WASHINGTON—When 24-year-old Albert Jermaine Payton wielded a knife in front of the police in this city’s southeast corner, officers opened fire and killed him.

Yet according to national statistics intended to track police killings, Mr. Payton’s death in August 2012 never happened. It is one of hundreds of homicides by law-enforcement agencies between 2007 and 2012 that aren’t included in records kept by the Federal Bureau of Investigation.

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