As catalogs of gravitational-wave transients grow, new records are set for the most extreme systems observed to date. The most massive observed black holes probe the physics of pair instability supernovae while providing clues about the environments in which binary black hole systems are assembled. The least massive black holes, meanwhile, allow us to investigate the purported neutron star-black hole mass gap, and binaries with unusually asymmetric mass ratios or large spins inform our understanding of binary and stellar evolution. Existing outlier tests generally implement leave-one-out analyses, but these do not account for the fact that the event being left out was by definition an extreme member of the population. This results in a bias in the evaluation of outliers. We correct for this bias by introducing a coarse-graining framework to investigate whether these extremal events are true outliers or whether they are consistent with the rest of the observed population. Our method enables us to study extremal events while testing for population model misspecification. We show that this ameliorates biases present in the leave-one-out analyses commonly used within the gravitational-wave community. Applying our method to results from the second LIGO--Virgo transient catalog, we find qualitative agreement with the conclusions of Abbott et al, ApJL 913 L7 (2021). GW190814 is an outlier because of its small secondary mass. We find that neither GW190412 nor GW190521 are outliers.
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