With a convincing win in the New Jersey primary, Hillary Clinton has clinched the Democratic Party nomination and become the first woman nominated by a major party for the presidency. This is, obviously, a major historical moment, but her campaign has generally struggled to articulate exactly why it matters beyond symbolism.

But there is clear and convincing evidence that it does matter. Enormously. In the aggregate, women do govern differently than men, even when you control for partisan affiliation and the ideological composition of the election. But there aren't many women in the governing class. More than 80 percent of the members of the US House of Representatives are men, as are just over 75 percent of state legislators and 88 percent of governors.

Electing Clinton would be a break from that pattern, but it would also drive further breaks. Studies show that when women achieve high office, female advancement in politics "trickles down," with a woman governor or senator inspiring a downstream boost in women state legislators. These victories would themselves carry important symbolic value, but beyond that they would generate concrete changes in the governance of the country — including more attention to issues related to child care, family life, women's health, and the needs of the neediest.

Women in office inspire other women to run

We can't directly study the impact of putting a woman in the White House, but Amelia Showalter, a political consultant specializing in data and analytics, has studied the impact of electing women to statewide offices. Her research project initially focused on the impact of state-level efforts to recruit women to run for office — things like a state version of Emily's List — and she found that the presence of a recruiting campaign could increase the share of women in the state legislature by about a percentage point.

That's not nothing. But she found that electing a woman to statewide office could have double or more the impact of a recruiting campaign.

We don't know exactly why this is, but Showalter suggests that "women in powerful positions aren't just serving as role models to little girls — they’re normalizing politics for adult women who may need a little nudge."

Jennifer L. Lawless and Danny Hayes, authors of Women on the Run: Gender, Media, and Political Campaigns in a Polarized Era, help us understand the precise causal mechanism. There are lots of reasons women may be disinclined to run for office — including the broad suite of structural barriers related to gender norms and family life that tend to limit women's access to all manner of high-powered jobs — but one reason is the perception that a woman candidate is likely to be unfairly disadvantaged.

The election of a high-profile woman gives other women residing in the state confidence that a woman can win, making them more likely to run and making more women likely to win.

The "Clinton effect" would likely be large

Electing a woman governor has a bigger impact than electing a woman attorney general. That is no surprise — the governor is a higher-profile, more important role. But even so, governors are relatively obscure. A 2007 Pew survey showed that just 66 percent of the public could correctly name their state's governor.

The president, by contrast, enjoys near-universal name recognition. Indeed, even back in 2007 Clinton was correctly identified by 93 percent of respondents. Consequently, you would expect an impact that's quite a bit larger than the gubernatorial one.

But the impact would also exert itself across a much larger scale, since the president covers the entire country and not just one state. It would also have knock-on effects, since if a Clinton presidency inspired one additional woman senator she, in turn, would inspire more down-ballot runs.

Clinton's campaign itself includes many women in important roles, many of whom, on a personal level, seem to feel this demonstration effect is important, though they've hesitated to make it an explicit element of campaign messaging.

But beyond the campaign, Clinton's inner circle of close advisers includes many more women than we've seen for any past president. There has never been a woman Treasury secretary or chief of staff or secretary of defense, and Clinton is very likely to tap a woman for at least one and possibly all three of those jobs — offering a wider range of role models of political success.

More women in office would make a difference on policy

Even all this would be of pretty narrow concern except for the fact that it turns out that gender is a significant driver of legislators' behavior.

John Sides summarized some of the research last March:

For one, women are more likely than men to advocate for issues often associated with women’s interests — child care, women’s health, abortion, pay equity and the like. There are many studies, but see Michele Swers’s two books to start with. This shows up, for example, in in floor speeches and legislative debates, where women are more likely to discuss issues in terms of women’s interests. (Women are also more likely than men to give floor speeches, period.) [...] Other research suggests that women may be more effective legislators than men. Craig Volden, Alan Wiseman and Dana Wittmer find that, within the minority party, women are able to get their sponsored bills further through the legislative process. Sarah Anzia and Christopher Berry have shown that women sponsor and co-sponsor more bills than men do, and deliver about 9 percent more funding to their districts.

What's even more striking is that these differences seem to grow with scale rather than shrink.

Tali Mendelberg, Christopher Karpowitz, and Nicholas Goedert show that "when women are many, they are more likely to voice women’s distinctive concerns about children, family, the poor and the needy." What's more, when women are more numerous and therefore more vocal on these topics, men become more vocal too, and "these effects are associated with more generosity to the poor."

Personal identity matters in politics

The idea that women would govern in a distinctive way may seem far-fetched to some, but there's actually extensive evidence that personal identity matters to the behavior of professional politicians.

Nicholas Carnes's research on occupational background and policymaking finds that lawmakers with a working-class background — a group that is even more underrepresented in politics than women — amass a voting record that is more supportive of labor market regulation and less pro-business than lawmakers with a white-collar background, even when controlling for party identity and constituency.

In other words, it's not just that working-class legislators are more likely to have left-wing views and decide to be Democrats and represent liberal districts. Working-class legislators vote in a more left-wing manner on economic issues than other legislators from the same party who represent similar districts.

Barry Burden's book Personal Roots of Representation finds that legislators' life experiences impact political behavior on everything from tobacco regulation to school vouchers.

Rob Portman was the first Republican senator to come out in favor of marriage equality for gay and lesbian couples, and it's no surprise that he has a gay son.* Sarah Palin is an avowed proponent of small government and spending cuts but makes an exception for programs designed to help special needs children and their families — families like hers.

Viewed in this context, the idea that legislatures full of women would govern differently than the current ones looks more like common sense. Politicians face significant constraints determined by party politics and electoral incentives. They also subscribe to broad ideological worldviews that prescribe stances on many issues. But within those constraints there is substantial wiggle room, specifically when it comes to the question of what to emphasize or try to place on the agenda.

Women lead different lives than men, and would consequently govern differently if more of them were in office. And the evidence strongly suggests that electing women to high-profile jobs inspires more women to run for and win lower-profile jobs. The presidency is by far the highest-profile job in American politics, meaning a Clinton presidency would likely have a meaningful downstream impact on women's representation for years to come — with far-reaching ramifications for public policy at both the state and national level.

Clinton's campaign is focused at the moment on making the case against Donald Trump, which is obviously an important aspect of their strategy. But at the end of the day, to turn out voters you need to make them feel excited about the idea of being part of something larger and interesting. For Clinton, the woman angle is by far the most promising route to accomplishing that.

And it's not something they should feel sheepish about saying or that others should dismiss as symbolism. The life experience and personal identity of the governing class matters enormously, and a Clinton presidency would shift the composition of the American government in a profound and unprecedented way.

*Correction: An earlier version of this article described Portman as "one of the first" Republican senators to favor marriage equality, which is technically true, but he was the very first to do so and should be described as such.