By sheer statistical coincidence, the percentage of Americans who self-identify as independents (approximately 40 percent) is nearly identical to the percentage of Americans who self-identify as moderates. This coincidence has nurtured a common misunderstanding: that independents are mostly moderates who want members of both parties to move toward the center.

But a growing body of literature studying the precise political attitudes of the public – and especially moderates and independents – is challenging the conventional wisdom: The research of David Broockman and Doug Ahler, for example, has revealed that so-called moderates tend to harbor specific opinions that are more extreme than previously thought, and Barbara Norrander's research shows that the electorate in open primaries is more extreme – not more moderate – than the electorate in closed primaries.

These findings are corroborated by the results of the presidential primaries in the 2016 election cycle. Rather than open primaries benefitting the more moderate candidates in both parties, such as Hillary Clinton and John Kasich, the contests have actually benefitted the more "extreme" candidates in both races: Bernie Sanders and Donald Trump.

On the Democratic side, for example, the Sanders campaign received an early boost after winning a landslide victory in New Hampshire's open primary. In that contest, Sanders won nearly 75 percent of the vote among self-described independents. Sanders also won decisively in his home state of Vermont, which also held an open primary.

Recent exit poll data from 27 primary contests (that is, excluding caucus states) show that self-identified independents backed Sanders in 24 of those primaries. Significantly, the data also show that independents in five states were decisive in giving Sanders the lead in the popular vote in those primaries. Among these five states (Indiana, Michigan, Oklahoma, West Virginia, and Wisconsin), four held open primaries: only Oklahoma's contest was closed.

It is not only Sanders' popularity among non-Democrats that shows the ideological diversity of independent voters. On the Republican side, Donald Trump's success has in part been a result of his appeal among voters outside the party.

An analysis by CNBC in March showed that Trump's electoral performance has been stronger in open primaries rather than closed primaries. Indeed, Trump lost only a handful of open contests, and two of those losses occurred in the home states of his opponents (Cruz's Texas and Kasich's Ohio). In closed contests, on the other hand, Cruz was often the party favorite.

It is tempting to see independents' support for Sanders and Trump as symptomatic of these voters' larger frustration with the political process – as protest votes against a status quo characterized by partisanship and gridlock.

After all, Sanders, the longest-serving independent in congressional history and an open critic of the Democratic Party, has campaigned on his opposition to the influence of big money in Washington, and Trump, who has held four different partisan affiliations and is largely out of sync with Republican orthodoxy, has won over many voters given his perceived ability to fix a "broken" system, make deals, and "get things done."

But while Sanders and Trump may be thriving on voters' anger and anxiety, they are also receiving support based on their policy positions as well – including positions that are seldom touted by establishment partisans.

Approximately 40 percent of the public supports the kind of single-payer healthcare system touted by Sanders, for instance, and nearly half approve of a temporary ban on Muslims entering the U.S. – a level of support that jumps to 84 percent among backers of Trump, who first advocated the ban.

These findings show that when independents have the chance to make their voices heard in primary elections, they do not uniformly support moderate candidates espousing moderate proposals. While there are undoubtedly independents who have supported more moderate candidates like Clinton and Kasich – not to mention independents who are searching in vain for candidates with whom they agree – the results of the 2016 presidential primaries serve as strong evidence that many independents, if not most, are not ideological moderates.

These findings upend not only the conventional wisdom that opening primaries will make primary electorates more moderate and produce more centrist candidates, but also the supposition that, in a general election, the major party nominees will undoubtedly increase their appeal among independents by pivoting to the "mythical center."

It is for these reasons that independents of all ideological persuasions, and even some traditional partisans, are looking for ways to reform the electoral process to make it more responsive to the preferences of a diverse public craving more choices.