Society of Actuaries (SOA) PA Practice Exam

When plotting an elbow plot, what happens as k increases?

1. The proportion of variance explained generally increases

2. The proportion of variance explained decreases

3. The plot stabilizes at a fixed value

4. The complexity of the plot becomes irrelevant

The correct answer is: The proportion of variance explained generally increases

As k increases in an elbow plot, the proportion of variance explained generally increases. This is because, as you increase the number of clusters (k) in a clustering analysis, the model has more flexibility to fit the data points more closely. Each additional cluster can capture more variance within the dataset, leading to a lower within-cluster sum of squares (WCSS), which translates to a higher explained variance. The elbow plot visually represents the trade-off between the number of clusters and the variance explained. Initially, as k increases, you will notice a significant drop in the WCSS, indicating that adding more clusters is beneficial. However, this increase will eventually slow down, leading to a point where additional clusters contribute less and less to the variance explained, which is illustrated by the "elbow" in the plot. This characteristic behavior of the plot underscores the importance of determining an optimal number of clusters where the marginal gain in variance explained diminishes – this is often the point at which adding more clusters does not improve the model substantially anymore.