For every $100 earned by an average white family, how much do you think is earned by an average black family? $0-$25 $26-$50 $51-$75 $76-$100 $100+ Wrong. You’re actually too pessimistic. Black families in America earn $57.30 for every $100 in white family income. Wrong. But you’re close. Black families in America earn $57.30 for every $100 in white family income. Right. Black families in America earn $57.30 for every $100 in income earned by white families. Wrong. The country isn’t as equal as you think. Black families in America earn $57.30 for every $100 in white family income. Wrong. Black families earn much less — not more — than whites, just $57.30 for every $100 in white family income.

For every $100 in wealth accumulated by an average white family, how much wealth has the average black family accumulated? $0-$25 $26-$50 $51-$75 $76-$100 $100+ Right. You’re taking a bleak view — and an accurate one. For every $100 in white family wealth, black families hold $5.04. Wrong.Your guess is too optimistic. For every $100 in white family wealth, black families hold $5.04. Wrong. The reality is far worse. For every $100 in white family wealth, black families hold $5.04. Wrong. Blacks and whites are far from equality here: For every $100 in white family wealth, black families hold $5.04. Wrong. Black families actually hold just a fraction of the wealth white families do: $5.04 for every $100 in white wealth.

The Yale researchers suspected that many people would not get the answers right.

“I’m a person who studies inequality, who should really know how inequality looks,” said one of the psychologists, Michael Kraus, who researches the behaviors and beliefs that help perpetuate inequality. “And I look at the black-white gap, and I’m shocked at the magnitude.”

Black families in America earn just $57.30 for every $100 in income earned by white families, according to the Census Bureau’s Current Population Survey. For every $100 in white family wealth, black families hold just $5.04.

If Mr. Kraus, of all people, is taken aback by these numbers, what are the odds that most Americans have a good understanding of them? The answer, he and his colleagues fear, has broad implications for how we understand our society and what we’re willing to do to make it fairer.

Americans, and higher-income whites in particular, vastly overestimate progress toward economic equality between blacks and whites, the psychologists reported Monday in the Proceedings of the National Academy of Sciences. Americans believe that blacks and whites are more equal today than they truly are on measures of income, wealth, wages and health benefits. And they believe more historical progress has occurred than is the case, suggesting “a profound misperception of and unfounded optimism” regarding racial equality.

“It seems that we’ve convinced ourselves – and by ‘we’ I mean Americans writ large – that racial discrimination is a thing of the past,” said Jennifer Richeson, who was another of the study’s authors, along with Julian Rucker, a doctoral student. “We’ve literally overcome it, so to speak, despite blatant evidence to the contrary.”

To understand how people have perceived that progress, the researchers asked blacks and whites of varying income levels to estimate answers to the questions above in both recent years and historically. They also asked about how much black workers with a high school diploma but no college degree earn relative to whites of the same education level, and how the earnings of blacks and whites with a four-year college degree compare.

The present-day results, aggregated across several surveys used in the study, are compared here with actual government data:

Perception Reality Family wealth Family income College wages High school wages $0 $50 $100 Perception Reality Family wealth Family income College wages High school wages $0 $20 $40 $60 $80 $100 Perception Reality Family wealth Family income College wages High school wages $0 $20 $40 $60 $80 $100 Perception Reality Family wealth Family income College wages High school wages $0 $20 $40 $60 $80 $100 For every $100 in white family wealth, income or individual wages, respondents were asked how much they believe average black families and workers have.

Why would people get these questions so wrong — and consistently in the direction of too much optimism? (This study was also conducted after the 2016 election.) Blacks overestimated equality, too, but the biggest effects were among wealthy whites.

The researchers suspect that the answer in part has to do with how little exposure Americans have to people who are unlike them. Given how economically and racially segregated the country remains, many Americans, and especially wealthy whites, have little direct knowledge of what life looks like for families in other demographic groups.

But the pattern this study identifies isn’t simply about lack of access to accurate information. As Mr. Kraus points out, popular videos and charts regularly circulate on social media highlighting the startling levels of inequality in America. And yet, many people who click on them forget about the severity of inequality just long enough to be surprised by it again in the future.

“Despite this information being out there, we don’t really take it in,” Mr. Kraus said. This happens “in a way that suggests that maybe we’re motivated to forget it, or motivated to distort it in our own minds.”

He and Ms. Richeson suspect that we also overgeneralize from other markers of racial progress: the election of a black president, the passage of civil rights laws, the sea change in public opinion around issues like segregation. If society has progressed in these ways, we assume there’s been great economic progress, too.

We’re inclined, as well, to believe that society is fairer than it really is. The reality that it’s not — that even college-educated black workers earn about 20 percent less than college-educated white ones, for example — is uncomfortable for both blacks who’ve been harmed by that unfairness and whites who’ve benefited from it.

“It’s very difficult to consider the possibility that some of what we’ve achieved or gained is due to forces that aren’t our own individual hard work,” Ms. Richeson said. “That’s hard to grapple with, especially in American society. We really believe in egalitarianism and meritocracy.”

These findings suggest that the motivation to see the world as fair may be even stronger in this context than stereotypes white Americans hold, for instance, equating blacks with poverty.

The researchers found in some additional surveys that whites answer these questions more accurately when they’re first asked to consider an America where discrimination persists. If we want people to have a better understanding of racial inequality, this implies that the solution isn’t simply to parrot these statistics more widely. It’s to get Americans thinking more about the forces that underlie them, like continued discrimination in hiring, or disparities in mortgage lending.

It’s a myth that racial progress is inevitable, Ms. Richeson said. “But it’s also dangerous insofar as it keeps us blind to considerable inequality in our nation that’s quite foundational,” she said. “Of course we can’t address it if we’re not even willing to acknowledge it.”