WCSS

Within-cluster sum of squares (WCSS) is a measure used in cluster analysis to quantify the compactness of clusters produced by partitioning algorithms such as k-means. It is defined as the sum, over all clusters, of the squared distances between each data point and the centroid of the cluster to which it belongs. For a dataset partitioned into K clusters with centroids μ1, μ2, ..., μK, WCSS = sum_{j=1}^K sum_{x_i in C_j} ||x_i − μ_j||^2, where ||·|| denotes the Euclidean norm. In many software packages, this quantity is called inertia.

WCSS serves as a criterion for evaluating clustering quality: lower values indicate more compact clusters. It is commonly used in the elbow method to help determine an appropriate number of clusters by plotting WCSS against the number of clusters and looking for a point where the marginal gain decreases markedly.

Key considerations include its dependence on the scale and distribution of the data. Since WCSS uses squared distances, features with larger scales can dominate the measure, so standardization or normalization is often recommended before computing WCSS. WCSS is also influenced by outliers, which can increase the value and distort comparisons. Moreover, WCSS is not a standalone validity index; while useful for comparing solutions on the same data and the same feature space, it should be augmented with other metrics or qualitative assessment when judging clustering quality across datasets or methods.