The statistical paradox
of police killings
In the numbers of fatal encounters with the cops, one kind of discrimination masks another.
A protester in Philadelphia wore a shirt with the names of people who died in encounters with the police.TYGER WILLIAMS / PHILADELPHIA INQUIRER/ASSOCIATED PRESS
There is overwhelming evidence of racial bias in the criminal justice system, in everything from policing to sentencing. Nonetheless, the ongoing protests against racism and police brutality have prompted a familiar, fallacious reply from armchair statisticians in op-eds, social media, and police departments: that racial bias in the use of force by police is a myth, easily debunked with statistics.
The people making this argument don’t dispute the fact that police kill Black people at disproportionate rates. A Black person in America is roughly three times more likely than a white person to be killed by police. But according to this argument, the disparity is rooted in crime rates and more frequent encounters with police, not racism. In 2018, the rate of arrests for violent crime was 3.6 times higher for Black people than white people. So actually, the argument goes, Black people are underrepresented as victims of police killings, after controlling for the number of encounters.
Could the reaction to high-profile killings like those of Breonna Taylor and George Floyd be a matter of confirmation bias? Could the narrative of police racism be disproved with a tweet-sized calculation?
These statistics are consistent with excessive use of deadly force against Black people, due to a mathematical phenomenon called Simpson’s Paradox.
The key point is that not all encounters with police are equally deadly. In any given kind of encounter with the police, a Black person can be likelier to be killed than a white person even if the overall rate of deaths per encounter appears lower for Black people. This would happen because Black people have many more interactions with police in non-deadly situations — a dynamic exacerbated by racism. And all those extra encounters dilute the rate.
Consider two extremes of police encounters: traffic stops and active shooter scenarios. Suppose, hypothetically, that a white suspect is killed by police in one out of 100,000 traffic stops and nine out of 10 shootings. And imagine that Black suspects are killed by police after 20 out of 1,000,000 traffic stops and in 10 out of 10 active shooter incidents. In each kind of incident, Black suspects are killed more often than white suspects. In aggregate, though, the percentage is higher for white people: 10 out of 100,010 white people are killed vs. 30 out of 1,000,010 Black people, because the white people tend to encounter the police in more grave situations.
There are, of course, more than two types of police encounters in reality, and whether any of them involves deadly force will depend on many factors, such as whether the suspect is armed and making threats, how many officers are on the scene, and so on. The actual data is far more complex than in this simplified example, and there isn’t consensus over whether clear evidence of encounter-specific racial bias exists. There are just too many variables for the data to be definitive on its own.
That’s why one study, frequently cited as evidence that Black people are killed just as often (or less often) as others in similar situations, has been critiqued by other researchers who noted that “its approach is mathematically incapable of supporting its central claims."
The inflated number of non-lethal encounters Black people experience due to racial profiling could be what shifts the balance, perversely using one kind of discrimination, over-policing, to mask another: the greater use of deadly force against Black suspects. Simpson’s Paradox predicts these counterintuitive results whenever data is averaged over inconsistent group sizes. Here, the inconsistency lies in the types of interactions Black and white people have with police. Since these are distributed differently, the pooled numbers can get the story backwards.
Aubrey Clayton is a mathematician living in Boston and the author of the forthcoming book “Bernoulli’s Fallacy.” Follow him on Twitter @aubreyclayton.