In the words of her detractors during the 2016 presidential race, Hillary Clinton was abrasive and shrill. She was aloof. She was unlikable.

It’s not a coincidence that some of these adjectives are now bubbling up in discussions of Senators Elizabeth WarrenKirsten Gillibrand and Kamala Harris as they campaign for the 2020 Democratic nomination.

Few Americans acknowledge they would hesitate to vote for a woman for president — but they don’t have to, according to researchers and experts on politics and women and extensive research on double standards in campaigns. Reluctance to support female candidates is apparent in the language that voters frequently use to describe men and women running for office; in the qualities that voters say they seek; and in the perceived flaws that voters say they are willing or unwilling to overlook in candidates.

The influence and impact of these stereotypes and double standards are about to play out in uncharted territory: a presidential race featuring six women running for the Democratic nomination, all vying to compete against President Trump, who has a history of making sexist comments.

“For 20 years, we’ve heard participants in our focus groups say they would vote for a woman, just not that woman,” said Amanda Hunter, research and communications director at the Barbara Lee Family Foundation, which supports women in politics. “That promise will be put to the test in some cases, because folks may not be able to hide their bias behind that excuse when there’s multiple women on the debate stage.”

Senator Amy Klobuchar of Minnesota is the latest woman to join the 2020 field, kicking off her campaign on Sunday. In the run-up to her announcement, she faced headlines about her well-known reputation in the Senate as a difficult boss. Whether she is held to a different standard than men remains to be seen: Male politicians pressure and berate staff members too, and, like Ms. Klobuchar’s Senate office, President Trump has had high employee turnover.

How much sexism ultimately influences votes is a matter of debate. In general elections, partisanship beats everything else, said Kathleen Dolan, a political scientist at the University of Wisconsin-Milwaukee, whose research shows that most voters stick with their party’s candidate regardless of gender.

But there has been little comparable research on primaries, where partisanship isn’t in the equation. And the Democrats will have a wide-open presidential primary in 2020 with multiple leading female candidates.

What is not a matter of debate is the array of ways that sexism can manifest on the campaign trail, affecting not only how voters perceive candidates but how candidates present themselves to voters.

The likability trap

The very first question of Ms. Gillibrand’s campaign was about her likability. “A lot of people see you as pretty likable,” a reporter told her. Did she consider that a “selling point”?

Ms. Warren’s evaluation arrived with similar speed, in a Politico articlethat asked how she could “avoid a Clinton redux — written off as too unlikable before her campaign gets off the ground.”

One of the most amorphous qualities candidates are judged on, likability is also deeply influenced by gender bias, researchers say. Voters look for it in men, too — consider the “who would you rather have a beer with” question in campaigns — but only in women, research shows, do they consider it nonnegotiable.

Women also tend to be viewed as unlikable based on their ambition. Harvard researchers found in 2010 that voters regarded “power-seeking” women with contempt and anger, but saw power-seeking men as stronger and more competent. There is often some implication of unscrupulousness in descriptions of female candidates as “ambitious” — an adjective that could apply to any person running for president but is rarely used to disparage men. Within 24 hours of Ms. Harris’s campaign kickoff, some critics were bringing up her onetime relationship with a powerful California politician, Willie Brown — a common tactic faced by women that sexualizes them and reduces their successes to a relationship with a man.

And if a narrative of unlikability takes hold, it can influence voters without their even realizing it.

“I actually heard so often that people didn’t like her and she was unlikable that I started to think, ‘I don’t know if I really like her so much,’” Heather Pasqualino Weirich, a Democrat from Phoenixville, Pa., said of Mrs. Clinton, whom she later supported. “I realized I had no reason why I didn’t like her.”

Masculinity and femininity

The qualities voters tend to expect from politicians — like strength, toughness and valor — are popularly associated with masculinity. This often means that from the moment a man steps onto the campaign trail, he benefits from a basic assumption that he is qualified to run, while a woman “has to work twice as hard to show that she’s qualified,” Ms. Hunter said.

“People just start off assuming that you care about the soft issues, you care about hearth and home, and that you can’t know anything about finance or military,” said former Senator Carol Moseley Braun, who ran for the Democratic presidential nomination in 2004.

