WCSS)
• B. Total Sum of Distances
• C. Between-Clusters Sum of Squares (BCSS)
• D. Average Inter-Centroid Distance

Answer: (C)

This item tests understanding of how cluster separation is quantified in a variance-based view of clustering. The metric defined as the sum of squared distances between each cluster centroid and the global centroid is the Between-Clusters Sum of Squares. It sums how far each cluster center sits from the overall mean, weighted by cluster size, typically expressed as BCSS = sum_k n_k * ||c_k − c_T||^2, where c_k is a cluster centroid, n_k its size, and c_T the global centroid. A larger BCSS indicates that the cluster centers are more spread out from the overall mean, signaling clearer separation between clusters. For contrast, the within-cluster sum of squares (WCSS) measures how tightly data are clustered by summing squared distances of points to their own cluster centroids, which speaks to compactness rather than separation. In the standard decomposition, total sum of squares equals BCSS plus WCSS, tying together overall variability with both separation and compactness. The other option mentioned, average inter-centroid distance, isn’t the same squared, size-weighted measure tied to the grand mean and is not used to quantify the same separation concept.

This item tests understanding of how cluster separation is quantified in a variance-based view of clustering. The metric defined as the sum of squared distances between each cluster centroid and the global centroid is the Between-Clusters Sum of Squares. It sums how far each cluster center sits from the overall mean, weighted by cluster size, typically expressed as BCSS = sum_k n_k * ||c_k − c_T||^2, where c_k is a cluster centroid, n_k its size, and c_T the global centroid. A larger BCSS indicates that the cluster centers are more spread out from the overall mean, signaling clearer separation between clusters.

For contrast, the within-cluster sum of squares (WCSS) measures how tightly data are clustered by summing squared distances of points to their own cluster centroids, which speaks to compactness rather than separation. In the standard decomposition, total sum of squares equals BCSS plus WCSS, tying together overall variability with both separation and compactness. The other option mentioned, average inter-centroid distance, isn’t the same squared, size-weighted measure tied to the grand mean and is not used to quantify the same separation concept.