Abstract: The uniform index of arbitrary distribution function was first defined, then the axioms of ‘uniform index independence’ and proposition of ‘uniform distribution is the most uniform’ were put forward. The similarity between uniform index and entropy was clearly illustrated, and the inner mechanism of interpreting chaos with uniform index was revealed thereby. According to the independence axiom, k step expectation uniform index does exist, and can be estimated by k step average uniform index. As the arbitrary distribution function F can be regarded as a transformation of uniform distribution by F^-1, numerical calculation indicates the the nonlinear degree of distribution function is the direct cause for k step average uniform index reduction. Entropy is a varable to describe the uncertainty of distribution, meanwhile, it also describes the disperse degree of samples, which means the degree of uniformity and vice versa. Certainty of distribution signifies the concentration degree of samples, which also implies the non-uniformity and vice versa. This is the common ground of entropy and uniform index. In ecology, the fact of most structures are clustering is caused by the nonlinear transformation existing in the nature.

Key words: chaos, entropy, uniform index, k step chaometry