CONCORD — All eyes may be on Iowa and New Hampshire, but many of them are rolling.

Despite efforts to evict the two states from the front of the presidential calendar, both managed to hang on for another election cycle that culminates with the Iowa caucuses on Thursday and the New Hampshire primary on January 8. As a year of media attention reaches its crescendo, voters in other states are saying enough is enough.

According to a national survey conducted for The Associated Press and Yahoo News, just over half of all voters said New Hampshire and Iowa have an extraordinary amount of influence over who wins the two parties nominations.

“They have way too much — WAY too much — say,” Kevin Thomas of Tacoma, Wash. said “California’s a big state and they don’t have any say, and Iowa’s not even half the size of California. It really makes me as a voter wonder what’s going on.”

Fewer than one in five voters said they favor the current system that allows Iowa and New Hampshire to hold the first contests, while nearly 80% would rather see other states get their chance at the front of the line.

“I think they should take turns, maybe take it to a small state like Rhode Island that doesn’t have a whole lot of voting power,” Mr. Thomas said.

Both states have been criticized as unrepresentative of the country given their size and lack of racial diversity. Iowa — population 3 million — is 95% white; New Hampshire — population 1.3 million — is 96% white. Democrats tried to inject more diversity into the process by adding early contests in Nevada and South Carolina, but Iowa and New Hampshire moved even earlier.

The system became so scrambled last year that the New Hampshire secretary of state, Bill Gardner, was prepared to move the primary into December to keep ahead of other states that scheduled their own early primaries and caucuses. If anything, the front-loaded calendar made Iowa and New Hampshire more important.

Mr. Gardner and other defenders of New Hampshire say the country — and the candidates — are well-served because the primary requires close contact with voters, not just a big advertising budget and name recognition.

“It gives the little guy a chance,” Mr. Gardner said.

He wasn’t surprised by the poll results and negative reaction toward the early states given that most of the country knows nothing about the primary’s history or the state’s uniquely inquisitive and democratic culture.

Governor Lynch of New Hampshire, a Democrat who has not endorsed any candidate, argues that New Hampshire’s retail politics cannot be duplicated anywhere else in the country.

“We have made it possible for the so-called unknown candidates to make their case without having millions of dollars in the bank. And in turn, we demand that candidates move beyond the rope line and scripted town hall meetings, and directly answer the hard questions from voters,” he said. “As a result, the voters, the candidates and the political process all benefit from the New Hampshire primary.”

Unsurprisingly, every one of the 21 Iowans who participated in the AP-Yahoo survey think their state and New Hampshire have just the right amount of influence over the presidential selection process. Not so in New Hampshire.

There, two of the five participants said the two states don’t have enough power.