In this study we have investigated the electroencephalogram (EEG) background activity in patients with Alzheimer's disease (AD) using non-linear analysis methods. We calculated the Lempel-Ziv (LZ) complexity - applying two different sequence conversion methods - and the central tendency measure (CTM) of the EEG in 11 AD patients and 11 age-matched control subjects. CTM quantifies the degree of variability, while LZ complexity reflects the arising rate of new patterns along with the EEG time series. We did not find significant differences between AD patients and control subjects' EEGs with CTM. On the other hand, AD patients had significantly lower LZ complexity values (p<0.01) at electrodes P3 and O1 with a two-symbol sequence conversion, and P3, P4, O1 and T5 using three symbols. Our results show a decreased complexity of EEG patterns in AD patients. In addition, we obtained 90.9% sensitivity and 72.7% specificity at O1, and 72.7% sensitivity and 90.9% specificity at P3 and P4. These findings suggest that LZ complexity may contribute to increase the insight into brain dysfunction in AD in ways which are not possible with more classical and conventional statistical methods.