Linguistic multi-attribute group decision making with linguistic distribution assessment (LMAGDM-LDA) problems have been widely investigated. However, in the existing studies, the risk attitudes of decision maker are seldom considered. Inspired by this, in this paper, we employ the idea of fractional stochastic dominance to develop a consensus reaching framework for LMAGDM-LDA with bounded risk preference attitudes. Then, we design an interactive learning algorithm for obtaining the bounded risk preference threshold. Next, we analyze the deception behaviors and average behaviors of experts, and develop the minimum adjustment cost consensus model with bounded risk preference attitude (MACC-BRP). And we justify the connection between the MACC-BRP and the minimum adjustment cost consensus model with other risk attitudes. Furthermore, we provide a distance-based method to obtain the rankings of alternatives. Furthermore, a numerical analysis is provided to show the computation process of the proposed method, and a comparison analysis and a simulation analysis are further conducted to show the advantages and features of the proposal. The simulation results show the effects of the risk preference threshold, the intrinsic individual weight and the average threshold. The comparison results show that the proposed minimum adjustment cost consensus models have clear advantages.