The paper explores the possibility to generate experimental independent andidentically distributed data sets starting from the logistic map and discusses its applicability in cryptography. In order to reveal the statistical independence in the context of a chaotic signal, and to come up with firm and accurate results, the paper combines usual statistical methods with an original test procedure useful in the case of continuous random variables of unknown probability law. The overall theoretical approach may be viewed as a guide to generate independent andidentically distributed samples starting from the logistic map, and also to design chaotic sequences useful for cryptographic purposes.