racial disparities in police killings using bayes theorem

The Washington Post has been tracking police killings across the nation. Last week, Peter Aldhous published an analysis of these data.1 He figured that blacks suspects were 37.8% of all unarmed suspects killed by police. White suspects made up a nearly similar percentage of unarmed suspects killed by police, despite the fact that there are almost five times as many whites in the United States as blacks.

This does not provide the best evidence to adjudicate racial disparities in police violence, however. Aldous writes:

Video of McDonald’s last moments, shot 16 times by a white officer, made a stark contrast with images of a handcuffed Robert Lewis Dear, the white suspect in the shooting at a Planned Parenthood clinic in Colorado Springs — as activists were quick to point out.

Rather than figure out the probability that an unarmed suspect was black, it would be important to know the probability that a black suspect shotkilled by police was unarmed. We care less whether an unarmed victim was black as we do whether a black victim was unarmed. That would be more in line with, though not exactly equivalent to, what Aldhous wrote.2

Below, I try to explain how we can use rules of probability to explain this problem to an introductory statistics class.

In my statistics class, I teach students that probability equals the number of events that we care about divided the total number of events for which that event is possible. Let’s start by figuring out the probability that the police shootkill an unarmed black man. The numerator — the events we care about — would be the number of unarmed black men shot. The denominator — the total number of events — equals all police shootings that kill a suspect, which comprises 867 police shootings.

The numerator is a little trickier. Since we want to know how many unarmed black men are shotkilled, our best estimate absent any additional data would be the overall probability that a suspect killed by the police was unarmed. This prior probability equals 9.46% since 82 of 867 suspects in the data were unarmed. But, fortunately, we have more data — we know the probability that an unarmed person was black. The data comprised 82 unarmed suspects, of whom 31 were black. The likelihood, therefore, that an unarmed suspect was black is 37.8%, which is the probability that Aldhous calculated. This represents the rate at which unarmed suspects killed by police are black.

We finish calculating the numerator by multiplying these two values together. We multiply the probability of a suspect being unarmed by the rate at which unarmed victims are black, 9.46% (or 82 of 867) by 37.8% (31 of 82). The result of that multiplication represents the probability that a suspect was both black and unarmed, which was 31 out of 867 cases or 3.58%. If you hear about a police shooting that killed the suspect, about one out of every 28 times the police have shot an unarmed black person.

That does not sound too bad, but remember that almost 4% of all shootingspolice killings are of unarmed black men. We change the denominator from being all possible events — in this case 867 police shootings — to include only those that involve black men. In other words, we divide the 3.58% of all shootings by the probability that a police shooting involved a black suspect. Of the 867 shootings, 235 involved black suspects, making the probability of police killing a black 27.1%.

Dividing 3.58% by 27.1%, we find that about 13% of black men shotkilled by police are unarmed. This means that if you hear that the police shot a black person, you can guess that about one out of every eight times the suspect would be unarmed.

That provides a discouraging picture of what it means to be black and be policed by in America. But police work is difficult and not every unarmed suspect should be considered safe to officers or the public. It could well be true that the unarmed suspect posed an imminent danger to public safety.

If that is the case, however, then we would expect unarmed white suspects to be killed at a similar rate to blacks. Using the same statistical principles that I did above, which mathematicians call Bayes’ Theorem, we can figure out the analogous rate for whites. When we do, I calculate that unarmed white victims make up 29 of the 867, or 3.34%, of all shootings, while 443, or 51.1%, of the shootingspolice killings involved whites. This results in a probability that a white suspect killed by police was unarmed, or about one in every 16 shootings of white suspects.

These calculations mean that the police are twice as likely to have killed an unarmed suspect if that suspect is black rather than white.a black suspect killed by the police is twice as likely to be unarmed as a white suspect to be killed by the police. I don’t know that I have much else that I can add — the number, in this case, kind of does speak for itself.


  1. The data that I use here are those posted by Aldhous and are, thus, out of date with the most current Washington Post analysis of officer-involved shooting deaths, which sadly continues to increase rapidly.↩

  2. UPDATE As Andrew Wheeler pointed out, I used "shot" and "killed" interchangeably, which is incorrect. The dataset only includes suspects who have been killed by the police. I should have been more careful in the language that I used. I have edited the post to be consistent with the probability that I solved. I regret the error.↩

  3. UPDATE 2: This will teach me to try to post while my mind is fried grading. One more edit for correction to get the conditional correct in the last paragraph.

