Uncertainty quantification of deep neural networks has become an active field of research and plays a crucial role in various downstream tasks such as active learning. Recent advances in evidential deep learning shed light on the direct quantification of aleatoric and epistemic uncertainties with a single forward pass of the model. Most traditional approaches adopt an entropy-based method to derive evidential uncertainty in classification, quantifying uncertainty at the sample level. However, the variance-based method that has been widely applied in regression problems is seldom used in the classification setting. In this work, we adapt the variance-based approach from regression to classification, quantifying classification uncertainty at the class level. The variance decomposition technique in regression is extended to class covariance decomposition in classification based on the law of total covariance, and the class correlation is also derived from the covariance. Experiments on cross-domain datasets are conducted to illustrate that the variance-based approach not only results in similar accuracy as the entropy-based one in active domain adaptation but also brings information about class-wise uncertainties as well as between-class correlations. The code is available at https://github.com/KerryDRX/EvidentialADA. This alternative means of evidential uncertainty quantification will give researchers more options when class uncertainties and correlations are important in their applications.