The Math Behind Race, Crime and Sentencing Statistics

<i> John Allen Paulos, a professor of mathematics at Temple University, is the author of "A Mathematician Reads the Newspaper" and the forthcoming "Once Upon a Number."</i>

Who can forget the morass of statistics used in the O.J. Simpson double-murder trial? What did the cited DNA probabilities mean? Did the jury and the public grasp the mathematics undergirding the numbers?

More recently, the issues of race, death and mathematics have again subtly entertwined. This time, the misunderstanding arises because the technical meaning of a common phrase differs substantially from its informal meaning. What at first glance may seem like semantic nitpicking has significant consequences for public policy and perceptions.

In a study published in The Times, there appeared a potentially inflammatory, although ostensibly correct, statement. In reporting on death sentences in Philadelphia, the study asserted that the odds of blacks convicted of murder receiving a death sentence were four times the odds faced by other defendants similarly convicted. The Times article, as well as accounts in other newspapers, then transmuted that statement into the starkly inequivalent one that blacks were four times as likely to be sentenced to death as whites. The author of the study used the technical definition of odds, not the more familiar idea of probability, and, as a consequence, most readers were seriously misled.


The difference between “probability” and “odds” is crucial. The odds of an event is defined as the probability it will occur divided by the probability that it will not occur. Consider a coin flip. The probability of it landing heads is one-half, or .5, and the probability of not landing heads is also one-half, or .5. Hence: The odds of the coin landing heads is 1 to 1 (.5 divided by .5). Now consider rolling a die and having it land on 1,2,3,4 or 5. The probability of this event is five-sixths, or .83, and the probability of the die not landing on 1,2,3,4,, or 5 is one-sixth, or .17. Hence: The odds of the die landing on one of these five numbers is 5 to 1 (.83 divided by .17). More serious discrepancies between probabilities and odds occur for events with higher probabilities.

What’s the relevance of this to murder statistics and death penalties? To most readers, the phrase “four times the odds” means that if, say, 99% of blacks convicted of murder were to receive the death penalty, about 25% of whites similarly convicted would receive the same penalty.

Yet, when the technical definition of “odds” is used, the meaning is quite different. In this case, if 99% of blacks convicted of murder received the death penalty, then a considerably less unfair 96% of whites similarly convicted would receive the death penalty. Why? Using the technical definition, we find that the odds in favor of a convicted black murderer receiving the death penalty are 99 to 1 (99/100 divided by 1/100). The odds in favor of a convicted nonblack murderer getting death are 24 to 1 (96/100 divided by 4/100). Thus, since 99 is roughly four times 24, the odds that a convicted black murderer will receive the death penalty are, in this case, approximately four times the odds that a convicted nonblack murderer will receive the same sentence.

As Arnold Barnett and others have shown, similarly misleading claims were made in the 1987 U.S. Supreme Court decision in McClesky vs. Kemp. The issue concerned the effect of a murder victim’s race on death sentencing in the state of Georgia, but the confusion is the same.

By dissecting the phrase “four times the odds,” I don’t mean to deny that racism exists, that there are racial disparities in sentencing or that the death penalty is morally wrong. Rather, I mean to deflate the likely-to-be-inferred magnitude of racial disparities in the sentencing for murder and other violent crimes. The difference between 99% and 96%, for example, is much less egregious than that between 99% and 25%. Still, whatever they are, the raw percentages are troubling enough without the tendentious and easily misinterpreted phrase “four times the odds.”