When President Trump didn’t like the weak jobs numbers that were released on Friday, he fired the person responsible for producing them.

It was a move with few precedents in the century-long history of economic statistics in the United States. And for good reason: When political leaders meddle in government data, it rarely ends well.

There is the case of Greece, where the government faked deficit numbers for years, contributing to a debilitating debt crisis that required multiple rounds of bailouts. The country then criminally prosecuted the head of statistical agency when he insisted on reporting the true figures, further eroding the country’s international standing.

There is the case of China, where earlier this century the local authorities manipulated data to hit growth targets mandated by Beijing, forcing analysts and policymakers to turn to alternative measures to gauge the state of the country’s economy.

Perhaps most famously, there is the case of Argentina, which in the 2000s and 2010s systematically understated inflation figures to such a degree that the international community eventually stopped relying on the government’s data. That loss of faith drove up the country’s borrowing costs, worsening a debt crisis that ultimately led to it defaulting on its international obligations.

It is too soon to know whether the United States is on a similar path. But economists and other experts said that Mr. Trump’s decision on Friday to fire Erika McEntarfer, the Senate-confirmed head of the Bureau of Labor Statistics, was a troubling step in that direction.

Janet L. Yellen, the former Treasury secretary and chair of the Federal Reserve, said the firing was not what is expected from the most advanced economy in the world.

“This is the kind of thing you would only expect to see in a banana republic,” Ms. Yellen said.

Essential Data

The Bureau of Labor Statistics is officially part of the Labor Department, whose secretary is a member of the president’s cabinet. But the agency operates independently, producing detailed, nonpartisan data on employment, prices, wages and other topics.

Economists say that reliable, independently produced statistics are critical to good decision making in both the public and private sector. Officials at the Federal Reserve rely on government-collected data on inflation and unemployment to decide how to set interest rates, which affect how much Americans must pay to get a mortgage or a car loan.

“Good data helps not just the Fed, it helps the government, but it also helps the private sector,” Jerome H. Powell, the Fed chair, said at a recent news conference. “The United States has been a leader in that for 100 years,” he added, “and we really need to continue that in my view.”

Experts on government statistics say data from the Bureau of Labor Statistics and other agencies is unlikely to deteriorate dramatically overnight. The acting commissioner named to replace Dr. McEntarfer on a temporary basis, William J. Wiatrowski, is a longtime employee of the agency who is widely respected by experts inside and outside government. The career employees who collect and analyze the data remain in place, using the same methods and procedures they used before Dr. McEntarfer was pushed out.

But experts who just days ago were defending the integrity of the statistical agencies now find themselves asking uncomfortable questions about the trajectory of economic data in the United States.

“If the poverty numbers come in and look great, is the director of the Census going to get a raise?” said Amy O’Hara, a former Census Bureau official who is now a professor at Georgetown University. “If the household income numbers don’t look great what happens then? What about G.D.P.? What about C.P.I.?”

Andreas Georgiou knows the challenges of standing up to such political pressure.

After he took over Greece’s statistical agency in 2010, he found that the country has been severely understating its budget deficits. Those findings ran afoul of the Greek authorities, who spent years trying to prosecute him on a variety of charges related to his work, despite independent reviews that supported his conclusions. (He fared better, though, than Olimpiy Kvitkin, the Soviet census official who was arrested and executed when his population count came in lower than Joseph Stalin had announced.)

Mr. Georgiou refused to bend. Reliable statistics are important for policymaking, he said. But they are also essential to democracy.

“Official statistics, government statistics are a mirror that society holds up to itself,” he said. If that mirror is distorted, or broken entirely, then the accountability that is central to a democratic system cannot work.

“If society cannot see itself clearly, then it cannot identify its problems,” he said. “If it cannot identify its problems, then it cannot find the right solutions. It cannot find the right persons to solve these problems.”

Data Integrity at Risk

Mr. Trump said he fired Dr. McEntarfer because the numbers produced by her agency were “rigged” to hurt him politically.

Experts on the government statistics, including former commissioners in both Democratic and Republican administrations, have called foul on that accusation. The commissioner, who is the bureau’s sole political appointee, does not control the numbers that the agency publishes, or even see them until they have been finalized by a staff of career technocrats whose careers typically span multiple presidential administrations.

Erica Groshen, who led the bureau under President Barack Obama, recalled getting resistance from the agency’s staff when she tried to liven up the language of the monthly jobs reports. The bureau’s staff insisted that the agency’s job wasn’t to say whether the glass was half-full or half-empty, only to report that, “It is an eight-ounce container with four ounces of liquid.” Ms. Groshen relented.

That is not to say political interference would be impossible. Government statistics rely on hundreds of methodological decisions, many of them judgment calls with no obviously correct answer. A sufficiently sophisticated agency head might, over time, be able to nudge the data in a politically advantageous direction, without any single decision being so egregious that it led to a mass resignation of career employees.

“I could imagine a new commissioner coming in and trying to make changes to those methods and procedures that try to move those numbers one way or the other,” said Katharine G. Abraham, who led the bureau during the Clinton and George W. Bush administrations. “They would have to know a lot in terms of where to put the finger on the scale.”

Private Alternatives

There are also blunter approaches. In Argentina in 2007, the government of then-President Néstor Kirchner pushed out the mathematician in charge of the country’s consumer price data, then released an inflation figure that was dramatically lower than the one the mathematician had calculated.

The public wasn’t fooled. Nor were international bond investors, who ultimately turned to alternative sources of inflation data, calculated by researchers outside the government.

But such alternative sources are inherently limited, said Alberto Cavallo, a Harvard economist who developed one of the most widely used private inflation indexes in Argentina.

“Private alternatives can complement official statistics, but they are not a substitute,” Mr. Cavallo wrote in an email. “Government agencies have the resources and scale to conduct nationwide surveys — something no private initiative can fully replicate.”

Recently, Mr. Cavallo has been publishing data on consumer prices in the United States, which has shown the impact of Mr. Trump’s tariffs more quickly than the government’s data. But while such real-time sources are valuable, they don’t carry the “institutional credibility” of government data.

The trouble is that once that credibility is eroded, it is hard to repair — particularly at a time when partisans on both sides of the political aisle are skeptical of numbers put out by members of the opposing party.

Nancy Potok, a former Census official who served as chief statistician of the United States during the first Trump administration, said that in the past there had been strong bipartisan support for the statistical system in Congress and the business community. But partisanship seems to have eroded that support at a moment when a combination of political pressures and longstanding budget challenges are making it most necessary.

“There were some people who really understood the value of the economic data, and now that’s not the conversation and those champions aren’t there that were there in the past,” she said. “There’s no one leading the charge to make these kind of investments.”