"When Mexico sends its people, they're not sending their best... They're sending people that have lots of problems... They're bringing drugs. They're bringing crime. They're rapists. And some, I assume, are good people."

- Donald Trump, presidential announcement speech, June 16.

"I said tremendous crime is coming across. Everybody knows that's true. And it's happening all the time. So, why, when I mention, all of a sudden I'm a racist?"

- Trump on Fox News, July 5.

"The Mexican Government is forcing their most unwanted people into the United States. They are, in many cases, criminals, drug dealers, rapists, etc."

- Trump statement issued July 6.

Several readers asked the Washington Post to fact-check Trump's controversial comments.

What do the data tell us about the criminal threat of immigrants?

Though the data are incomplete, a range of studies show no evidence that immigrants commit more crimes than native-born Americans. In fact, first-generation immigrants are predisposed to lower crime rates.

Immigration and crime levels have had inverse trajectories since the 1990s: Immigration rose as crime fell. Some experts say the influx of immigrants actually helped fuel an overall drop in crime rates.

In his July 6 statement, Trump clarified that he was referring to cases where undocumented immigrants commit violent crimes or smuggle drugs. He pointed to the recent arrest of an undocumented immigrant - a felon who had been deported five times to Mexico - in a San Francisco killing.

Trump's campaign pointed to data from the U.S. Sentencing Commission, which tracks offenders in federal prisons by "primary offense" - the charge with the longest maximum sentence when a person is convicted of multiple charges. Of 78,022 primary-offense cases in fiscal year 2013, 38.6 percent involved illegal immigrants. Most of their cases (76 percent) were immigration-related. Of total primary offenses, 17.6 percent of drug-trafficking cases and 3.8 percent of sex abuse cases involved illegal immigrants. Of 22,878 drug-crime cases, 17.2 percent were illegal immigrants.

But these numbers don't reflect wider crime trends among noncitizens. Federal prisoners made up 10 percent of the nation's total incarcerated population in 2013.

The Congressional Research Service (CRS) found that the vast majority of unauthorized immigrants who are jailed for federal crimes are not charged with murder, drug trafficking or illegal gun trafficking. CRS also found that noncitizens make up a smaller slice of the inmate population in state prisons, compared with their portion of the total U.S. population.

An analysis of 2010 Census data in a forthcoming report from the American Immigration Council, a pro-immigrant group, says 1.6 percent of immigrant males 18 to 39 years old were jailed, compared with 3.3 percent of native-born males. That disparity has been consistent in the decennial census since 1980, the report says.

Are countries like Mexico "not sending their best"?

Immigration offenses account for most of the federal convictions of immigrants (the majority of whom were from Mexico), followed by drug and traffic violations. Sex offenses made up just 1.6 percent of total crimes in 2013.

Inmate legal status is not always tracked at local jails or state prisons. The Government Accountability Office's 2011 analysis collected reports from 2003 to 2009 to the Justice Department's State Criminal Alien Assistance Program, through which states and localities get reimbursed for convicting and incarcerating inmates of illegal or unknown immigration status (mainly from Mexico).

The GAO found that drug offenses made up the majority of convictions in fiscal year 2008 in the five states (Arizona, California, Florida, New York, and Texas) with the largest populations of such inmates.

Cartel- and gang-related arrests along the Texas border fell after a surge in 2014, the Houston Chronicle reported. In 2013 the Center for Investigative Reporting found four out of five drug-smuggling arrests involved U.S. citizens.

The theory is that immigrants generally have a strong incentive to stay out of trouble - especially undocumented immigrants, who risk deportation. And those who legally are here (or are pursuing legal status) must pass criminal background checks.

"Immigrants in general - unauthorized immigrants in particular - are a self-selected group who generally come to the U.S. to work. And once they're here, most of them want to keep their nose down and do their business, and they're sensitive to the fact that they're vulnerable," said Marc Rosenblum of the nonpartisan Migration Policy Institute.

Interestingly, crime rates rise as generations of immigrants assimilate into America. Second-generation immigrants have similar crime rates as native-born Americans.

Drug smuggling and violent crimes do exist, but the cases don't reflect larger trends in the immigrant population. What we do know about crime rates show Trump's assertions about a crime wave are not accurate.