For many years, female candidates tried to adopt the characteristics voters wanted to see — to act, stereotypically speaking, like men. This worked for some but also brought pitfalls. For one thing, it did not challenge the premise that masculinity is better suited for leadership. It also opened women up to a familiar double standard: A man who speaks authoritatively might be confident or opinionated, while a woman who does the same is arrogant or lecturing. Most pressingly, it created a backlash among some voters who saw women acting “like men” and deemed them inauthentic.

Nichole M. Bauer, an assistant professor of political communication at Louisiana State University, found that when women played up stereotypically masculine qualities, voters — regardless of party — rated them better in terms of leadership ability, but voters in the opposing party rated them significantly lower in terms of likability. There was no similar backlash to male candidates who defied gender stereotypes.

Dr. Bauer said that in all her research, she had found no way for women to win the support of voters in the opposing party. It’s a basic psychological phenomenon, she said: If a Republican starts out disliking a Democratic woman, or vice versa, “they’ll use gender stereotypes about women to maintain that perceived negative relationship” no matter what the woman does.

Superficial judgments

Perhaps the most obvious way female candidates are judged differently is on their appearance: not only how “attractive” they are and how they dress, researchers say, but also their facial expressions, their body language and their voice. Accused in 2015 of “shouting” about gun violence, Mrs. Clinton responded: “First of all, I’m not shouting. It’s just when women talk, some people think we’re shouting.”

Women are conscious that small elements of how they present themselves are subject to scrutiny.

Representative Madeleine Dean — one of four Democratic women elected to the House last year from Pennsylvania, whose congressional delegation was previously all-male — said an aide would stand in the back of the room during her campaign events, holding up a cardboard sign with a smiley face to remind her to shift the serious expression she naturally wore while listening to voters.

She was also coached, “though I did not take his coaching, not to cross my arms in front of myself because then you look mad,” Ms. Dean said.

These sorts of criticisms were common in the 2016 campaign, not only against Mrs. Clinton but also against Carly Fiorina, who ran in the Republican primary. “Look at that face,” Mr. Trump said at one point, openly mocking Ms. Fiorina’s appearance. “Would anyone vote for that? Can you imagine that, the face of our next president?” (Mr. Trump subsequently claimed he had been talking about her persona. Ms. Fiorina, who did not respond to interview requests for this article, said at the time, “I think women all over this country heard very clearly what Mr. Trump said.”)

Because these judgments are so superficial, and their gendered nature so obvious, they draw substantial backlash. But that doesn’t mean they stop.

“When we take a look at states that have had female governors, when voters see women succeed in executive office, it opens the door for more women in that role because it does literally change the face of leadership,” Ms. Hunter, of the Barbara Lee Family Foundation, said.

Precisely because women running for president have lacked models for how to do it, what they are left with, Ms. Walsh said, is “very prescribed: ‘Wear a little suit, talk about your résumé but don’t talk about your personal life, don’t talk about your kids because then people are going to ask who’s taking care of your kids.’”

In 2018, of course, many female candidates ran unabashedly as themselves, bucking the public images and political messages that women traditionally adhered to in campaigns. Some of them appeared with their young children; others proudly showed off their tattoos; and many talked in personal terms about discrimination and harassment as well as a range of policy issues.

The presence of six women in the 2020 race, Ms. Walsh argued, is “going to give women who run for office at every level more leeway, and a path to navigate that may be not quite so narrow.”

“The women who run are still going to be, I think, more scrutinized about their appearance,” said Debbie Walsh, director of the Center for American Women and Politics at Rutgers University. “I would love to think that they won’t get the kind of comments that Hillary Clinton got about, ‘Why is she yelling at me?’ ‘Why doesn’t she smile more?’ I’d love to think that that’s all gone now, but I don’t believe that to be true.”

Who is a leader?

History influences what voters see as normal. And for 230 years in the United States, presidential leadership has been male.

For many decades, this was a self-sustaining cycle: Female candidates were outside the norm, so they didn’t win, so they remained outside the norm. But the history of women’s representation at other levels of government shows that the norm can shift.