Regular network data envelopment analysis (DEA) models deal with evaluating the performance of a set of decision-making units with a two-stage construction in the context of a deterministic data set. In the real world, however, observations may display a stochastic behavior. To the best of our knowledge, despite the existing research done with different data types, studies on two-stage processes with stochastic data are still very limited. This article proposes a two-stage network DEA model with stochastic data. The stochastic two-stage network DEA model is formulated based on the satisficing DEA models of chance-constrained programming and the leader–follower concepts. According to the probability distribution properties and under the assumption of the single random factor of the data, the probabilistic form of the model is transformed into its equivalent deterministic linear programming model. In addition, the relationship between the two stages as the leader and the follower, respectively, at different confidence levels and under different aspiration levels, is discussed. The proposed model is applied to a real case concerning 16 commercial banks in China in order to confirm the applicability of the proposed approach.