A methodology relying on relating terms of a two-port error network to the scattering (S-) parameters of a two-port network or device is applied to extract its full S-parameters. The new methodology has only one sign ambiguity problem (two solutions) in evaluation of S11 (and thus S22) while the similar methodology in the literature has two sign ambiguity problems (4 solutions). This ambiguity problem of our method is shown to be eliminated by applying an approach based on continuity of the argument of S11 in the frequency domain. On the other hand, our methodology, as compared with the thru-reflect-line calibration technique, does not necessitate usage of any reflection standard in determining full S-parameters of a two-port network or device. Finally, it gives information about error networks. For its validation, S-parameters of a microwave phase shifter and a polyethylene sample flushed with the left/bottom terminal of the coaxial cell were extracted.