Let's take your example:
The scales we use are 0.1 resolution but our tolerances are usually +6g or +\- 10g. Doing a Type 1 on a simple 200g weight is likely not going to show any variation.
So the values are:
Tolerance range: -10g ... +10g
Required Cgk >= 1.00 using K=20%
Resolution: 0.1g -- note that this is 0.5% of the (total) tolerance range and therefore negligible
The first question I would ask myself is: Suppose have bias of n*SD, what is the SD I need to achieve?
Answer: Just use the formula Cgk = 0.2 * tol/(6*SD) - bias/(3*SD) and enter some "standard" numbers. Here my expectations:
* bias = 1*SD and SD = resolution = 0.1 => Cgk = 6.3. Thus, you really do not need a higher resolution than 0.1g
* bias = 2*SD and SD = 2*resolution = 0.1 => Cgk = 2.6. Thus, even this setting is fine.
Using these values, we can calculate how many runs our type 1 analysis needs. It turns out, that we need only 11 measurements. The acceptance region is Cgk>=1.61, the confidence is 95% and the power of the qualification check is 90%.
Coming back to your original question, let's change the setup
Tolerance range: -10um ... +10um
Required Cgk >= 1.00 using K=20%
Resolution: 1um -- note that this is 5% of the (total) tolerance range. Using K=20% this corresponds to 25% of the (measurement) range!!!
Now enter some numbers
* bias = 1*SD and SD = resolution => Cgk = 0.33
If we use the "probable error" PE = 0.77*SD instead, we get
* PE = resolution/2 and bias =1*PE => Cgk = 0.77
* PE = resolution/2 and bias = PE/2 => Cgk = 0.90
From these numbers we see that the qualification will be hard. Redoing the qualification several times, we might be able to achieve a run where Cgk>1.00. However, this result is not (!) reproducible, and this method would simple show our lack to understand quality.
I hope we all agree, that the assumptions of the underlying theory are not met. E.g. the errors are be normally distributed. Thus, you might be able to argue with the quality department that you need different methods and come up with something like a
tolerance interval. However, if they won't listen, here is a possible work-around: For non-normal distributions the Cpk value is defined using the percentiles 0.1349898% and 99.86501%. Thus, if you take N=100 measurements and then you apply the Cgk formula for non-normal distributions ... that might do the job.
If you manage to get all readings within [-1, 1]um the first term in the Cgk formula is expected to be 2. Thus, you have "space" for the bias term.
I would apply this work-around only if the "quality person" is unwilling to listen (and react) to my arguments. If he/she just wants to see a certain number like Cgk>1, then this method might be for you. And yes: This is all "math" and no insight