Fatal shootings by police hover around 1,000 in U.S.

Fatal shootings by police are the rare outcomes of the millions of encounters between police officers and the public. Despite the unpredictable events that lead to the shootings, in each of the past four years police nationwide have shot and killed almost the same number of people: nearly 1,000.

Last year police shot and killed 998 people, 11 more than the 987 they fatally shot in 2017. In 2016, police killed 963 people, and 995 in 2015.

Years of controversial police shootings, protests, heightened public awareness, local police reforms and increased officer training have had little effect on the annual total.

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.

“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 project relies on news accounts, social media postings and police reports.

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.

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.

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.

Spiegelhalter 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 is a statistical tool that has been used for hundreds of years to predict rare events.

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 U.S. 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.

Andrew Wheeler, a criminologist and statistician at the University of Texas at Dallas, said spikes in news coverage or on social media can make it seem like fatal shootings by police 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 the number changes much.