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Measuring Cylindricity

We are running a part with a 3" cylinder. The cylinder has a .0002 cylindricity call out. For 10 years we have been using a air Gage to measure this feature, we zero the Gage & spin it throughout the cylinder & the total reading is what we use as our result. Anyway, I was asked to start measuring this on our CMM, I get the cylindricity reading at around .001 every time, I measured a master ring the nominal size of this cylinder & also get the cylindricity around .001. Can I hear everyone's input on measuring cylindricity please.

1.) Which way is more accurate, CMM or air gage (I have tried multiple sized probes & stylus lengths.
2.) Is cylindricity of a cylinder the same as roundness of a circle??
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