Error Propagation Calculation I have a few machines that are used to calibrate each other.
Machine 1 has is accurate to 0.025%
Machine 1 is used to calibrate Machine 2, which has an accuracy of 0.005%
Machine 2 is used to calibrate Machine 3, which has an accuracy of 0.025%
Machine 3 is used to calibrate Machine 4, which has an accuracy of 0.04%
Using the root of the sum of the squares gives an error for Machine 4 of just over 0.052%, but I need it to be below 0.05%.  Is there any games I can play (like bootstrapping, maybe) to get this error down?
In other words, I can get all sorts of empirical calibration trials...Can I use that data somehow to bring down that error propagation?
 A: If you absolutely can't measure the same samples with Machine 1 and Machine 4, then I don't see what can be done here.
Machine 4's precision is going to be 0.04% no matter what. Its accuracy is going to be $\sqrt{0.025\%^2+0.005\%^2+0.025\%^2}=0.036\%$.
If you could measure the same sample with Machine 1 and 4, then you could adjust for the bias and increase accuracy, me thinks. You could increase accuracy to the precision of Machine 1 in this case.
UPDATE: The way I see this is that Machine 4 has a bias (accuracy) of ~0.036%, which was introduced by Machines 1-3. When you measure a sample with it, in average the mean will be at the bias level, while the dispersion (precision) will be 0.04%.
Machine 3 has both bias and precision ~0.025%. Since you can measure the same sample with 3 and 4, there has got to be a way to bring the accuracy of Machine 4 to that of Machine 3. Maybe you could compute an average difference between MAchine 4 and Machine 3 on a sub sample, then subtract it for all subsequent measurements of Machine 4, which would bring the accuracy up to ~0.025%. Again, I don't think anything can be done with a precision of Machine 4 (0.04%)
