When Federal Reserve Chair Janet Yellen wanted to illustrate how hard it is for unemployed Americans to find jobs, she recounted the tales of two Chicago residents whose struggles were perhaps a little too real.

At a conference this week in the Windy City, Yellen told the story of Dorine Poole, who was unemployed for two years after losing her job processing medical insurance claims. But Yellen didn’t mention that Poole had also been convicted of felony theft 16 years ago.

Yellen also invoked Jermaine Brownlee. He was an apprentice plumber and skilled construction worker before the recession, but his wages dropped as he struggled to find work after the economy collapsed. What Yellen did not say was that Brownlee was arrested two years ago and convicted of narcotics possession, the latest black mark on a rap sheet dating to 1997, according to public records.

The plainspoken speech was an unorthodox move for a leader of the nation’s central bank, where jargon-laden Fed-speak has been the law of the land. But after taking note of Yellen’s message about the persistent weakness of the nation’s labor market, news reports soon focused on Poole and Brownlee, implying that they did not reflect the U.S. workforce and that their criminal records made them less deserving of employment.

Neither assertion could be further from the truth.

A 2012 study in the medical journal Pediatrics found that nearly one-third of Americans have been arrested by age 23. In other words, people with criminal histories make up a significant portion of the population. The number of people who have been convicted of a crime is significantly smaller, with estimates ranging from 12 million to 14 million. But ex-offenders make up a substantial number of the nation’s jobless – enough to depress the male employment rate by 1.5-1.7 percent in 2008, according to analysis by the Center for Economic Policy Research.

The takeaway is that workers with criminal histories are not an anomaly in the labor force: They are an integral part of it. Any discussion of how to help the unemployed includes addressing the challenges facing people with records.

Does their checkered past make Poole and Brownlee less qualified for jobs? There are certain places where the answer is yes, including child-care centers. But perhaps the best answer lies in a point that was lost in Yellen’s speech: Both are now gainfully employed.