The anarchy principle leading to the seesaw ensemble is studied analytically with the usual tools of random matrix theory. The probability density function for the seesaw ensemble of $N \times N$ matrices is obtained in terms of a multidimensional integral (where $N$ corresponds to the number of generations). This probability density functions is then used to extract information on the relevant physical parameters of the neutrino sector of a seesaw-extended Standard Model. For $N = 3$, a probability test is introduce to help characterize the type I-III and type II seesaw ensembles using numerical integration methods. A systematic comparison between the two ensembles is performed to point out the fundamental differences between them. It is found that the type I-III seesaw ensemble is better suited to accommodate experimental data. Moreover, the results indicate a strong preference for the mass splitting associated to normal hierarchy. However, because of the decoupling of the probability density function for the light neutrino masses and the neutrino mixing angles and phases, all permutations of the singular values are found to be equally probable for a particular mass splitting, which implies that redictions regarding the hierarchy of the mass spectrum remains out of reach in this particular framework.