How do we know if what we see is really there? When visualizing data, how do we avoid falling into the trap of apophenia where we see patterns in random noise? Traditionally, infovis has been concerned with discovering new relationships, and statistics with preventing spurious relationships from being reported. We pull these opposing poles closer with two new techniques for rigorous statistical inference of visual discoveries. The "Rorschach" helps the analyst calibrate their understanding of uncertainty and "line-up" provides a protocol for assessing the significance of visual discoveries, protecting against the discovery of spurious structure.