This paper analyzes the underlying complexity and non-linearity of electroencephalogram (EEG) signals by computing a novel multi-scale entropy measure for the classification of seizure, seizure-free and normal EEG signals. The quality factor (Q) based multi-scale entropy measure is proposed to compute the entropy of the EEG signal in different frequency-bands of interest. The Q -based entropy (QEn) is computed by decomposing the signal with the tunable-Q wavelet transform (TQWT) into the number of sub-bands and estimating K-nearest neighbor (K-NN) entropies from various sub-bands cumulatively. The optimal selection of Q and the redundancy parameter (R) of TQWT showed better robustness for entropy computation in the presence of high- and low-frequency components. The extracted features are fed to the support vector machine (SVM) classifier with the wrapper-based feature selection method. The proposed method has achieved accuracy of 100% in classifying normal (eyes-open and eyes-closed) and seizure EEG signals, 99.5% in classifying seizure-free EEG signals (from the hippocampal formation of the opposite hemisphere of the brain) from seizure EEG signals and 98% in classifying seizure-free EEG signals (from the epileptogenic zone) from seizure EEG signals, respectively, using the SVM classifier. We have also achieved classification accuracies of 99% and 98.6% in classifying seizure versus non-seizure EEG signals and the individual three classes, namely normal, seizure-free and seizure EEG signals, respectively. The performance measure of the proposed multi-scale entropy has been found to be comparable with the existing state of the art epileptic EEG signals classification methods studied using the same database.