A department spokesman, James Rockas, noted that Mr. Ross was seeking nearly $750 million for advertising and outreach programs to persuade members of hard-to-reach groups to participate. The Obama administration had “severely underestimated” both the cost and technical challenges of moving to a digital census, he said. Outside experts disputed that charge, noting that Congress had ordered the Census Bureau to spend no more than $13 billion on the 2020 census, and then cut even more from Obama administration budget requests that sought to meet that mandate.

Consternation about pulling off an accurate count has been part of the run-up to past censuses, especially regarding funding challenges. During the last census, worries ranged from undercounting military personnel and their families on bases to fairly accounting for large inmate populations in rural Republican districts.

A bungled count could have profound consequences. Data from the census — which aims to count everyone, whether citizens or not — dictate the distribution of more than $600 billion yearly in grants and subsidies to state and local governments. Demographic data from the count are the bases for surveys that are benchmarks for major businesses, governments and researchers.

The census results also will determine which states will gain or lose seats in the House of Representatives and how those lines are drawn when redistricting begins in 2021. Serious undercounts would invite lawsuits that could hogtie that process, some experts said, and sap public trust in one of the government’s core functions.

The census is the gold standard of data collection not just in the United States but in the world, said Phil Sparks, a director of the Census Project, a network of organizations promoting an accurate head count in 2020. “The last thing we want to do in this current debate,” he said, “is to make this a base metal.”

The bureau has been working on the 2020 count since the 2010 census was completed. The complete overhaul now underway seeks to shrink the count’s costliest and toughest task: sending hundreds of thousands of enumerators to find and interview the millions of people who fail to fill out their census forms.