7 thoughts on “racial disparities in police killings using bayes theorem”

  1. Mike,

    I’m having a hard time following along. This dataset is only of police shootings in which someone was killed. Your final statement, “These calculations mean that the police are twice as likely to kill an unarmed suspect if that suspect is black rather than white.” I would take to mean in probability notation:

    P(Kill|Unarmed,Black) = 2*P(Kill|Unarmed,White)

    What you end up calculating in the end is:

    [P(Black|Killed,Unarmed)*P(Unarmed|Killed)]/P(Black|Killed) = P(Unarmed|Killed,Black)

    and ditto for white victims, so we have

    P(Unarmed|Killed,Black) = 2*P(Unarmed|Killed,White)

    So I would rewrite your last sentence as “Given the race of a person killed in a police shooting, a black person is twice as likely to be unarmed compared to white person.”

    You could probably make reasonable guesstimates to get P(Kill|Unarmed,Race), but it is not possible with just this dataset. This baseline problem is confusing in the exposition in various places, so

    “Rather than figure out the probability that an unarmed suspect was black, it would be important to know the probability that a black suspect shot by police was unarmed.”,

    again can’t answer this question with this dataset. You have to change “black suspect shot by police” to “black suspect killed in a police shooting”.

    Also “If you hear about a police shooting, about one out of every 28 times the police have shot an unarmed black person.” you would need to change the general “police shooting” to “police shootings in which someone died” to be accurate.

    In general, I just have hard time following textual narratives like this. I much prefer algebra, and for those who like figures here are some examples I think are good, http://understandinguncertainty.org/using-expected-frequencies-when-teaching-probability.

    ——

    For a review, Bayes Theorem is:

    P(A|B) = [P(B|A)*P(A)]/P(B)

    And for this data here Mike gives us:

    P(Unarmed|Killed) = 82/867
    P(Black|Killed) = 235/867
    P(White|Killed) = 443/867
    P(Black|Killed,Unarmed) = 31/82
    P(White|Killed,Unarmed) = 29/82

    So that

    [P(Black|Killed,Unarmed)*P(Unarmed|Killed)]/P(Black|Killed) = P(Unarmed|Killed,Black) ~= 13%
    [P(White|Killed,Unarmed)*P(Unarmed|Killed)]/P(White|Killed) = P(Unarmed|Killed,Black) ~= 6%

    Which is where we get the 2:1 ratio.

    Notice that Killed never leaves the conditional – it can’t with this dataset, as it is only a dataset of police shootings that resulted in someone being killed.

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    1. Andrew — Thank you for your comments. You are absolutely correct that the entire dataset is predicated on those killed by police officers so, all probabilities are P(.|.,killed). Thank you for pointing out the ambiguity in the language. I have tried to fix the problematic wording above.

      Your probability statements at the bottom are exactly what I used to solve the problem — thanks for including them. I was trying to provide a narrative explanation that I could use alongside the formal probability statements since I know that many students freak out when they see probability notation.

      Although embarrassing, hopefully, it will encourage others not to be afraid to work out problems like this.

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  2. Thoughtful and useful, though I really dislike the language of “unarmed suspect” that is used by the media and scholars alike. What is at issue in terms of the justifiable use of lethal force — as you hint at — is whether the individual faces a reasonable (as determined by the legal system) threat of death or grave bodily injury.

    So, whether or not the suspect is “armed” with a weapon (gun, knife, bat, vehicle) is less important legally AND in the moment than whether they pose a lethal or grave threat. According to LEOKA, of 72 LEOs feloniously killed in 2011, 3 were killed with their own firearm. 21 of 72 LEOs killed in 2011 were 0 to 5 feet from their offenders. Someone within 5 feet of you who is appears “unarmed” can still pose a lethal or grave threat. Indeed, someone within 20 feet of you can pose such a threat since they can close that gap in a matter of seconds.

    Some of the “unarmed” individuals killed by police in the Washington Post dataset had physically assaulted the LEO, and we know that about for 6% of civilians murdered every year, their killer was “armed” with their hands, fists, feet, etc. (FBI UCR data). So, the possibility of being beaten to death or to have your weapon taken and used against you is a reality for LEOs.

