In 1991 the Central Laboratory for Radiological Protection and the Institute of Nuclear Physics (INP) began a country-wide study of radon concentration in dwellings using diffusion cups with CR-39 track detectors. In the region of South-Eastern Poland assigned to the INP, a geo-morphologically diversified area constituting 17% of Poland's territory with 20% of the country's inhabitants, two survey series were performed in which 350 cups were placed in randomly selected dwellings. The detectors, after their exposure, were chemicaly etched and counted visually. It was found that the radon concentration over the surveyed area exhibited an approximately log-normal distribution with an arithmetic mean of 75 Bq m^-3. In about 5% of monitored houses the concentration of radon exceeded 200 Bq m^-3. The annual average effective dose to the population due to radon daughter inhalation was estimated to 1.6 mSv. About 4% of the population receive doses exceeding 5 mSv per year. Dwellings with elevated indoor radon concentration have been found mainly in the southern, mountainous part of the surveyed area. The analysis of sources of uncertainty and of their ranges leads to the conclusion that low radon concentrations were estimated with a SD relative uncertainty of 55% while moderate and high concentrations with a SD of less than 20%.