  This study presents the Fisher information for the position-dependent mass
Schr\"odinger equation with hyperbolical potential {$V(x)=-V_0{\rm
csch}^2(ax)$}. The analysis of the quantum-mechanical probability for the
ground and exited states $(n=0, 1, 2)$ has been obtained via the Fisher's
information. This controls both chemical and physical properties of some
molecular systems. The Fisher information is considered only for $x>0$ due to
the singular point at $x=0$. We found that Fisher-information-based uncertainty
relation and the Cramer-Rao inequality holds. Some relevant numerical results
are presented. The results presented shows that the Cramer-Rao and the
Heisenberg products in both spaces provide a natural measure for anharmonicity
of {$-V_0{\rm csch}^2(ax)$}