    Which is not to say there aren’t racial disparities in policing. And which is to say nothing of the Chicago shooting (or any other LEO shootings deemed unjustified) and that particular officer is being charged with murder. But the language of “unarmed” seem less useful to me legally and analytically than whether the person poses a lethal threat, even though using that measure makes these analyses even more complicated.

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  3. Typo in Andrew’s comment: [P(White|Killed,Unarmed)*P(Unarmed|Killed)]/P(White|Killed) = P(Unarmed|Killed,Black) ~= 6% ** SHOULD BE Killed,White

    Re David Yamane’s comment (Hi, David) I’ve talked to a lot of law enforcement folks, including a bunch of deputy sheriffs I was on a “body camera” committee with this year. One of the unsettling realizations is that law enforcement’s weapons are a danger to law enforcement, as well as to civilians, especially when they shoot people they know are unarmed because they fear loss of control of their own weapon. If they were not armed, this fear would not be there. It is also however important to recognize that officers are shot with their own weapons by unarmed people they are grappling with about 4 times a year, while police shoot several hundred unarmed people a year. Police are trained to get something over fast, and that is pretty scary.

    Other points the deputies made is that they are not rambo, not in shape, and cannot subdue someone by wrestling with them. So the implication is they shoot people not just because they fear losing control of their weapon, but because they cannot win a physical struggle and, once they begin a confrontation, they feel they absolutely cannot back down. The guys I talked to said this: I cannot let you disobey an order. (!)

    I and the other “citizen” members of the committee bonded with the deputies (at least the guys we got to know well on our subcommittee) and like them, but this does not make me any more comfortable with the level of armament carried by law enforcement nor the training or whatever that leads them to believe they should escalate confrontations.

    The deputies also stressed that most shots fired by police miss, and that things happen so fast they don’t really have time to reflect. I also talked to a retired police officer (black) who killed someone, who talked about how fast it happened and the consequences for him. His experience in law enforcement does not make him less concerned about racial disparities in arrests, incarceration, and police killings.

    We also had conversations about training in deescalating situations, which they said they think they don’t get enough of. Lots of range practice with the weapons, a lot less on how to deal with situations.

    I’m also really concerned about the “put my life on the line every time I go out” rhetoric, which I think fuels the fear. The police in my community say this but the last time a police officer was shot here was 1932. In fact, police don’t even make the top 10 in deaths on the job. AND most of the deaths on the job are vehicular accidents.

    Another point back to the OP: this racial difference in the probability of being unarmed if killed (or of being killed) is not necessarily due to individual level differences, but probably is place/jurisdiction related: the police in some places (regardless of their individual traits or dispositions) are more likely to kill people than in other places, and these places are more likely to be places where there are more black people.

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    1. olderwoman (Pam, is that you?) – What a valuable experience to serve on that committee. Really profound insights into some current problems with policing, made more acute due to the militarization of the police, poor recruitment and training, lax standards, the blue wall of silence, implicit and explicit racism – all of the things your local LEOs told you about. Even given my social class privileges, I do not like to interact with police because of the feeling that if I make one wrong move things could go bad quickly. That is sad, and I can only imagine how much worse it is for others.

      Your final comment is most important. I have often noted that when it comes to felonious homicide in general, some places in the United States are as safe as England and Australia and some places are as dangerous as parts of Central and South America (cf. Papachristos). Do police shoot more unarmed civilians in the latter parts of the US than the former? I would guess the answer is yes.

      It would be great to have the data to model the effect of overall rates of violent crime on the likelihood of police shootings, alongside race (extremely difficult to disentangle, as you note). But I would still prefer that data to measure reasonable threat of death or grave bodily harm than the more blunt indicator of “unarmed.”

      Even if most unarmed citizens do not pose a reasonable threat to LEOs, some do. And the risk assessments are very complex in situations where something is very unlikely to happen but if they do happen the consequences are extreme. So, at the same time I am critical of certain police officers, police actions, and styles of policing, I’m not changing places with them even in my England-like safe neighborhood.

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      1. Belated thanks for sharing this article link, pseudonymous olderwoman! It does seem to answer the question of the effect of race (both as an individual and communal measure) and being “armed” on the likelihood of being shot, NET OF the local crime rate. I’m looking forward to reading it beyond the abstract.

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