A development in method of evaluating the measurement precisions using approximate F tests and variance components from expected mean square (EMS) is investigated. The research proposes three-factor mixed measurement models with the fixed and random factors. Unrestricted and unconstrained design work was rarely studied, while restricted and constrained designs have been significantly discussed. The unrestricted and unconstrained designs assume to be an independence of interaction. The proposed evaluation method about the measurement precisions can be extended to four-factor random measurement model or mixed measurement model. The study also presents the three evaluation indexes of precisions such as R&RTR (Reproducibility & Repeatability-To-Total Precision Ratio), PTR (Precision-To-Tolerance Ratio), and SNR (Signal-To-Noise Ratio). Numerical examples are proposed to evaluate the approximate F tests with Satterthwaite degrees of freedom and three indexes using the measurement precisions from EMS.