For decades, scholars have acknowledged that local crime rates cannot be predicted by officer strength and police budgets. Sometimes a boost for policing is followed by a drop in crime; sometimes it isn’t.

History shows that homicides fell after more officers were hired 54 percent of the time, according to Aaron Chalfin, a criminologist at the University of Pennsylvania who has studied ways of driving down crime.

“Crime goes up and down for a million reasons that are completely independent of the police,” Dr. Chalfin said. “But we know, on average, if you look across many cities for many years, there is an effect.”…

Overall, more Black lives were saved than white lives when police officers were added, but in Southern cities with larger Black populations the homicide rate did not budge, according to an early draft of the paper.