  The exact nature of the chiral phase transition in QCD is still under
investigation. In $N_f=2$ and $N_f=(2+1)$ lattice simulations with staggered
fermions the expected O($N$)-scaling behavior was observed. However, it is
still not clear whether this behavior falls into the O(2) or O(4) universality
class. To resolve this issue, a careful scaling and finite-size scaling
analysis of the lattice results is needed. We use a functional renormalization
group to perform a new investigation of the finite-size scaling regions in
O(2)- and O(4)-models. We also investigate the behavior of the critical
fluctuations by means of the $4^{\text{th}}$-order Binder cumulant. The
finite-size analysis of this quantity provides an additional way for
determining the universality class of the chiral phase transition in lattice
QCD.
