A 3 percent positive rate in coronavirus testing is a critical threshold for New York City. It is the point at which the mayor shut down public schools last week. The governor says that a sustained 3 percent level in the city will result in banning indoor dining, closing gyms and hair salons, and placing a 25-person cap on attendance at houses of worship even as the holidays approach.

But as important as that 3 percent rate is, it seems the city and the state can’t agree on whether it has been reached.

That conflict has played out over the past week, with Mayor Bill de Blasio saying 3 percent has been breached, while Gov. Andrew M. Cuomo said the positive test rate was well below that. Each relies on his own statistics, which are compiled and reported in different ways. And, it turns out, the state and city also can’t agree on which tests to include in the calculation.

The discrepancy can be striking: On Saturday, for instance, the city said its seven-day average was 3.11 percent. Mr. Cuomo’s office, however, put the city’s rate at more than half a point lower, at 2.54 percent.

It is the latest discordant message between two rivals that has played out over the entire pandemic, adding a level of dysfunction and confusion to the response.

The cause of the discrepancy lies in both the tests that are included and the time frame in which statistics are reported. The state treats a new case as arising on the day the test result comes in. The city dates each new case to when the sample was provided.

So if an infected person is tested on Monday and the result is reported to the health authorities on Wednesday, the state would include the positive test in Wednesday’s tally of new cases, while the city would add it to Monday’s column.

Because the 3 percent threshold is based on a seven-day rolling average, it matters which day a new case is registered.

Another factor: antigen tests, while generally faster, are less likely to detect an infection in people with a low viral load. New York State includes antigen test results in its official metrics. The city does not. It relies only on the more sensitive test known as a polymerase chain reaction. That’s why the state — which is counting both antigen and P.C.R. tests — may have a higher tally for overall cases in New York City but a lower percentage of positives.

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