Welcome! In today's lesson, we'll delve into cluster validation. We will interpret and implement the Silhouette Score, and learn how to visualize clusters for validation in Python. All of these concepts form a unified understanding that we'll explore.

Cluster validation, a key step in Cluster Analysis, involves evaluating the quality of the outcomes of the clustering process. Proper validation helps avoid common issues such as overfitting or misjudging the optimal number of clusters.

One metric that plays a crucial role in cluster validation is the Silhouette Score. This measure quantifies the quality of clustering, providing an indication of how well each data point resides within its cluster. The Silhouette Score $s(i)$ for a sample $i$ is formulated as:

$s(i$