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Alzheimer’s disease is one of the main challenges of modern medicine since no cure has been found yet, the scientific community still does not fully understand the reasoning behind it, and any interventions found can delay the progress for only a limited amount of time. Over the years, research has shifted from attempts for curing the disease to efforts towards understanding the mechanisms behind it as well as finding tools that will speed up diagnosis many years before its clinical manifestations, when the brain deterioration begins. One of the many promising tools towards this direction is electroencephalography. Electroencephalography employs a variety of different measures that can be used as biomarkers for early diagnosis and differentiation of Alzheimer’s disease from other neurodegenerative disorders. Literature has produced a number of methods that have established reliable correlation between electroencephalography signals and structural abnormalities in Alzheimer’s disease. To that end, the present work proposes the combination of Tsallis Entropy and Higuchi Fractal Dimension within a common classification framework using machine learning techniques for classification among healthy, Mild Cognitive Impairment, and probable Alzheimer’s disease. The proposed methodology is applied on 75 subjects with different feature utilisation scenarios, reaching to an accuracy of 98.03% when classifying a signal epoch, following a 10-fold cross validation, as compared with other similar studies. Nevertheless, in a leave-one-out scenario with the same approach, the average accuracy drops significantly, suggesting that this method could complement other diagnosis approaches but cannot be used on each own.