  For small volume of the quark-gluon plasma formed in heavy ion collisions,
the observable near criticality must obey finite-size scaling. According to the
finite-size scaling, there exists a fixed point at the critical temperature,
where scaled susceptibility at different system sizes intersect. It also exists
at the transitional temperature of a first order phase transition and can be
generalized to the region of the crossover. In order to quantify the feature of
the fixed point, we introduce {\it the width of a set of points}. When all
points in the set are in their mean position within error, the width reaches
its minimum, and all points merge into the fixed point. Using the observable
produced by the Potts model, we demonstrate that the contour plot of the width
defined in this study clearly indicates the temperatures and exponent ratios of
fixed point, which could correspond to either the critical point, the points on
the transition line, or the crossover region. This method is therefore
instructive to the determination of QCD phase boundary by beam energy scan in
relativistic heavy ion collisions.
