Diversity of cluster size has been used as a measurement of complexity in s
everal systems that generate a statistical distribution of clusters. Using
Monte Carlo simulations, we present a statistical analysis of the cluster s
ize diversity and the number of clusters generated on randomly occupied lat
tices for the Euclidean dimensions 1 to 6. A tuning effect for diversity of
cluster size and critical probabilities associated with the maximum divers
ity and the maximum number of clusters are reported. The probability of max
imum diversity is related to the percolation threshold and several scaling
relations between the variables measured are reported. The statistics of cl
uster size diversity has important consequences in the statistical descript
ion of the Universe as a complex system. [S1063-651X(99)08709-7].