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Interpreting group differences observed in aggregated data is a practice that must be done with enormous care. Often the truth underlying such data is quite different than a naive first look would indicate. The confusions that can arise are so perplexing that some of the more frequently occurring ones have been dubbed paradoxes. This article describes two of these paradoxes—Simpson's paradox and Lord's paradox—and illustrates them in a single dataset. The dataset contains the score distributions, separated by race, on the biological sciences component of the Medical College Admission Test (MCAT) and Step 1 of the United States Medical Licensing Examination™ (USMLE). Our goal in examining these data was to move toward a greater understanding of race differences in admissions policies in medical schools. As we demonstrate, the path toward this goal is hindered by differences in the score distributions which gives rise to these two paradoxes. The ease with which we were able to illustrate both of these paradoxes within a single dataset is indicative of how widespread they are likely to be in practice. [ABSTRACT FROM AUTHOR]