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Four out of 10 people surveyed by the Pew Research Center said women have to do more than their male peers to succeed, and nearly as many said the country is "not ready" to see women in power.

“Why would people think the country is not ready? It's a question I don't really have an answer to," said Kim Parker, lead author of Pew's new report on women and leadership and director of the center's Social Trends Research.

People are slightly more optimistic about female leadership in politics—which probably has something to do with a woman named Hillary Clinton—than in business. (Exceptions include Sheryl Sandberg, chief operating officer at Facebook, and Mary Barra, the CEO of General Motors.)

"Somehow in business, these barriers seem pretty broad and persistent," Parker said.

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One reason is that the business world may be a space where risk-taking is rewarded, and that's one area, the survey revealed, where the public thinks men still come out on top.

A third of those surveyed say men in top executive positions are better at taking risks than women, while just five percent say women are better. Women have an edge when it comes to honesty and ethics.

But there are reasons for hope. Pew points out that the pipeline for female leaders appears to be widening, and women outnumber men when it comes to college completion.

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How do we make more progress?

"We haven't found in our studies that the majority of young women are motivated to reach those leadership positions," Parker said. "Part of it may be an attitude adjustment on the part of women to be willing to go for it."

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It might also help to get more men on board. While men think women are capable, they are less likely to think that putting more women in senior management positions benefits all women. They are also less likely to think that women face discrimination. Forty-eight percent of men see gender discrimination in today's society, according to the survey.

Stereotypes continue to play a detrimental role, too. Most people, women included, say men would do a better job than women at running a professional sports team or a large oil company, while women would be more suited than men to run a major hospital or retail chain, according to the survey.

"We may see some changes going forward," Parker said, "but it seems like we're a little bit stuck."

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Emily DeRuy is a Washington, D.C.-based associate editor, covering education, reproductive rights, and inequality. A San Francisco native, she enjoys Giants baseball and misses Philz terribly.