How much variability was there in the O=16 atomic mass standard, and how does it compare to the shift to the current carbon standard? This is a follow-up question to this excellent answer by David Hammen on the reasons and history of the choice of carbon as the element used to define the atomic mass unit.
As related in that answer, and particularly in this reference, for a very long time the calibration of the atomic mass numbers was actually fixed by setting the atomic mass of oxygen at 16 (also called the O=16 standard, back in the when); this makes a lot of sense as oxygen is plentiful and reactive, so it's possible to make it combine with just about anything.
However, this was brought to a halt by the discovery that oxygen has three different stable isotopes, $^{16}\mathrm{O}$, $^{17}\mathrm{O}$ and $^{18}\mathrm{O}$, with different masses - and, more importantly, by the fact that the isotopic ratio between them will vary depending on where the sample comes from, which is obviously a killer for precision metrology.
In response to this dilemma, the obvious choices are to set a fixed isotopic ratio for the definition of the atomic mass (a metrological nightmare) and changing the standard to specify that it's $^{16}\mathrm{O}$ that has an atomic mass of $16$, but the latter would have meant a large jump in the definition of the atomic mass. Instead, a compromise was chosen - setting the mass of $^{12}\mathrm{C}$ as the standard, which meant a smaller change (42 ppm) with respect to the isotopic-mixture standard than a change to a $^{16}\mathrm{O}$ standard would have (275 ppm).
However, I feel that that comparison isn't quite the correct one to take, and it just isn't all that informative. Instead, to really evaluate how much of a change was brought in by the switch to the $^{12}\mathrm{C}$ standard, the real yardstick one should use is the uncertainty brought into the standard by the variable isotopic composition of oxygen. So: for the sources of water used for metrology at the time of the shift (rain water, sea water, river water, etc., from different relevant locations), what is the range of the variability in the average atomic mass of oxygen, and how does this compare to the change brought by the switch to the carbon standard?
 A: The presuposition of your question is not quite correct. The problem with the atomic mass unit (amu) was more complicated.
1 multiple definitons
The amu had two definitions which were incompatible:

*

*Physics definition: amu = $ m(^{16} O)/16$

*Chemistry definition: amu = $ \bar m(O)/16 = \sum f_i \cdot m(^i O)/16$
where $f_i$ is the naturally occuring fraction (or abundance) of the oxigen isotope $ ^i O$.
Wheras it is true that the chemists' amu is poorly defined, it was more the fact that the two amu are quite different, that became a problem.
The big question is: why did science not just settle on the physics definition? Which brings us to the second problelm:
2 Chemists often don't use mass units
For some strange reason, chemists to this day often don't use mass units in in their writing about atomic scale. For example, they would usually write someting like "CO2 has a mass 44" instead of "CO2 has a mass of 44 amu".
This implies that in chemistry it is not possible to change the atomic scale units, as you would never know what units were meant. This is why it was not possible for chemists to switch to the physics amu definition. Their only choice was to switch to a definition that was close enough to their original amu so that the difference would not be noticed.
For physicists this was much easier. They just had to give up the amu and replace it by any new unit. When reading a physics text you would know which unit was used because physicist generally don't have the bad habit of omitting units.
3 Resolution
This is why a new unit, the "unified atomic mass unit" u was defined as

*

*u $= m(^{12} C)/12$
It has a new name and it is reasonably close to the "unnamed" chemistry amu.
Unfortunately, to this day, many chemists have not learned the lesson and continue to:

*

*write text where the units are omitted

*using the ill defined amu insted of the u

4 Future
Two things will or should happen in the future:

*

*The clumsy name "unified atomic mass unit" with the symbol "u" will be substituted by the unit "dalton" and its symbol "Da".

*There sould also be a new name and symbol for the "atomic charge unit" and its symbol "e". It has been suggested to use the name "stoney" and the symbol "St".

Stoney was the inventor/discoverer of the unit electron. (Note that the electron was the name of a unit before the name was hijacked by the elementary particle now known as electron. The unit only survived in the derived unit electronvolts).
