A Consultant's Casebook
about me: Andres Inn, Ph.D.
Shootings by Police
The first problem I encountered when I examined the data was the relatively low base rate of police shootings. Imagine, my surprise when I found that, in a city the size of Dallas, with over 2,000 uniformed police, there were only 80 times police officers fired shots during a 3-year period.
It is a fact that distribution shapes limit regression analyses and correlations. If data distributions do not match, correspondence of high scores in one distribution to high scores in another are impossible. Therefore, if I were to develop a selection test to predict racism or overly aggressive response potential, it would likely be normally distributed (the typical bell-shaped curve). But with police shootings showing such a skewed distribution (that is most people never involved in a shooting, and only a few involved in one), it would be statistically difficult to achieve prediction. Making the world safer by selecting out rogue officers appeared unlikely to succeed.
It appeared especially unlikely, because the distribution of shootings closely approximated the Poisson distribution. First described by Emile’ Poisson to describe random events, the Poisson distribution has been used to predict the frequency of being kicked to death by mules in the Prussian army, the frequency of suicides in Paris districts, as well as the likelihood of German V-2 bombs falling on London neighborhoods during the Second World War. The Poisson distribution appears to accurately describe the likelihood of shootings by police officers as well!
The Poisson formula is simple and is presented below. The only parameter that needs estimation is lambda (l), the arithmetic mean or average of the Poisson distribution. In the case of the shooting data, there were 63+2*7+3*1=80 shootings divided by the number of officers 335, and l=.24. Substituting in the equation below, we can determine the probability of an officer being involved in zero shootings, p(0) shootings as .79, and the expected number of officers involved in zero shootings would be .79*335, and so on.
p(x) = e-llx/x!
Clearly, the number of shootings exactly matched the Poisson distribution – a random distribution. This process describes what to expect if we were to assemble all 335 officers in a group and while blindfolded threw 80 darts at the group, each dart representing one shooting incident. By chance, we would hit some officers with two darts (7), and one officer thrice. Sixty-three (63) officers would be hit once, and most officers would remain un-hit.
I performed a number of analyses all of which yielded much the same results. No racial bias could be concluded from the data. But, as might be expected, shooting incidents appeared to relate to the degree of hazard faced by the officers; that is, officers fired more shots at suspects who were armed and firing at them. And, unexpectedly, officers appeared no more or less likely to wound a suspect in any case.
These results were not at all what I considered possible when I began the research. And, these results remained counter-intuitive. How could it be possible that all police officers have the same probability of being involved in shootings (only one l for the entire department) when officers are assigned to different precincts and beats, some with markedly different types of crime?
One additional feature of these data remained unexplained. Over a larger sample, there was a positive correlation between complaints against police officers and their being involved in shooting incidents – the more complaints, the greater the likelihood an officer had used his weapon. More surprising was that there was a similar positive correlation between commendations and shooting incidents – the more commendations, the greater the likelihood an officer had used his weapon!
 Among 335 police officers with continual patrol assignments over the entire 3 year time period.
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