The exact statistical models for the observation phase noise (OPN) and their approximations in time-domain, phase-based estimation of the frequency and phase of a noisy single sinusoid are summarized, and the underlying physical interconnections are highlighted. The application of the various OPN models to the design of the weighted phase averager (WPA) estimator is developed. Simulation results show that using the time-varying OPN variance information which can intelligently indentify those phase samples with large errors, the performance of the WPA estimators can be improved, especially for low SNR.