Automatic sleep stage scoring is an ambitious task that gives clinical information on diagnosing patients with sleep disorders. The objective of this paper is to establish an automatic sleep stage scoring technique by utilizing the Discrete Wavelet Transform (DWT) and feature extraction algorithm. The DWT is to decompose the EEG signal into different frequency rhythms: Gamma, Beta, Alpha, Theta and Delta for each 30 second epoch. The Hilbert transform which generates spectra in the frequency or time-frequency domain is applied on the decomposed rhythms to extract the envelope based statistical features. The most relevant features are preferred by looking at the probability distributions for each metric conditioned on the sleep stages and identifying the features which gives greatest separation between the sleep stages. The Gaussian Mixture Model with Expectation Maximization (GMM-EM) based classification technique is adopted on the relevant feature vectors to assign each epoch one of the six possible sleep stages: Wake, Rapid Eye Movement (REM) and Non-Rapid Eye Movement (N-REM) stage1 (S1), N-REM stage2 (S2), N-REM stage3 (S3), N-REM stage4 (S4). Automating the scoring process offers a way to diagnose and treat sleep disorders in a more robust manner.