By Charlotte Alter
January 14, 2015

Correction appended, Jan. 14

The glass ceiling may not have shattered, but it is a lot less invisible than it used to be. Women make up only 20% of the Senate, 10% of governors, and 5.2% of Fortune 500 Business leaders. But at least Americans are beginning to understand why: sexist doubts about women’s competence are being slowly replaced by acknowledgement of the tough hand dealt to women leaders.

Not everybody feels that situation should change: A new Pew study says only 20% of women who identify as Republican consider it “personally important” to see a female president in their lifetime.

Mostly, however, Americans attribute the gender gap in leadership roles to structural inequalities, and sexist perceptions of female performance are actually relatively uncommon, says the study: 43% said that double standards keep women from achieving top business positions, and 38% said that kept them from political office. Meanwhile, fewer than 10% of people think women aren’t achieving these positions because they’re not tough enough or they make bad managers.

When it comes to gender equality in the business world, there’s a wide perception that women perform just as well as men, if not better. Most people said they saw no difference between how men and women perform in top business positions, but those that did see a difference tended to give women the edge: 31% said they thought women were more ethical and 30% said women would give fairer pay (although 34% said they thought men would be more likely to take risks.) Respondents said they thought women would do a better job running a hospital, a retail chain, or a large bank, while men would have better skills for running a professional sports team, an energy company, or a tech company.

But “soft skills” aside, 52% of women say the reason there aren’t more women running companies is because female CEOs are held to to higher standards than men. And it’s unclear whether these standards will ever disappear completely: 53% of respondents said men will dominate corporate leadership forever, while 44% said the gender gap at the top will eventually close.

Politics is a little different. A full 73% of Americans expect to see a female president in their lifetime, and while most respondents still thought men and women would do equally well in top political positions, 34% say women are better at working out compromises and are more ethical than men. Yet the double standard persists—47% of women believe that women don’t achieve high political office because they’re held to higher standards than men.

Unsurprisingly, the political opinions break down among party lines as well. Democrats are consistently more likely to attribute positive leadership skills to women: 40% of Democrats said women politicians are more honest than men while only 31% of Republicans agreed.

In both business and politics, women seem to be much more aware of the challenges facing female leaders than men are. Almost three quarters of women think it’s easier for men to become CEOs, compared to 61% of men, and 73% of women think it’s easier for men to be elected to political office, compared to 58% of men. Women were also much more likely to recognize gender discrimination by almost a 2o point margin (65% of women, compared to 48% of men,) but of that group, only 15% of women said there was “a lot” of discrimination. Both men and women said they saw more discrimination against gay people, African-Americans, and Hispanics: 28% said there is “a lot” of discrimination against gays and lesbians.

When asked whether more women in leadership roles would approve the quality of life for all women, 38% of women said that more female politicians would help them “a lot.” Only half as many men agreed.

Correction: The original version of this story incorrectly described one of the findings of the Pew study. It found that 20% of Republican women “personally hope” to see the United States elect a female president during their lifetime.

Write to Charlotte Alter at charlotte.alter@time.com.

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