Summary: Aiming at the information description and aggregation of group decision-making (GDM), this study develops an innovative GDM method based on Z-numbers and their aggregation techniques. The Z-number is a powerful tool for describing real-life information and reflecting information reliability. However, Z-numbers have a complicated three-dimensional structure, and many existing studies did not manage Z-numbers appropriately. Besides, no study has investigated Z-number aggregation considering Z-numbers’ potential probability distributions. To remove the above defects, many valid techniques are introduced. An optimisation model is constructed to determine the potential probability distributions involved in Z-numbers. Then, a mean function for comparing Z-numbers is presented, and a series of Z-number operations are defined based on Archimedean \(t\)-norms and \(t\)-conorms. Moreover, a Z-number Bonferroni mean aggregation operator for integrating Z-number information is proposed. To test the applicability and validity of the developed Z-number GDM method, a new energy investment selection problem is addressed, and the sensitivity analysis and comparison discussion are conducted. The sensitivity results show that the developed method possesses favourable stability and effectiveness. In addition, the comparison results demonstrate that it outperforms other existing methods, and it can handle existing defects effectively.