FIG. 1: (Color online) S$_{q}$/k$_{B}$N of ideal gases as a function of q for N = 100 calculated by the q average ( q -av.: the dashed curve) and the normal average (N-av.: the solid curve), and the difference between the two entropies (the bold curve) ( D = 1, T/T$_{0}$ = 1 . 0: T$_{0}$ = h $^{2}$/k$_{B}$ m ). The inset shows the validity range of the two averages in the q - N space: the MEMs with the q - and normal averages are valid for 0 < q < q$_{U}$ and q > q$_{L}$ , respectively, and for q$_{L}$ < q < q$_{U}$ both the averages are valid: the dashed line denotes N = 100. FIG. 2: (Color online) S$_{q}$/k$_{B}$N as functions of q and N obtained by the q -average ( D = 1, T/T$_{0}$ = 1 . 0): note the logarithmic scale in the ordinate and abscissa for N . FIG. 4: (Color online) S$_{q}$/k$_{B}$N as functions of T and N obtained by the q -average ( D = 1, q = 0 . 8): note logarithmic scales in the ordinate and abscissa for N . FIG. 5: (Color online) ˜ S$_{q}$/k$_{B}$N as functions of q - and N obtained by the normal average ( D = 1, T/T$_{0}$ = 1 . 0): note the logarithmic scale in the abscissa for N . FIG. 6: (Color online) ˜ S$_{q}$/k$_{B}$N as functions of q and T obtained by the normal average ( D = 1, N = 100). FIG. 3: (Color online) S$_{q}$/k$_{B}$N as functions of q and T obtained by the q -average ( D = 1, N = 100): note the logarithmic scale in the ordinate. 26