Years of controversial police shootings, protests, heightened public awareness, local police reforms and increased officer training have had little effect on the annual total. Everyone agrees — criminal justice researchers, academics and statisticians — that all of the attention has not been enough to move the number.

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Mathematicians, however, say that probability theory may offer one explanation. The theory holds that the quantity of rare events in huge populations tends to remain stable absent major societal changes, such as a fundamental shift in police culture or extreme restrictions on gun ownership, which are unlikely.

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“Just as vast numbers of randomly moving molecules, when put together, produce completely predictable behavior in a gas, so do vast numbers of human possibilities, each totally unpredictable in itself, when aggregated, produce an amazing predictability,” said Sir David Spiegelhalter, a professor and statistician at the University of Cambridge who studies risk and uncertainty.

The Washington Post began tracking the shootings after Michael Brown, an unarmed black man, was killed in 2014 by police in Ferguson, Mo. A Post investigation found that the FBI’s tracking system undercounted fatal police shootings by about half, because of the fact that reporting by police departments is voluntary and many departments fail to do so. The ongoing Post project relies on news accounts, social media postings and police reports.

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In the wake of the findings by The Post and similar reporting by the Guardian, the FBI in 2015 committed to improving its tracking and last month launched a system to track all police use-of-force incidents, including fatal shootings. The new system, however, is still voluntary.

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The Post’s reporting shows that both the annual number and circumstances of fatal shootings and the overall demographics of the victims have remained constant over the past four years.

The dead: 45 percent white men; 23 percent black men; and 16 percent Hispanic men. Women have accounted for about 5 percent of those killed, and people in mental distress about 25 percent of all shootings.

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About 54 percent of those killed have been armed with guns and 4 percent unarmed.

“We’ve looked at this data in so many ways, including whether race, geography, violent crime, gun ownership or police training can explain it, but none of those factors alone can explain how consistent this number appears to be,” said Geoffrey Alpert, a criminologist at the University of South Carolina who has studied police shootings for more than three decades.

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Mathematicians say that the fact that the number of shootings is stable even though each one is a complex, isolated event can be explained through a fundamental principal of statistics coming out of probability theory. This was used notably to examine the accuracy of German bombings of London during World War II, according to Spiegelhalter.

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He said that every year there are a huge number of instances in which police interact with civilians. In each, there is a tiny probability that a shooting will occur, usually as the result of a chain of unpredictable events.

“If this probability stays reasonably constant from year to year, then the beautiful theory of probability says that the number of police shootings will be a ‘Poisson’ random variable,” he said.

Poisson’s random variable, named after Siméon Denis Poisson, a 19th-century French mathematician, is a statistical tool that has been used for hundreds of years to predict rare events, such as how often Prussian soldiers died from horse kicks.

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Spiegelhalter used this probability theory — the chance that a given event will occur — to analyze homicides in London and whether several plane crashes in one weekend meant flying has become less safe. It has not.

In almost every case, a police shooting is an individual, unrelated event that can’t be predicted, he said.

But because the data covers the entire United States and millions of police-civilian interactions, he said statisticians can make predictions about the pattern of shootings, based only on knowing the overall number over four years. Then they can see if the prediction fits the observed pattern.

With about 1,000 shootings each year, Spiegelhalter said he would expect the number to range between around 940 and 1,060 annually, as long as no major systematic change occurs, like a dramatic reduction in crime rates.

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Andrew Wheeler, a criminologist and statistician at the University of Texas at Dallas, agreed with Spiegelhalter’s theory about the number of shootings and said spikes in news coverage or on social media can make it seem like they are happening more often, even when the overall rate is unchanged.

He likened the phenomenon to flipping a coin and having it land on heads three times in a row. One may conclude heads is more likely, but if the coin is tossed 1,000 times, the 50-50 probability becomes clear, he said. Even if the total number of fatal shootings fluctuates yearly, it doesn’t mean that the number is changing in a meaningful way.

“Most things like homicide and crime rates don’t change by a lot, and when they do, it’s often consistent with random chance,” Wheeler said.

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This is one of the reasons experts, including Wheeler, have cautioned against using fatal police shootings as a measure of police conduct overall: They are rare events given all the interactions with the public.

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“Looking at other more common applications of the use of force is much more telling,” Wheeler said. “Since things like Taser usage or open hand strikes are more common, it is easier to see when a police department or individual officer is an outlier.”

But other statisticians believe it may be possible to reduce the fatal shootings through policy changes, for example, by preventing high-risk individuals from having access to guns.

Daniel Nagin and David Choi, statisticians at Carnegie Mellon University in Pittsburgh, said they are exploring another theory based on the binomial distribution. That model considers the 1,000 fatal shootings to be a subset of 1,300 highly charged interactions annually between police and civilians. The 1,300 is an estimate derived from four years of The Post’s data.

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An advantage of the model is that it acknowledges that some interactions are more likely to lead to a fatal shooting, such as when a police officer encounters someone armed with a gun, Nagin said. He also believes it suggests there is less variation in the annual shooting totals, about 30 cases a year.

The model, however, may underestimate the total number of times police encounter highly charged events — police shooting data made public by large cities shows that officers shoot and wound people more often than they kill, he noted.

Nagin said the binomial model is important to consider because, unlike the Poisson model, it suggests that the outcomes of the many individual steps that lead up to a shooting can be changed.

“The distinction between these two models is important because of their very different policy implications,” Nagin said. “If policy can reduce the number of high-risk events, by better training for example, much would be gained.”

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Criminologists would like to believe the number of police shootings is not immutable. Their focus has been on what societal factors may affect the toll.

