# Will the silhouette not approach 1 when increasing k to n?

My understanding of silhouette in the context of k-means clustering is that its equal to:

$$\frac{\mbox{Av. distance to subjects in hearest neighboring cluster} - \mbox{Av. distance to subjects in my cluster}}{\mbox{max. of the two above}}$$

So, assume really good clustering, Av. distance to subjects in neighbouring cluster is really big and average distance to subjects in my cluster is really small (numerator becomes approx. the first term), divide by the max, so the silhouette becomes 1 --> Good clustering.

But if I have as many clusters as I have items, then will not the average distance to subjects in neighbouring cluster always be greater than the distance to the current cluster (since the item is sitting on it)? Then I should always get silhouette = 1, no?

• By convention, the silhouette index for any singleton object is set to 0. Alternatively, you may set it to missing (no value). We can never say how well a singleton is clusrtered because there are no other points in its cluster, to compare the distance to them with that to the neighbouring cluster. Sep 3, 2016 at 21:42
• @ttnphns intuitively that convention doesn't make much sense to me. The distance to every point in a singleton cluster seems like it should be zero, because the distance between any point and itself is zero. Sep 3, 2016 at 22:28
• @ssdecontrol, In clustering indices such as silhouette, comparing within and between cluster densities, the distances and their averages serve as indicators of density. Singleton object is formally a cluster, yet it has undefined (and not zero or 1) density within. It is difficult to say if a singleton object alone is clustered "well" or "bad". Therefore Silhouette index is set to 0 (neither any good nor completely bad), or is set to missing. It makes clear sense. Sep 3, 2016 at 22:48