The main objective of this study was to enhance the performance of sleep stage classification using single-channel electroencephalograms (EEGs), which are highly desirable for many emerging technologies, such as telemedicine and home care. The proposed method consists of decomposing EEGs by a discrete wavelet transform and computing the kurtosis, skewness and variance of its coefficients at selected levels. A random forest predictor is trained to classify each epoch into one of the Rechtschaffen and Kales' stages. By performing a comprehensive set of tests on 106,376 epochs available from the Physionet public database, it is demonstrated that the use of these three statistical moments has enhanced performance when compared to their application in the time domain. Furthermore, the chosen set of features has the advantage of exhibiting a stable classification performance for all scoring systems, i.e., from 2- to 6-state sleep stages. The stability of the feature set is confirmed with ReliefF tests which show a performance reduction when any individual feature is removed, suggesting that this group of feature cannot be further reduced. The accuracies and kappa coefficients yield higher than 90 % and 0.8, respectively, for all of the 2- to 6-state sleep stage classification cases.