The paper presents a theoretical introduction to the cryptographic hash function theory and a statistical experimental analysis of selected hash functions. The definition of hash functions, differences between them, their strengths and weaknesses are explained as well. Different hash function types, classes and parameters are described. The features of hash functions are analyzed by performing statistical analysis. Experimental analysis is performed for three certified hash functions: SHA1-160, SHA2-512 and SHA3-512. Such an analysis helps understand the behavior of cryptographic hash functions and may be very helpful for comparing the security level of the hashing method selected. The tests may serve as a basis for examination of each newly proposed hash function. Additionally, the analysis may be harness as a method for comparing future proposals with the existing functions.