roughly 2 pieces per 100 that fail a specific measurement
Maybe you will find something useful in my story.
I recall one close-tolerance machining problem with a reject rate consistently around 200 ppm, which appeared to be random. What sample size would you choose to investigate a 200 ppm problem? How big a DOE study would you lay out, when the occurrence is 200 ppm?
The engineers did not make much progress until someone (me) arrived at 4:30 am to hand-collect the first 20 pieces off the machine. I knew to look for Red X variation where there are discontinuities in the process. It wasn't until I measured parts for my multivari study that we proved ALL out-of-tolerance parts were made in the first 5 minutes of a 16-hour run. This was not a case of tool wear, but expansion of the cutting tool before the cutting oil reached equilibrium temperature.
The cause was hidden from most observers because after machining, the first 20 parts were tumbled with hundreds of following parts in order to clean them, and then a few hundred parts were feeder-bowl fed to the assembly process, in guess what? random order.
The problem was easy to solve after we gained this insight. We looked at oil heaters to maintain 140°C, but it was cheaper (at a penny a piece) to throw away the first 20-30 pieces of every day's run. In your shop, you will have to come up with your own viable solution.
This is when I learned, randomness is more a statement of our ignorance rather than any useful information about the nature of the problem.