News organizations widely report that there are 11 million unauthorized immigrants living in the United States. But where does this figure come from?

Donald Trump has falsely asserted: “It could be three million. It could be 30 million. They have no idea what the number is.”

In the third debate, Hillary Clinton said, “We have 11 million undocumented people. They [undocumented parents] have 4 million American citizen children. 15 million people.”

The confusion is warranted. After all, the U.S. Census Bureau does not ask people about their immigration status, so how can we know much about the unauthorized foreign-born population?

Well, demographers have figured out a simple and effective way to estimate the number of unauthorized immigrants. In the last five years, my colleagues Frank D. Bean, James D. Bachmeier and I have conducted a series of studies that evaluate this method and its assumptions. Our research on the methods used to estimate the size of this group indicates that these estimates are reasonably accurate.

Here’s how it works.

A simple formula

Beginning in the late 1970s, a group of demographers consisting primarily of Jeffrey Passel, Robert Warren, Jacob Siegel, Gregory Robinson and Karen Woodrow introduced the “residual method” for estimating the number of unauthorized immigrants living in the country. At the time, Passel and his collaborators were affiliated with the U.S. Bureau of the Census and Warren with the Office of Immigration Statistics of the Immigration and Naturalization Service. Much of this work was published in the form of internal reports, but some of it appeared in major journals.

The residual method uses an estimate of the total foreign-born population in the country (F), based on U.S. Census data. Researchers then subtract from it the number of legal immigrants residing here (L), estimated from government records of legal immigrants who receive “green cards” minus the number that died or left the country. The result is an estimate of the unauthorized population (U):

F – L = U

Various adjustments are typically made to this formula. Most adjustments are minor, but a particularly important one adjusts for what researchers call “coverage error” among the unauthorized foreign-born. Coverage error occurs when the census data underestimate the size of a group. This can occur when people live in nonresidential or unconventional locations – such as on the streets or in a neighbor’s basement – or when they fail to respond to the census. Coverage error could be particularly high among unauthorized immigrants because they may be trying to avoid detection.

Currently, the Department of Homeland Security and the Pew Hispanic Center are the two major producers of estimates of the unauthorized foreign-born population. This report, compiled by Passel, who now works at Pew, summarizes many of the estimates. It shows that the estimated number increased steadily from 3.5 million in 1990 to 12.2 million in 2007, but declined between 2007 and 2009 and has since stabilized at around 11 million.

How accurate are the estimates?

The residual method has been widely used and accepted since the late 1970s. Within a reasonable margin of error, it predicted the number of unauthorized immigrants to legalize under the Immigration Reform and Control Act of 1986, which, among other things, granted permanent residency status to unauthorized immigrants who had been living in the country since 1982. The residual method predicted that about 2.2 million met the residency requirement and the actual number to come forward was about 1.7 million.

Both Department of Homeland Security and Pew have used the residual method to produce estimates of the unauthorized population since 2005. Despite using slightly different data and assumptions, Pew’s and the Department of Homeland Security’s estimates have never differed by more than 600,000 people, or 5.5 percent of the total unauthorized population.

Nevertheless, many skeptics question a key assumption of the residual method, which is that unauthorized immigrants participate in census surveys. Both Pew and the Department of Homeland Security inflate their estimates to account for the possibility that some unauthorized immigrants are missing from census data. Pew inflates by 13 percent and the Department of Homeland Security by 10 percent. But is this enough?

AP Photo/Damian Dovarganes

My colleagues and I estimated coverage error among Mexican immigrants, a group that composes 60 percent of all unauthorized immigrants. Even if they are not counted in a census, populations leave “fingerprints” of their presence in the form of deaths and births. Because people give birth and die with known regularity regardless of their legal status, we were able to use birth and death records of all Mexican-born persons to determine the number of the Mexican-born persons living in the U.S. We also looked at changes in Mexican census data between 1990 and 2010 to gauge the size of Mexico’s “missing” population, most of whom moved to the United States.

We then compared these estimates based on births, deaths and migration with the number of estimated Mexican immigrants in census data.

Based on this analysis, we found that the census missed as many as 26 percent of unauthorized immigrants in the early 2000s. We speculated that this could have been due to the large numbers of temporary Mexican labor migrants who were living in the United States at the time. Because many worked in construction during the housing boom and lived in temporary housing arrangements, it may have been particularly difficult to accurately account for them in census surveys. However, when the Great Recession and housing crisis hit, many of these temporary workers went home or stopped coming to the U.S. in the first place, and coverage error declined. By 2010, the coverage error may have been as low as 6 percent.

If current levels of coverage error for all unauthorized immigrants were as high as 26 percent, then the number living in the country could be as high as 13 million. But if coverage error were as low as 6 percent, then the figure could be as low as 10.3 million.

What this boils down to is that we have a pretty good idea of the number of unauthorized immigrants living in the United States. It most likely falls within a narrow range somewhere between 10.3 million and 13 million. If coverage error has declined as much as we think it has, then the truth is at the lower end of this range. Despite widespread beliefs, unauthorized immigration is not increasing out of control and certainly is not as high as 30 million. Instead, it has probably really has stabilized somewhere around 11 million.