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

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?

• Here is a similar question: physics.stackexchange.com/q/158589 Mar 20, 2018 at 23:43
• @PeaBrane thanks, will delete this question in 10 minutes as a duplicate. EDIT: Actually not, since the answers in that question doesn't directly deal with the question asked in this one. Mar 20, 2018 at 23:46
• $\pm$ usually denotes confidence interval of one $\sigma$, or 66% confidence. Mar 20, 2018 at 23:49

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

And a link to the meaning of uncertainty.

• Thanks, couldn't find the definition page for some reason. Taking this to mean my interpretation is correct, still not sure if innisfree's interpretation is equivalent though. Mar 21, 2018 at 0:00
• @Allure Yes, I think that fits with what innisfree wrote. One should be aware of the conditional "if results are distributed as a Gaussian" and this includes assumptions that different methods should give results that cluster around the same average. If that is not the case, it becomes very difficult for the metrologists to estimate uncertainty.
– user137289
Mar 21, 2018 at 8:01

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).