What is the meaning of the bracketed numbers in reference values for constants?

Question

For example, the NIST reference value for Newton's gravitational constant is $$ G = 6.674 08(31) \times 10^{-11}\,\mathrm{m^{3} kg^{-1} s^{-2}}. $$ What is meant by the $(31)$?

I have generally understood this as the uncertainty in the results. That is, there's a 68% chance (1 sigma) that the true value of $G$ is between $6.67377 \times 10^{-11}$ and $6.67439 \times 10^{-11}$, and a 95% chance (2 sigma) that it is between $6.67346 \times 10^{-11}$ and $6.67470 \times 10^{-11}$.

However in a comment to another question, innisfree said that this is a confidence interval, and that "In 68% of the hypothetical repeat experiments, an interval produced in that manner would include the true $G$, were the model true".

Which is correct, or are they equivalent?

Answer 1

Actually on that same page, there is a line for the standard uncertainty.

And a link to the meaning of uncertainty.

Answer 2

You mention two possible interpretations of a confidence interval at $68\%$:

1. There is a $68\%$ chance that the true value lies in the interval.
2. The confidence interval would contain the true value in $68\%$ of an infinite number of repeat experiments.

They are not equivalent and only the second one is correct. The distinction follows from two interpretations of probability; Bayesian and frequentist. The first definition is usually called a credible region to distinguish it from a confidence interval.

In the first definition, the true value is unknown and we talk about the probability or plausibility of it lying inside an interval. In the second definition, the data and thus confidence interval is a random variable, but the true value, whilst unknown, is fixed, and we do not speak of the probability that the true value lies in such and such an interval.

In practice, confidence intervals and credible regions tend to be similar, though the latter would depend upon a prior for the true value. Confidence intervals - with their arcane definition - are very often wrongly interpreted as credible regions by scientists (see. e.g., Robust misinterpretation of confidence intervals, or google misinterpretation of confidence intervals).