Distribution system state estimation (DSSE) has been developed for real-time monitoring of distribution systems. As a weighted least square (WLS) based method, DSSE relies on measurement variances to properly weigh the reduction of the measurement residuals. However, traditional DSSE considers an approximated modeling of measurement uncertainty/variance, which can limit the accuracy and quality of state estimates. The main contribution of this paper is the adoption of a new approach to represent uncertainties in loads and measurements more accurately. The paper shows this approach improves the quality of state estimates using DSSE. Several statistical metrics, like bias, quality, and error rate are used to define the quality and accuracy of the state estimates. Comparative Monte Carlo analysis using an IEEE test distribution feeder and an example distribution feeder based on a real feeder is provided to illustrate the improvement from modeling measurement uncertainty more accurately.