Complexity measurement using Lempel and Ziv algorithm (LZ) has been used to analyze physiological data. This work shows that the Lempel and Ziv complexity measurement of EEG signals using wavelets transforms is independent of electrode position and dependent on cognitive tasks and brain activity. EEG database with 122 subjects from the public EEG dataset was used in this study. This database have spontaneous EEG and evoked potential (EP) data from a 64-multielectrode array under a variety of conditions collected at several centers in the United States, sponsored by the National Institute on Alcohol Abuse and Alcoholism (NIAAA) project. Two experiments were performed with this database. The first experiment was to test the dependency of electrode positions into LZ complexity measures and the second experiment was to analyze if the LZ complexity was sensitive to the EEG acquired from control and alcoholic subjects. The results show that the complexity measurement is dependent on the changes of the pattern of brain dynamics and not dependent on electrode position.