In a traditional signal processing system sampling is carried out at a frequency which is at least twice the highest frequency component found in the signal. This is in order to guarantee that complete signal recovery is later on possible. The sampled signal can subsequently be subjected to further processing leading to, for example, encryption and compression. This processing can be computationally intensive and, in the case of battery operated systems, unpractically power hungry. Compressive sensing has recently emerged as a new signal sampling paradigm gaining huge attention from the research community. According to this theory it can potentially be possible to sample certain signals at a lower than Nyquist rate without jeopardizing signal recovery. In practical terms this may provide multi-pronged solutions to reduce some systems computational complexity. In this work, information theoretic analysis of real EEG signals is presented that shows the additional benefits of compressive sensing in preserving data privacy. Through this it can then be established generally that compressive sensing not only compresses but also secures while sampling.