High interest rates pushed Virginia's housing starts down by 13 percent last year, the University of Virginia reported this week.

Building permits for nearly 46,000 homes--including mobile homes, single-family homes, duplexes and multiple-unit structures--were authorized in the state in 1980, a 7,000-unit drop from 1979, university economists said.

In Northern Virginia, meanwhile, single-family home sales, dropped 47 percent last month to 966 from their level of 1,831 in September 1980, the Northern Virginia Board of Realtors reported. The year-to-date total was 17 percent lower than last year, at 12,893 through September.

Average prices of single-family homes rose 5 percent over the year, from $90,842 to $95,892, the group reported.

Trying to put the best face on the dismal figures, the group's analysis points out that nearly 1,000 homes were sold by Realtors in September. "Those area citizens who depend solely on media interpretations of economic conditions probably would not believe this figure, yet there it is," the analysis states.

"None of us in real estate and none who want to buy and sell are happy that September 1981 sales are 47 percent below those which were made in September 1980. However, transactions continue to be accomplished."

The University of Virginia report, prepared by Michael A. Spar and Julia A. Martin, also shows that 29,000 building permits for single-family homes were issued in Virginia last year, about 5,600 or 16 percent fewer than in 1979