Variance in predictions across different trained models is a significant, under-explored source of error in fair classification. In practice, the variance on some data examples is so large that decisions can be effectively arbitrary. To investigate this problem, we take an experimental approach and make four overarching contributions: We 1) Define a metric called self-consistency, derived from variance, which we use as a proxy for measuring and reducing arbitrariness; 2) Develop an ensembling algorithm that abstains from classification when a prediction would be arbitrary; 3) Conduct the largest to-date empirical study of the role of variance (vis-a-vis self-consistency and arbitrariness) in fair classification; and, 4) Release a toolkit that makes the US Home Mortgage Disclosure Act (HMDA) datasets easily usable for future research. Altogether, our experiments reveal shocking insights about the reliability of conclusions on benchmark datasets. Most fairness classification benchmarks are close-to-fair when taking into account the amount of arbitrariness present in predictions -- before we even try to apply common fairness interventions. This finding calls into question the practical utility of common algorithmic fairness methods, and in turn suggests that we should fundamentally reconsider how we choose to measure fairness in machine learning.