Unless I miss my guess, the FAA has decided that one person weighs 170 lbs. There's also a standard allowance for carry-ons. I don't know if checked baggage is weighed, or is there is a standard for that as well.
You are right that the numbers are very tightly monitored. I remember one flight in which the change in wind direction from flight planning to take off time made a flight no longer possible. I also remember one flight in which we sat in the run-up block for a few minutes, burning of the last few hundred pounds of fuel to make the flight legal.
Tom