IT may be a moot point if you always use the same chuck on the same machine, but measuring the chuck on the lathe spindle measures the mounted lateral deviation of the combination of the chuck and the spindle. If you are trying to analyze the chuck alone, you need a different method. If you have a D1 Camlock, you could use the "mounted" method, but switch the orientation of the chuck 60° in sequence. If you get the same runout at the same point on the chuck each time, you've got chuck runout. If putting the chuck pins in different holes give you different values of runout, you need to do some figuring to determine which runout is the spindle and mount, and which runout is the chuck.
The mandrel referred to above is a rod that has accurate center-drilled holes on each end. I think you'd put the rod into the chuck on the lathe, center it well, then remove the lathe and rod together, and chuck one end of the rod in a indexing head. Then you can set up the index head to get one jaw parallel with your surface plate (you are working on a surface plate, right), and measure the top and bottom height (near the center, and at the periphery both). Rotate four times to get another eight measurements. Then measure under the jaw - same thing with inner and outer measurements. Then move your surface gage with your indicator to the other side of the chuck and repeat.
You'll then get to do a boatload of calcs to ensure that the jaws are central. Anyway, that's what a mandrel is. A rod with end holes. A very precise rod with end holes. In fact, you may want to avoid standard mandrels (which have a small taper). You could create your own with TG&P rod.