  Mean field approximation is a popular method to study the behaviour of
stochastic models composed of a large number of interacting objects. When the
objects are asynchronous, the mean field approximation of a population model
can be expressed as an ordinary differential equation. When the objects are
(clock-) synchronous the mean field approximation is a discrete time dynamical
system. We focus on the latter.We study the accuracy of mean field
approximation when this approximation is a discrete-time dynamical system. We
extend a result that was shown for the continuous time case and we prove that
expected performance indicators estimated by mean field approximation are
$O(1/N)$-accurate. We provide simple expressions to effectively compute the
asymptotic error of mean field approximation, for finite time-horizon and
steady-state, and we use this computed error to propose what we call a
\emph{refined} mean field approximation. We show, by using a few numerical
examples, that this technique improves the quality of approximation compared to
the classical mean field approximation, especially for relatively small
population sizes.
