Sleep specialists often conduct manual sleep stage scoring by visually inspecting the patient's neurophysiological signals collected at sleep labs. This is, generally, a very difficult, tedious and time-consuming task. The limitations of manual sleep stage scoring have escalated the demand for developing Automatic Sleep Stage Classification (ASSC) systems. Sleep stage classification refers to identifying the various stages of sleep and is a critical step in an effort to assist physicians in the diagnosis and treatment of related sleep disorders. The aim of this paper is to survey the progress and challenges in various existing Electroencephalogram (EEG) signal-based methods used for sleep stage identification at each phase; including pre-processing, feature extraction and classification; in an attempt to find the research gaps and possibly introduce a reasonable solution. Many of the prior and current related studies use multiple EEG channels, and are based on 30 s or 20 s epoch lengths which affect the feasibility and speed of ASSC for real-time applications. Thus, in this paper, we also present a novel and efficient technique that can be implemented in an embedded hardware device to identify sleep stages using new statistical features applied to 10 s epochs of single-channel EEG signals. In this study, the PhysioNet Sleep European Data Format (EDF) Database was used. The proposed methodology achieves an average classification sensitivity, specificity and accuracy of 89.06%, 98.61% and 93.13%, respectively, when the decision tree classifier is applied. Finally, our new method is compared with those in recently published studies, which reiterates the high classification accuracy performance.