  The critical point is a fixed point in finite-size scaling. To quantify the
behaviour of such a fixed point, we define, at a given temperature and scaling
exponent ratio, the width of scaled observables for different sizes. The
minimum of the width reveals the position of fixed point, its corresponding
phase transition temperature, and scaling exponent ratio. The value of this
ratio tells the nature of fixed point, which can be a critical point, a point
of the first order phase transition line, or a point of the crossover region.
To demonstrate the effectiveness of this method, we apply it to three typical
samples produced by the 3D three-state Potts model. Results show the method to
be more precise and effective than conventional methods. Finally, we discuss a
possible application at the beam energy scan program of relativistic heavy-Ion
collision.
