  In the framework of the Tsallis statistical mechanics, for the spin-1/2 and
the harmonic oscillator, we study the change of the population of states when
the parameter $q$ is varied; the results show that the difference between
predictions of the Boltzmann--Gibbs and Tsallis Statistics can be much smaller
than the precision of any existing experiment. Also, the relation between the
privilege of rare/frequent event and the value of $q$ is restudied. This
relation is shown to be more complicated than the common belief about it.
Finally, the convergence criteria of the partition function of some simple
model systems, in the framework of Tsallis Statistical Mechanics, is studied;
based on this study, we conjecture that $q \leq 1$, in the thermodynamic limit.
