Studies have shown pulse oximeter measurements of blood oxygenation have statistical bias that is a function of race, which results in higher rates of occult hypoxemia, i.e., missed detection of dangerously low oxygenation, in patients of color. This paper further characterizes the statistical distribution of pulse ox measurements, showing they also have a higher variance for patients racialized as Black, compared to those racialized as white. We show that no single race-based correction factor will provide equal performance in the detection of hypoxemia. The results have implications for racially equitable pulse oximetry.