How should the optimal number of clusters K be selected in K-means clustering?

Selecting the optimal number of clusters K in K-means clustering primarily involves minimizing the total within-cluster variation, also known as inertia or within-cluster sum of squares. The goal of K-means is to partition the data into K distinct clusters where the data points in each cluster are as similar as possible, while data points in different clusters are as dissimilar as possible.

Minimizing total within-cluster variation ensures that the points in a cluster are close to the cluster centroid, which reflects the performance and effectiveness of the clustering solution. A lower total within-cluster variation indicates that the clusters are compact and well-separated, making it a crucial factor in determining an appropriate number of clusters.

Techniques like the Elbow Method or the Silhouette Score can be used to assess the impact of different values of K on total within-cluster variation. As K increases, the within-cluster variation typically decreases. However, one seeks to find an optimal K where the reduction in within-cluster variation begins to diminish significantly, which indicates the most meaningful number of clusters.

Choosing K to equal n (the number of observations) would result in each point being its own cluster, which defeats the purpose of clustering. Relying on an objective method is valid,