Results from gap statistics analysis of a random sample of the data. The expected and observed log (W
) values are shown in (a) for various levels of K (the number of clusters), where W
is the pooled within cluster sum of squares around the cluster means. The number of clusters versus Gap(K), the difference between the observed and expected values (mean value for 1000 random bootstrap permutations), is shown in (b). According to these results, three is the optional number of clusters for our data.