Abstract
<p>Every measure of racial disparity in imprisonment is a function of two quantities: the rate at which a state imprisons its Black residents and the rate at which it imprisons its white ones. Four summaries are in common use; because each collapses two dimensions into one, they can disagree on the same data about direction, not just magnitude. This paper makes the disagreement the object of study. It puts one question, whether more democratic U.S. states punish Black and white residents more equally, through the four measures and three further analytic choices: the comparison design, a clustering of states into types, and the reading of a voting-rights reform. The answer changes at nearly every fork, and the changes share one anatomy: white imprisonment falls with state democracy while Black imprisonment stays flat, so a one-number summary can deteriorate with nothing changing for Black residents. Rather than choose among summaries, I model the two rates jointly and read every measure off one posterior. Disagreement becomes an estimable quantity: the probability that stronger democracy goes with a higher ratio and lower white imprisonment is 0.97. Institutions that attach formula grants and sentencing law to these numbers are grading, in part, a denominator.</p>