The U.S. Department of Justice issued two reports last week, both growing out of the Ferguson shooting of Michael Brown.

The first report says in plain English what facts have been established by an autopsy on Michael Brown’s body; by DNA examination of officer Darren Wilson’s gun and police vehicle; by examination of blood stains on the street and by a medical report on officer Wilson.

The bottom line shows what a complete lie was behind all the stories of Michael Brown being shot in the back or being shot while raising his hands in surrender. Yet that lie was repeated, and dramatized in demonstrations and riots from coast to coast, as well as in the media and even in the halls of Congress.

The other Justice Department report, issued the same day — “Investigation of the Ferguson Police Department” — was a complete contrast. Sweeping assumptions take the place of facts, and misleading statistics are thrown around recklessly. This second report is worth reading, just to get a sense of the contrast with the first.

According to the second report, law enforcement in Ferguson has a “disparate impact” on blacks and is “motivated” by “discriminatory intent.”

“Disparate impact” statistics have for decades been used, in many different contexts, to claim that discrimination was the reason why different groups are not equally represented as employees or in desirable positions or — as in this case — in undesirable positions as people arrested or fined.

Like many other uses of “disparate impact” statistics, the Justice Department’s evidence against the Ferguson police department consists of numbers showing that the percentage of people stopped by police or fined in court is larger than the percentage of blacks in the local population.

The implicit assumption is that such statistics about particular outcomes would normally reflect the percentage of people in the population. But, no matter how plausible this might seem on the surface, it is seldom found in real life.

Blacks are far more statistically “over-represented” among basketball stars in the NBA than among people stopped by police in Ferguson. Hispanics are similarly far more “over-represented” among baseball stars than in the general population. Asian Americans are likewise far more “over-represented” among students at M.I.T. and Cal Tech than in the population as a whole.

You can find innumerable examples of such group disparities throughout recorded history.

An international study of the ethnic makeup of military forces around the world found that “militaries fall far short of mirroring, even roughly, the multi-ethnic societies” from which they come.

Even with things whose outcomes are not in human hands, “disparate impact” is common. Men are struck by lightning several times as often as women.

Since Ferguson is 67 percent black, the greatest possible “over-representation” of blacks among those stopped by police or fined by courts is 50 percent. That would not make the top 100 disparities in the United States or the top 1,000 in the world.