  In the study of rotating neutron stars the $I$-Love-$Q$ relations refer to
the existence of various approximate, equation of state-independent relations
involving the moment of inertia, the Love number and the quadrupole moment.
These relations are relevant for observational astrophysics, since they allow
(in theory) the inference of any two quantities within the $I$-Love-$Q$ triad
out of the third one alone. However, the quantities involved in the relations
are, in fact, normalized by a parameter $M_0$ that arises in the usual
perturbative analytical approach as the mass of the background configuration.
Since $M_0$ is not the mass of the rotating star $M_S$, it is not an
observational quantity, which may affect the application of the relations to
actual observations. This situation is usually ignored in most studies by
taking $M_0$ to be the mass of the star, an approximation that can, in some
cases, be inconsistent. In this paper we extract the value of $M_0$ using an
$\textit{extended}$ version of the universal relations that involve a fourth
parameter, $\delta M$, proportional to the difference $M_S-M_0$. We analyze to
which degree this extended set of relations yields a more precise inference of
compact star properties and equation of state parameters.
