Many things and phenomena in nature show extremely complex forms and are not as idealistic as they seem. Self-similarity or scale invariance is often expressed in a statistical way, that is, when the scale changes, some statistical characteristics contained in the scale are similar to the whole. This kind of fractal is a generalization of mathematical fractal and is called statistical fractal.

Mathematical fractal is an idealized situation, and two conditions must be met:

(1) Mathematical fractal curve must have infinite "hierarchical" structure, like Koch curve; Mathematical fractal must be a set of infinite points, just like Cantor set. Only with infinite hierarchical structure can self-similarity or scale invariance be established everywhere.

(2) Any local amplification of mathematical fractal is completely similar to the whole in shape, quantity and statistical distribution.

Mathematical fractal is a mathematical model for analyzing complex things in nature. In order to apply it to real natural phenomena, it should be summarized and revised: ① From infinite hierarchical structure to finite hierarchical structure, or from infinite set to finite set, there is a self-similarity or scale invariance problem in a certain range, that is, scale-free interval problem; ② Generalization from strict mathematical similarity to approximate statistical similarity.