Michigan had the highest unemployment in the nation in July, the Labor Department said this week, and South Dakota had the lowest.

Or did they?

While all surveys have some margin of error, the Bureau of Labor Statistics' reports on state and local unemployment have a considerable potential for inaccuracy, according to federal and local officials who work with them. And officials say they are becoming increasingly unreliable as a measurement of economic trends or as a device for distributing federal aid equitably.

The problem is such, according to BLS officials, that Michigan's unemployment rate, reported as 14.6, could have been as high as 16.3 percent or as low as 12.9 percent. In an average of one out of 20 months, the rate could have been ever further off.

In late 1981, budget cuts forced the BLS to trim the size of the survey it takes for determining unemployment from 72,000 households to 60,000. While the BLS says that hasn't affected the accuracy of the nationwide unemployment rate substantially, it has reduced the reliability of the state and local figures even more. In Michigan's case, the margin of error was increased by a tenth of a percentage point; the margin is greater in smaller states and localities where the BLS uses a smaller sample.

"There is really no easy way to nail down the number any further using these new statistics," said Fred Cronkhite, chief of research and methods for the BLS. "This new sample has the lottery effect of creating a larger possibility of error. That means the data is less sensitive to the problems of a given community."

Matthew F. Shannon, acting director of the District of Columbia's unemployment services department, said of the federal government's unemployment figure for the District, "We don't believe it has integrity . . . . It is not sufficient for us to rely on."

Pat Arthur, who handles labor market information for the Virginia Employment Commission, said: "There's definitely a disparity between the adequacy of the sample and the quantity of money that depends on it."

Just over $5 billion in federal funds were distributed in fiscal 1982 in programs that required a state or locality to have a certain unemployment rate to qualify. These include job programs under the Comprehensive Employment and Training Act, grants and loans from Commerce's Economic Development Administration, urban development action grants and contracts to areas designated as labor surplus areas. With the exception of the latter program, all are being abolished or cut back by the Reagan administration.

"The use of these new statistics will result in less equity in the distribution of federal funds," said Ray A. Konstant, chief of the BLS local area unemployment statistics division. "Some communities will be excluded that should have been included. If there is no bias in the system, then everything may wash out, but individual areas will not have been treated properly."

"The local unemployment statistics program is one which clearly needs more work," said BLS Commissioner Janet L. Norwood. She said the agency has been wanting to redesign the 22-year-old program, but won't have the money available until at least fiscal 1984.

For example, Florida's unemployment rate moved erratically from 7.7 percent last January to 7.1 percent in Feburary, 8.6 percent in March and then 7.9 percent in April. BLS officials say that those jumps could have been caused by "chance variations" instead of true changes in unemployment.

"If a state's rate moved from 7.0 to 7.5 percent and then back to 7.0 percent throughout the year, this would tell us nothing about employment trends," said Cronkhite. "It would just be chance."

Florida -- as one of the 10 largest states -- is supposed to have figures which are more reliable than the other 40 states, where the survey sample is smaller, but federal officials said the margin of error for a given monthly figure is 1.3 percent up or down.

With the smaller sample size, these "spikes" can occur more frequently, federal officials said. For example, the unemployment rate in Michigan moved from 16.1 percent in February to 17 percent in March and then back to 15.5 percent in April. While this spike could have reflected a true increase, Cronkhite said, much of it could have also have been caused by chance.

Under the BLS system, each state counts the number of people who filed for unemployment insurance in a month and follows a 43-step mathematical formula a so-called handbook number. BLS then adjusts that number, using data collected through the Current Population Survey, the program that has been cut back. The 43 steps are taken to account for people who wouldn't normally be filing for unemployment compensation, such as women and young people looking for their first job.

"Given the fact that you're going to use that unemployment statistic to allocate federal dollars, the sample should certainly not be reduced," said the VEC's Arthur. "If anything, it should be increased."

"It's tragic that the household survey, which is a basic source for our unemployment data, is getting this back-of-the-hand treatment from the administration," said Markley Roberts, an economist with the AFL-CIO and the chairman of the BLS's Labor Research Advisory Council.

Norwood, who has fought both Congress and the administration to stop further cuts, said, "It would be nice to double or triple or quadruple our budget . . . but given the budget constraints under which we operate, we do a pretty good job with what we have."