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While reading publications in astrophysics I've encountered this kind of notation (fictitious example): $$M = 6.1^{+0.6}_{-0.1} \times 10^{30} \textrm{ kg}.$$

Several examples are here, p. 2.

I assume that the superscript and subscript refer to the right and left uncertainty of the main magnitude. I have three questions:

  1. Can someone explain to me what these three numbers (main magnitude, superscript, subscript) refer to, in terms of the probability distribution for the true value? For example, are they the mean and 10% and 90% quantiles? Or maybe the median and 25% and 75% quantiles? Or have I misunderstood them completely?

  2. Can someone provide early (or the first) examples of this particular superscript-subscript notation? I only knew the "$x \pm \triangle x$" notation.

  3. Can someone provide a written reference – textbook, manual, article, or similar – where this notation is explained?

I checked this question, but it didn't really help me. I also checked the Guide to the Expression of Uncertainty in Measurement, published by the Joint Committee for Guides in Metrology (JCGM), formed by various organizations (BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, OIML). It is a standard for nomenclature and practice in reporting uncertainty, and makes important distinctions, for example between "coverage intervals" and "confidence intervals" (see JCGM 101:2008 definitions 3.12–3.16). But I can't find anything about the superscript-subscript notation there.

Cheers!

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When quoted like this they would usually be the most likely value (peak of the probability distribution) or the median value (where half the distribution is above or below), and then the superscripts and subscripts would be increments that would contain $\sim 34$% of the probability distribution. In the M87 papers, the numbers are the 50th percentile (median) and lower (upper) limits corresponding to the 16th (84th) percentile.

The total range corresponds to about a 68% confidence interval. The reason for this is just that it corresponds to $\pm$ one sigma in a normal (Gaussian) distribution.

It is good practice to say what your error bars mean, but in the absence of that I would assume the above.

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    $\begingroup$ Thank you! Do you happen to have a reference to a manual, or journal guidelines, or similar, about this? $\endgroup$ – pglpm Apr 13 at 14:05
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    $\begingroup$ Mmm wait, are you assuming that that's their meaning, or do you know that that's their meaning? :) $\endgroup$ – pglpm Apr 13 at 14:08
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    $\begingroup$ @pglpm You do not have to quote asymmetric limits. Any 68% confidence interval is as good as another, but one normally likes to indicate the most likely value too. $\endgroup$ – Rob Jeffries Apr 13 at 14:09
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    $\begingroup$ Thank you for the new extended answer. Still, there must be some text around (ISO or NIST standards, or author guidelines, or astrophysics textbook, or something) where this convention is stated... $\endgroup$ – pglpm Apr 13 at 15:17
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    $\begingroup$ @pglpm The point is, there is no convention, so it wouldn't make sense to try to state it anywhere but the current text in which you are using some asymmetric uncertainties. Because in some other work, another author uses a different convention. Both mode, median, and even mean is often used as the central value. Personally, I prefer the median with the 16th and 84th percentiles, as you can then be sure that the central value is between the lower and upper errors, which is not the case if you use mode or mean. $\endgroup$ – pela Apr 13 at 21:13

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