Even before former President Donald J. Trump emerged as the winner in the presidential race, the numbers on election night showed signs of a victory … for polling.

Over the final weeks of the campaign, pre-election polls consistently pointed to a tight race between Mr. Trump and Vice President Kamala Harris, in both the popular vote and the battleground states. And though the current results are preliminary, and many states are still counting ballots, a close race is what has unfolded.

Last month, Chief Judge Albert Diaz of the U.S. Court of Appeals for the Fourth Circuit, in Richmond, Va., writing for six judges, said that approach had created “a labyrinth for lower courts, including our own, with only the one-dimensional history-and-tradition test as a compass.”

By Wednesday morning, The New York Times’s forecast model had estimated that Mr. Trump was on track to win the national popular vote by about 1.5 percentage points, meaning the pre-election polls — which, on average, had shown Ms. Harris with a one-point lead — were probably off by roughly two and a half points.

Mr. Trump was also on track to sweep all seven swing states, according to Times estimates, giving him a more comfortable victory in the Electoral College. But that is well within the range of outcomes the polls had suggested. His margins of victory in most of those swing states appeared to be relatively narrow, as the polls had suggested they might be.

Of course, the polls weren’t perfect; they never are. And they sometimes showed Ms. Harris with a slight edge when it turned out that Mr. Trump had the narrow advantage. But a lead of one point in polling is best interpreted as, effectively, a tie, and the polls’ real value in this election cycle was showing that the race was, in fact, very tight.

In an analysis of the preliminary results in states where enough pre-election polls had been conducted to calculate an average, and where a winner had been called by Wednesday morning, the polls were off by an average of three percentage points — within the typical margin of error. They were even closer among battleground states where races had been called, varying about two percentage points from the preliminary results.

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