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Increasing the parameter K in k-means clustering directly influences the number of clusters formed. As K rises, more clusters can be created, leading to data points being grouped into smaller, more localized clusters. This typically results in centroids that are closer to their respective data points.

When there are more clusters, the average distance between each data point and the closest centroid generally decreases, as each centroid represents a smaller subset of the data. Consequently, with a larger K, the clustering error, which is often quantified as the sum of squared distances from each point to its assigned centroid, tends to decrease. Therefore, increasing K is associated with a reduction in clustering error because it allows points to be closer to their centroids.

In contrast, increasing K significantly beyond the true number of underlying clusters may lead to overfitting, but in general terms and considering the parameters as they are adjusted within reasonable bounds, the closest relationship is that a higher K leads to better representation of the data points relative to their centroids.