Some criminologists cite gun ownership, pointing out that in developed countries where guns are banned, police seldom shoot and kill people.

In England and Wales, with a combined population of about 70 million, police fatally shot six people in 2016, the most since 2004, according to an annual report from the Independent Office for Police Conduct in England.

Others say the nation’s violent culture leads to shootings, as police must routinely confront armed and dangerous people. Still others point to the wide discretion police are granted in using lethal force, saying the killings will not subside until police use more restraint, an enormous challenge that will require fundamental changes in the nation’s 18,000 police departments.

“Crime has nothing to do with it,” said Franklin Zimring, a criminologist at the University of California at Berkeley. “The number remains high because the police rules for shooting people don’t change, and police forces tolerate shootings of people armed with weapons that aren’t usually life-threatening to police.”

Justin Nix, a criminologist at the University of Nebraska Omaha, who worked with The Post to analyze the data, said police shootings appear to closely track three factors: the total number of police interactions with the public, the nation’s murder rate and the yearly number of arrests for violent crime.

Based on his research, Nix said that as those variables fluctuate, so too should the number of annual fatal shootings. But even the most dramatic changes in the former would produce a tiny shift in the latter, given how rarely they result from police-citizen interactions.

The most recent data on police interactions, drawn from a Bureau of Justice Statistics survey, shows that in 2015, officers had contact with the public on more than 50 million occasions. This included a range of encounters, including traffic stops, people seeking information and individuals reporting crimes.

Nix said those interactions led to fatal shootings about 0.00002 percent of the time.

He said, for example, a 2 percent drop in the arrest rate would probably lead to a drop of about 10 fatal police shootings per year.

Some cities seem to defy the formula. The District, for example, ranks in the top half of America’s most dangerous cities out of the nation’s 50 largest. In 2017, D.C. police seized more than 2,000 illegal guns. Yet, police in the District shot and killed only two people in 2017.

That is a dramatic decline from the 1990s, when a Post investigation showed that the District led the nation in fatal police shootings per capita. In that decade, police shootings in the city peaked at 15. The series prompted Justice Department intervention and an extensive retraining of officers. A steep and immediate drop in fatal shootings ensued.

David Klinger, a criminologist with the University of Missouri at St. Louis, said there is no official data set that captures all the elements needed to fully understand the factors that lead to fatal police shootings.

He pointed out that The Post’s data covers only fatal shootings — not the number of times police shot people without killing them. He also said the data fails to record the type of weapon used and the number of rounds fired, noting that police could be using rifles more frequently, or firing more bullets, two factors that would increase lethality.

There is limited historical context for the number, he added, because there is only four years of data.

“We need a much longer trend line and much more detail to show us where things really change,” Klinger said. “The question we can’t answer is, ‘Are we looking at the leading edge of a trend, or the end of one?’ ”

David Hemenway, a professor at Harvard’s T.H. Chan School of Public Health, said he was surprised at how many people were killed each year when he first heard The Post’s count in 2015, but today he is not surprised that it is largely unchanged.

The prevalence of guns is a key factor for Hemenway.

“Where there’s a lot of civilians that own guns, fatal police shootings happen more often,” Hemenway said.

He and other researchers compared the rates of fatal police shootings in states with the highest levels of gun ownership with rates in states with the lowest levels.

High gun-ownership states, such as Alaska, Georgia, Idaho, Kentucky and Louisiana, had 3.6 times more fatal police shootings than the low gun-ownership states, such as Connecticut, Hawaii and Massachusetts, the study found.

Zimring, the Berkeley professor, said his research has also led him to conclude that the saturation of firearms in the United States is an overwhelming factor in police shootings. Police, he said, tend to feel more threatened because so many people are armed.

In 2017, Zimring wrote “When Police Kill,” a book that analyzed when police are killed and when they kill. The annual number of officers killed has fallen by more than 65 percent since 1976, he said, but the number of people police kill remains relatively high.

“There are 100 million homes with guns in them and 10 million cars, so every time an officer walks into a home or pulls over a car they could encounter a gun,” Zimring said. “This is a characteristic of American life, which creates a unique set of hazards to police and more killings by police.”

Zimring said police also kill too many people who pose little threat. His review of shooting data in 2015 for armed people shot by police showed that 57 percent of those police killed were reportedly armed with a gun, but the other 40 percent were armed with things that usually don’t kill police officers.

“The rules police follow on when to shoot someone have not changed,” he said. “So the number of people they kill does not change either.”

Chuck Wexler, executive director of the Police Executive Research Forum, a Washington-based think tank that advises the chiefs on policy issues, said some police shootings are difficult to avoid, such as when an officer confronts a gunman. But he said many shootings can be avoided with better training, especially in cases involving mentally ill people and individuals armed with knives.

Wexler said New York City shows how training can reduce fatal police shootings.

In 1971, the city had 314 officer-involved shootings, 93 of which were fatal, he noted.

The next year the city passed a law prohibiting officers from shooting into vehicles, a practice still permitted by many police departments.

Within two years the city reduced police shootings to 121, with 41 fatal. By 2015, after a period when crime dropped enormously, the number had fallen to 23 people shot by police with eight killed. No New York City police officers have been seriously hurt by someone in a car in the 46 years since the ban in shooting into vehicles took effect, he said.

Wexler said his organization has made its mission to train police departments in how to reduce police shootings. But Wexler adds that they have been able to train only about 10 percent of the nation’s 18,000 departments.

“We have a long way to go,” he said, “but the shooting number will come down.”