Ms. Klobuchar’s question on whether a woman with Mr. Buttigieg’s experience would be on a presidential debate stage goes to both whether a woman with his experience would decide she was qualified to run — and whether donors and voters would find her to be a viable prospect. A woman’s decision might be distorted by self-doubt, said Kelly Dittmar, assistant professor of political science at Rutgers. But it is also influenced by a real understanding of the biases she is likely to face. A recent Ipsos poll found that 74 percent of independents and Democrats would personally be comfortable with a female president, but just 33 percent said the same of their neighbors.

“Men wake up in the morning, look in the mirror and say ‘I’d make a really good governor,’” said Professor Dittmar. “We haven’t seen that as much with women.”