How do I choose k when using k-means clustering with Silhouette function?

I've been studying about k-means clustering, and one big thing which is not clear is what Silhouette function really tell to me?

i know it shows that what appropriate k should be detemine but i cant understand what mean of silhouette function really say to me?

i read somewhere, if the mean of silhouette is less than 0.5 your clustering is not valid.

thanks for your answers in advance.

Solution

From the definition of silhouette :

Silhouette Value

The silhouette value for each point is a measure of how similar that point is to points in its own cluster compared to points in other clusters, and ranges from -1 to +1.

The silhouette value for the ith point, Si, is defined as

Si = (bi-ai)/ max(ai,bi) where ai is the average distance from the ith point to the other points in the same cluster as i, and bi is the minimum average distance from the ith point to points in a different cluster, minimized over clusters.

This method just compares the intra-group similarity to closest group similarity. If any data member average distance to other members of the same cluster is higher than average distance to some other cluster members, then this value is negative and clustering is not successful. On the other hand, silhuette values close to 1 indicates a successful clustering operation. 0.5 is not an exact measure for clustering.