Distribution state estimation (DSE) is an essential part of an active distribution network with high level of distributed energy resources. The challenges of accurate DSE with limited measurement data is a well-known problem. In practice, the operation and usability of DSE depend on not only the estimation accuracy but also the ability to predict error variance. This paper investigates the application of error covariance in DSE by using the augmented complex Kalman filter (ACKF). The Kalman filter method inherently provides state error covariance prediction. It can be utilized to accurately infer the error covariance of other parameters and provide a method to determine optimal measurement locations based on the sensitivity of error covariance to measurement noise covariance. This paper also proposes a generalized formulation of ACKF to allow scalar measurements to be incorporated into the complex-valued estimator. The proposed method is simulated by using modified IEEE 34-bus and IEEE 123-bus test feeders, and randomly generates the load data of complex-valued Wiener process. The ACKF method is compared with an equivalent formulation using the traditional weighted least squares (WLS) method and iterated extended Kalman filter (IEKF) method, which shows improved accuracy and computation performance.