Permutation patterns-based approaches, such as permutation entropy (PerEn), have been widely and successfully used to analyze data. However, these methods have two main shortcomings. First, when a series is symbolized based on permutation patterns, repetition as an unavoidable phenomenon in data is not took in to account. Second, they consider only the order of amplitude values and so, some information regarding the amplitude values themselves may be ignored. To address these deficiencies, we have very recently introduced dispersion patterns and subsequently, dispersion entropy (DispEn). In this paper, we investigate the effect of different linear and non-linear mapping approaches, used in the algorithm of DispEn, on the characterization of signals. We also inspect the sensitivity of different parameters of DispEn to noise. Moreover, we introduce frequency-based DispEn (FDispEn) as a measure to deal with only the frequency of time series. The results suggest that DispEn and FDispEn with the log-sigmoid mapping approach, unlike PerEn, can detect outliers. Furthermore, the original and frequency-based forbidden dispersion patterns are introduced to discriminate deterministic from stochastic time series. The computation times show that DispEn and FDispEn are considerably faster than PerEn. Finally, we find that DispEn and FDispEn outperform PerEn to distinguish various dynamics of biomedical signals. Due to their advantages over existing entropy methods, DispEn and FDispEn are expected to be widely used for the characterization of a wide variety of real-world time series.