I wanted to follow-up on my post with some more hard data. Especially in light of some folks in other forums calling the reports of inaccuracies in the compass and level as "pseudo-science."
In my physics lab, I have several air tracks which can be set to level, +/- 0.01°. All three of my recently calibrated 5S's were off. By 3-5 degrees on DIFFERENT sides of horizontal. In my specific case, that was a range of 10 degrees in the reporting of level by 3 iPhone 5S's. I also used a calibrated PRO SMARTLEVEL that is accurate to +/- 0.1°. Same results.
After being placed "perfectly" parallel and calibrated, the compasses on the three phones, which are harder to test relative to a known standard, varied by 12 degrees and seemed to jump all over the place, or drift over time. Sometimes I had to shake a phone around to get it to read even a somewhat reasonable value. (A phone would initially read 355 degrees when it should have been in the 110 degree range.) All compass experiments were done outside away from sources of distortion.
I reported these results to Apple on their Feedback site.
I really don't think we're at the pseudo science level. No pun intended.