I have an opinion on chucks that comes from a lifetime of using them, and measuring their run-out with a dial indicator. For what it's worth, here it is - -
1 Nowhere is it more true that you get what you pay for than with chucks.
2. The average run-out of a scroll chuck made for woodturning is in the range of 0.003" to 0.005", and it isn't difficult to find one that is closer to 0.010". You are extremely lucky if you have one that is better than that. There are scroll type chucks made with accuracy of 0.001", but they are for precision machine work and cost several hundred dollars.
3. The average Jacobs chuck will be within the same 0.003" to 0.005" when new, and it will deteriorate with age because of wear of the many moving parts inside of it. The run-out problem is compounded by the Morse Taper and the fact that it isn't at all difficult to find a lathe, in the price range most paid for it, whose internal MT is not in perfect alignment with the bore. A small angular difference can add up to several 1/1000" run-out at the end of the chuck.
The consistently most accurate Jacobs chuck I have found is the "Golden Goose" brand that is sold by Craft Supplies and Woodcraft stores. You can buy cheaper, but this is the best accuracy (0.003" or less) you will find for about $40.00. There are more accurate Jacobs chucks available, but they will cost several hundred dollars.
4. The Bealle and similar collet chucks will have a run-out on the order of 0.001". It is unusual to find a chuck with more than that, and if you did, you should be contacting the manufacturer or the point of purchase for a replacement.
The chuck that threads onto the spindle is more accurate than the one that uses the MT in the spindle because the threads were cut at the same time as the spindle was turned and they are more accurate than the MT (see 2 above) which are usually machined as a separate operation.
There are MT chucks available, but they are less accurate (see above) and work best when the srock being held is the nominal size of the collet because they are split from only one end. The ER 25 and ER32 collets are split from both ends and have equal accuracy and gripping over a range of about 1/64" larger to 1/32" smaller than the nominal size of the collet.
5. The best and the worst of all ways to hold a mandrel is the tapered adapter with the mandrel screwed into the end of a MT that fits in the lathe. First, there is the problem with the MT in the lathe, as noted above. Then there is the possibility that the drilled and tapped hole in the adapter in not centered or is not parallel with the lathe, and the possibility that the same is true for the threaded end on the mandrel. In the best of conditions, all tollerances are minimal or they cancel each other, and we have an accurate mandrel. The worst case is when the tolerance all add in the same direction, and we have the least accurate mandrel. Unfortunately, only 20% are the best. 20% are the worst, and the rest are somewhere in between.