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?

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    $\begingroup$ machine 1 has worse accuracy than Machine 2, yet it's used to calibrate the latter? $\endgroup$ – Aksakal Apr 9 '14 at 13:56
  • $\begingroup$ Yes. I don't like it either, but it's a matter of cost and size (machine 2 needs almost a crane to move, so sending it to be calibrated at a measurement facility is unfeasible). Machine 1 can fit in a suitcase, and therefore is what is sent out to be calibrated at a measurement facility. $\endgroup$ – testname123 Apr 9 '14 at 13:58
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    $\begingroup$ What's your data? is it measurements from Machine 5? $\endgroup$ – Aksakal Apr 9 '14 at 14:08
  • $\begingroup$ Yes...i can get measurement data from all machines $\endgroup$ – testname123 Apr 9 '14 at 14:10
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    $\begingroup$ Then why do you have to measure with Machine 4? Why not measure it all with Machine 2? I think I'm missing something in your setup $\endgroup$ – Aksakal Apr 9 '14 at 14:12

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%)

  • $\begingroup$ I see...How would we approach the last two sentences of your response? It looks now like Machine 4 already has accuracy+precision upper bounds less than .04%. I must have calculated that wrong. $\endgroup$ – testname123 Apr 9 '14 at 15:09

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