In the recent literature, many definitions of partial randomness of reals have been proposed and studied rather discretely. For instance, it is known that for a computable real ε ∈(0, 1), strong Martin-Löf ε-randomness is strictly stronger than Solovay ε-randomness which is strictly stronger than weak Martin-Löf ε-randomness. In the present work, we firstly give several new definitions of partial randomness - strong Kolmogorov ε-randomness and weak/strong DH-Chaitin ε-randomness. Then, we investigate the relation between ε-randomness by one definition and ε′-randomness by another. Finally, we show that all of the known definitions of ε-randomness are quasi-equivalent.