With the resurgence of openly hateful rhetoric from certain political candidates, racial profiling is once again a mainstream issue. While there are a plethora of arguments on why it is wrong, involving human dignity, human rights, negative externalities, etc, I’m going to ignore all of that. I’m not saying it’s not important or that those are not necessary issues to consider when thinking about racism in America, but in this case, they are not necessary to show the negative effects of this particular policy.

Before I even go in to this, I want to give a quick anecdote. Quite a long time ago, when I was in elementary school, we were supposed to be given a rudimentary sex ed class. It turned out to instead be a lecture on why racial profiling is okay, and should be encouraged. The argument went something along the lines of beginning by assuming people of some characteristics are more likely to commit crimes, and claiming policing those people more heavily will be of greater benefit to public safety. This is a pretty widespread view of the issue, and while I didn’t feel quite right with the idea at eight years old, I did not yet have the words to articulate my discontent. I now however, have something more important than words: mathematics(though the math presented here is by no means a formal proof, and should not be thought of as such). As always, this is a thought experiment designed to present questions, and make the case for more thorough examination.

In social sciences, it is always important to think about the data we have, as compared to the data we don’t have. There is quite a lot of data on arrests(though not nearly enough for the more qualitative aspects). This data on arrests however, is not the same thing as having data on crime. We only know that a crime has been committed because it has been observed, but given the nature of crime, we would be fools to assume that we observe all crime.

For the remainder of this piece, I’ll use as the definition of racial profiling: the systematic targeting of people of a given racial characteristics for a higher rate of investigation. I will also posit that a necessary condition for racial profiling is evidence that people with those characteristics are more likely to commit a crime.

Let’s start here in a perfect world. There are two groups in a society, group gA, and group gB. They are the exact same in every way except their grouping, and their number of members. Assume gA is a smaller group. Also assume they have a rate of criminality greater than zero, but equal across groups. Also assume there is no population growth, and that the number of police is fixed. Let’s consider what happens if we use profiling as a tool to decrease criminality.

If we do this in the simplest way possible, by setting the ratio of persons investigated equal to the ratio(as a share of population) of crimes committed, we have a basic process to look at. Let’s begin at year zero, where the ratio of people investigated will be equal to the population ratio. In this year, there will be some statistical variance between the populations, and it’s almost assured that one of them will appear to have a higher rate of criminality with respect to population. Then, in the next year, that group will have a higher chance of being investigated. This means there is a greater chance of finding criminal behavior across that group(even with the same rate of criminality), and as a share of crimes, it is likely they will appear higher, which means in the next year, they will be investigated at an even higher rate, which will continue the process. I should note here, that this process comes about by an error in statistical reasoning. The statistic being compared is the relative proportion of crimes recorded, but because we are investigating different populations at different rates, this does not reflect the actual rate of criminality. For this, we would need to look at crimes recorded as a share of investigations performed, which is something difficult to record(suspicion is an internal thing, and can only be self reported, which is precarious at best). Also, this assumption would not be reflective of reality. How many times do you hear someone talk about hit rates, compared to relative arrests?

In the preceding example, I should note that the relative probability of each group being targeted for profiling will be roughly equal(because the binomial distribution is mostly symmetric with a large n, though a hyper-geometric distribution would be more appropriate). We can however, consider a case with a few more assumptions, that would not yield the same result. Assume further that the profiling action is stronger for higher perceived levels of criminality than low values. This seems intuitive, because how often do you hear people talk about the low crime rates of groups? Here the result that follows is that the smaller group will be more likely to be profiled, even when the groups are the exact same in all ways but size. A mathematical fact of statistics is that a smaller group has more variance than a larger group. This means the lower population group will be more prone to high values than the higher population group. If we weigh higher deviations from the mean in the positive direction more strongly, we will find that the investigation rate for this group will be more likely to become higher. In this way, the profiling will fall on a group, simply because it has a smaller size. I should again note, that doing things in this manner would be a bad use of statistics. This assumes that the magnitudes that these are tested against is constant between groups(e.g. if gA shows a 10% higher crime rate as a share of population), but this would not occur if we instead tested for extreme values by setting up a confidence interval based on the number of the population that were actually sampled. Again however, how often do you hear police talking about confidence intervals?

The overall point here, is that the act of profiling invalidates the data that is used to justify profiling, thus invalidating the necessary condition posited earlier. Even in a perfect world, human judgement is imperfect, and we have to be very careful about using data to decide an action which will then alter the data. This is a pretty simple thought experiment(I’m working on a more formal model to look at this more carefully) but it does call into question how the bad use of data can really lead to negative effects with a policy like profiling. I’ll also note that even if the statistics used by police were perfect, there are a host of other arguments for why this is a bad policy. The next time you hear an argument about relative crime rates, I ask you to really think about what the data is being used, and how it is affected by the actions it is supposed to support.