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Alzheimers disease (AD) is a progressive disorder that affects cognitive brain functions and starts many years before its clinical manifestations. A biomarker that provides a quantitative measure of changes in the brain due to AD in the early stages would be useful for early diagnosis of AD, but this would involve dealing with large numbers of people because up to 50% of dementia sufferers do not receive a formal diagnosis. Thus, there is a need for accurate, low-cost, and easy-to-use biomarkers that could be used to detect AD in its early stages. Potentially, electroencephalogram (EEG) based biomarkers can play a vital role in early diagnosis of AD as they can fulfill these needs. This is a cross-sectional study that aims to demonstrate the usefulness of EEG complexity measures in early AD diagnosis. We have focused on the three complexity methods which have shown the greatest promise in the detection of AD, Tsallis entropy (TsEn), Higuchi Fractal Dimension (HFD), and Lempel-Ziv complexity (LZC) methods. Unlike previous approaches, in this study, the complexity measures are derived from EEG frequency bands (instead of the entire EEG) as EEG activities have significant association with AD and this has led to enhanced performance. The results show that AD patients have significantly lower TsEn, HFD, and LZC values for specific EEG frequency bands and for specific EEG channels and that this information can be used to detect AD with a sensitivity and specificity of more than 90%.