Prior research has suggested that clustered regression discontinuity designs are a formidable alternative to cluster randomized designs because they provide targeted treatment assignment while maintaining a high-quality basis for inferences on local treatment effects. However, methods for the design and analysis of clustered regression discontinuity designs have not been fully developed to address the array of core effects (e.g., main, moderation and mediation) typically examined in education studies. In this study, we complement prior design literature by developing principles of estimation, sampling variability, and closed-form expressions to predict the statistical power to detect mediation effects in clustered regression discontinuity designs. The results suggest that sample sizes typically seen in educational intervention studies (e.g., about 50 schools) can be sufficient to detect a mediation effect under some conditions when studies are carefully designed. We implement the results in software and a Shiny App (BLINDED FOR REVIEW).