All the above is entirely correct...however there is legal scope to weigh fat passengers if the airline suspects there are too many fat fokkers on board to up the average.
Unfortunately, so as not to make the pie-eaters feel lardy and hurt their poor fat feelings the requirement is that all passengers, even the little kiddies and grannies must be weighed. A time consuming problem.
There are many stories of aircraft taking an extra few thousand feet to get airborne.....particularly on flights to the good ol' US of Big Macs and Apple Pie.