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The uncertainty in the half life of uranium-238 is stated at 0.05% [1]. The same paper gives the date 251.941 myr ± 31 kyr. 251.941 $\times$ 0.05% = 125 kyr.

How are the authors justified in claiming an accuracy of ± 31 kyr when the uncertainty in the half life alone is ± 125 kyr? On top of that, they list two other kinds of analytical uncertainties, which would only increase the overall uncertainty.

[1] ​ Burgess, S. ​et al​, "High-precision timeline for Earth’s most severe extinction", Proceedings of the National Academy of Sciences USA​, Volume 111, 2014. https://www.pnas.org/content/pnas/111/9/3316.full.pdf

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I can't be entirely sure but I'll make an informed guess:

That value doesn't come for a single measurement. Therefore, if the error in the age of a single sample is $\pm125$ kyr, you just need to average 16 samples to get it down to $\pm31$ kyr.

The uncertainty in the addition (or substraction) of two or more quantities is equal to the square root of the addition of the squares of the uncertainties of each quantity (assuming they arise from random errors). For example, if we have quantity A with uncertainty $\sigma_a$ and quantity B with uncertainty $\sigma_b$, the error in the quantity $C=A+B$ would be:

$\sigma_c=\sqrt{\sigma_a^2+\sigma_b^2}$

And if we call M to the average between A and B. The uncertainty in the average is

$\sigma_m=\frac{\sqrt{\sigma_a^2+\sigma_b^2}}{2}$

So if we average 16 samples with $\sigma=125$ kyr, the uncertainty in the average would be

$\sigma_m=\frac{\sqrt{16 \sigma^2}}{16}=\frac{\sqrt{16 \times 125^2}}{16}=31$ kyr

Uncertainty propagation can be seen in the abstract of the article you refer to:

The extinction occurred between 251.941 ± 0.037 and 251.880 ± 0.031 Mya, an interval of 60 ± 48 ka.

Where the ±48 ka comes from?

$\sqrt{37^2+31^2}=48$

This treatment of uncertainties assumes that uncertainties are independent of each other. As @Mark pointed in the comments, this won't be the case if the uncertainty comes from "the length of your measuring stick (the half-life of U-238)".

This is: if you measure something with a "yard-stick" that have the wrong size, you can't reduce the resulting error by just averaging many measurements.

I don't know enough of geochronology to understand all the different errors they report. But the simple example in the abstract cite I presented above shows that they are indeed treating those errors as independent. Otherwise the reported interval error (±48 ka) would not make sense.

If one significant source of error is indeed the uncertainty in the half-life of $^{238}$U. However, it would be wrong to treat these errors as independent only if there exists an exact value for this half-life. Alternatively, maybe there is no exact value of the half-life, and what they meant is that the half-life value can truly vary a 0.05%. This is something to look into if you want to figure out if the treatment of errors they do is correct. However, after a quick google search I found that radioactive half-life can indeed vary by a small fraction due to environmental conditions, this article explains pretty well the phenomena. Here a short excerpt:

...radioactive half-life of an atom can depend on how it is bonded to other atoms. Simply by changing the neighboring atoms that are bonded to a radioactive isotope, we can change its half-life. However, the change in half-life accomplished in this way is typically small. For instance, a study performed by B. Wang et al and published in the European Physical Journal A was able to measure that the electron capture half-life of beryllium-7 was made 0.9% longer by surrounding the beryllium atoms with palladium atoms.

If the 0.05% uncertainty on the half-life of $^{238}$U comes from random environmental factors, it would indeed be acceptable to consider them as an independent source of error for each sample.

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    $\begingroup$ As far as I know, this only applies if the uncertainties are independent. If the uncertainty is in the length of your measuring stick (the half-life of U-238), the uncertainties are correlated and you can't reduce them by making more measurements. $\endgroup$ – Mark Mar 2 at 20:48
  • $\begingroup$ @Mark You are right. I've added something to my answer. Have a look. $\endgroup$ – Camilo Rada Mar 2 at 21:47
  • $\begingroup$ @Mark I just added something about the natural variation of radioactive decay half-life. I think that explain that they treat the errors as independent. $\endgroup$ – Camilo Rada Mar 2 at 22:05
  • $\begingroup$ The page you link to lists three potential mechanisms for changing the decay rate (and thus the half-life) of radioactive elements: time dilation, electron density change and bombardment with high-energy radiation. The first doesn't apply on Earth (well, technically it does, but the time dilation due to Earth's gravity is vanishingly small), the second only affects elements that decay via electron capture (Uranium doesn't), ... $\endgroup$ – Ilmari Karonen Mar 2 at 22:24
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    $\begingroup$ @IlmariKaronen Yes, but the OP doesn't seem to be content with just "a good approximation" as the whole question revolve around a 0.05% change in $^{238}$U half-life, a pretty tiny change. $\endgroup$ – Camilo Rada Mar 2 at 22:29

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