For reference, I am using a USB microscope calibrated against a microscope calibration slide with an estimated 1 sigma uncertainty of about 0.3 pts (about 15 microns or so, so when I quote that a card is 47.5, this means that there is about a 68.3% chance that the true value is between 47.8 and 47.2). Each point here is a card that was sent in with the error bars (the crosshairs) representing the 1 sigma uncertainty of each measurement. The cards that were sent in were almost exclusively English (with maybe 1 or 2 Japanese cards) modern from the Sw/Sh era onwards. I only recorded the lower of the two centering values (so the number is always lower than 50 and unfortunately I can't tell if a card is more right heavy or left heavy based on this graph).

The regions of interest highlighted here are the 45/55 mark and the 44.6/55.4 mark (which rounds up to 45/55)

Key takeaways:

1.  PSA's computerized centering system actually seems pretty precise and accurate. To be sure, all of the cards that were submitted had well-defined borders so your mileage might vary for cards that are less well centered.
2. Borderline cards measured nominally at 44.6 on either side were still considered for PSA 10s.
3. T:B centering is a requirement not a metric. A card with T:B centering of 44.5 is just as likely to get a 10 as a perfectly T:B centered card all things held equal. As long as T:B centering>44.5 you are good.
4. L:R centering is more differentiated on the lower end. A card with L:R centering of 45 is not judged the same as a card with L:R centering of 47 and above even when controlling for other defects (scratches, whitening etc on the card). There was no statistically significant difference between the PSA 10 rate for cards with L:R centering 47 and above. In other words, if L:R centering is 46 or worse, the card should almost be perfect. 
5. Even with all the measures in place to try to get as precise a measurement as possible, the process here still admits about a 0.3pt 1 sigma uncertainty. I think this is the bare minimum you need to differentiate cards that are borderline. As a ballpark estimate, I think the uncertainty of most centering apps should be about a 1pt or a bit higher. While the uncertainty of measuring by eye is about 2-3 pts. This means that a human 'by eye' grading can reliably tell if a card is centered around 47 - 50 or 48 - 50, but would fail in determining whether a card is exactly 45/55 which is the case that we should care most about.

I analyzed the centering on 186 cards I've submitted to PSA