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While people may be less likely to admit they don’t attend services often, the research institute found that people tended to answer equally as truthfully on the phone and online about their religious affiliation and beliefs.

“I think being attached to an institution is less important, it’s not viewed as negatively,” Mr. Cox said. “Simply saying you’re not attached to the Catholic Church, I don’t think there’s the same kind of weight.”

Even being labeled an atheist, is “becoming increasingly accepted,” Mr. Cox said. “Compared to where the United States was in the 1950s, considerably more people are at least tolerant.”

The researcher said he couldn’t pinpoint “this sort of tipping point where it’s OK to say I don’t believe in God, or religion is not that important,” but added that the change is likely to start among younger generations.

Nonetheless, people in the 18-29 age bracket were more likely to overreport their attendance. Despite the contradiction, the next generation is “the most diverse generation we’ve ever had; religiously, racially, ethnically, they’re exposed to different ideas. If it’s going happen anywhere, it’s going to happen there,” Mr. Cox said.