What is the average density of an average person?

Since the average person is predominantly made up of water, we are very close to the density of pure water - which is why we just barely float in swimming pools, hot tubs, etc. When you fill your lungs with air you float better, when you expel air, you start to sink. According to a 1966 study by the US Army (see the attached link), the average density when as much air as possible is expelled is about 1.062 g/cm3. Of course we don't go around with our lungs empty all the time, so most of the time our average density is less. People with more fat tend to be slightly less dense than people who have very low body fat. About 1% of all people are so lean that they will actually sink completely when placed in fresh water even without exhaling all the air they can.
Within all the variability posed by these factors, a reasonable estimate would be that the average person, fat and all, with their lungs inflated, has a density between 0.924 g/cm3 and 0.957 g/cm3. That averages out to about 0.94 g/cm3.