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According to Wikipedia, standard atomic weight is based on the published isotopic abundances of CIAAW:

The standard atomic weight $(A_\mathrm{r},$ standard) of a chemical element is the weighted arithmetic mean of the relative atomic masses $(A_\mathrm{r})$ of all isotopes of that element weighted by each isotope's abundance on Earth. The standard atomic weight of each chemical element is determined and published by the Commission on Isotopic Abundances and Atomic Weights (CIAAW) of the International Union of Pure and Applied Chemistry (IUPAC) based on natural, stable, terrestrial sources of the element.

However, this table of isotopic abundances appears to be incomplete — for example, it's missing carbon-14.

Are they only listing primordial isotopes, or maybe those above some threshold of abundance? Is there some other criteria?

Why is carbon-14 not listed?

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    $\begingroup$ I guess this is because we have very few of it. $\endgroup$ – Ivan Neretin Jun 25 at 12:00
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    $\begingroup$ According to this, the abundance is about a part per trillion, so the imprecision in the abundances of the two main isotopes dwarfs that. $\endgroup$ – Ed V Jun 25 at 12:39
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    $\begingroup$ Also, what was quoted also says stable ... $\endgroup$ – Ed V Jun 25 at 13:12
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    $\begingroup$ The CIAAW source specifically says under carbon: "Yet, this isotope never occurs in normal carbon sources in concentrations high enough to affect significantly the Ar(C) value." $\endgroup$ – Jon Custer Jun 25 at 13:59
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    $\begingroup$ On a final note, the OP’s linked table of isotopic abundances shows that no listed isotope has an abundance given to more than 6 decimal places, except for La and Ta, which both have 7 decimal place listed abundances. So carbon 14’s abundance is in the 12-th decimal place and it is entirely negligible. $\endgroup$ – Ed V Jun 26 at 16:28