The distribution function is an important tool for the study of the stochastic variances. The normal distribution is very popular in the nature and human society. The idea of membership functions is the foundation of the fuzzy sets theory. While the fuzzy theory is widely used, the completely certain membership function which has no any fuzziness at all has been the bottleneck of the applications of this theory. Cloud models are effective tools in transforming between qualitative concepts and their quantitative expressions. It can represent the fuzziness and randomness and their relations of uncertain concepts. Also cloud models can show the concept granularity in multi-scale spaces by the digital characteristic Entropy (E_n). The normal cloud models not only broaden the formation conditions of the normal distribution but also make the normal membership function be the expectation of the random membership degree. In this paper, the universality of the normal cloud model is proved,which is more superior and easier, and can fit the fuzziness and gentleness in human cognizing process. It would be more applicable and universal in the representation of uncertain notions.