<p>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 <span class="math-container">$\sim 34$</span>% 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. </p>

<p>The total range corresponds to about a 68% confidence interval. The reason for this is just that it corresponds to <span class="math-container">$\pm$</span> one sigma in a normal (Gaussian) distribution.</p>

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

CC BY-SA 4.0
2019-04-13T14:05:22.600Z
Thank you! Do you happen to have a reference to a manual, or journal guidelines, or similar, about this?
pglpm
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2019-04-13T14:07:20.733Z
There must be a reason why they choose asymmetric increments (skew distribution) and an algorithm to calculate them.
pglpm
CC BY-SA 4.0
2019-04-13T14:07:24.890Z
@pglpm No, I don't. But in an astrophysics journal, if nothing else was defined, then this is what I would assume.
ProfRob
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2019-04-13T14:08:45.587Z
Mmm wait, are you *assuming* that that's their meaning, or do you *know* that that's their meaning? :)
pglpm
CC BY-SA 4.0
2019-04-13T14:09:02.700Z
@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.
ProfRob
CC BY-SA 4.0
2019-04-13T15:17:54.423Z
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...
pglpm
CC BY-SA 4.0
2019-04-13T21:13:09.107Z
@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.
pela
CC BY-SA 4.0
2019-04-13T21:51:39.127Z
@pela I understand that there is no convention. Yet, Rob Jeffries says: "in an astrophysics journal, if nothing else was defined, then this is what I would assume". Well: why? how does he know? Someone must have told him or he must have read it somewhere. What I now presume, is that there must be many astrophysics articles out there where the authors *explicitly* say to be using the 16-50-84% quantiles, so Rob is just summarizing his statistical knowledge. It'd be nice to know if this was the source of his knowledge, or some other experience or text.
pglpm
CC BY-SA 4.0
2019-04-13T23:11:23.320Z
@pglpm I'm just guessing that the ways one might know without having seen an explicit source can include 1) written enough papers personally to see how journals treat notation, 2) read enough papers where the authors explain their method of determining asymmetric limits to know that journals are flexible as long as it's carefully explained in the manuscript. So "Someone must have told him or he must have read it somewhere." may not be exactly the case.
uhoh
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2019-04-13T23:42:03.740Z
The strongest convention is that the lower and upper bounds define a 68% confidence interval. I would assume this is standard in all physical sciences and would expect to see an explicit statement if otherwise. After that, it really doesn't matter that much what central value is quoted.
ProfRob
CC BY-SA 4.0
2019-04-14T00:35:33.993Z
1 SD is how I learned it in high school, and it does seem to be the convention. Here's [one source](https://www.webassign.net/question_assets/unccolphysmechl1/measurements/manual.html) that says it directly: "When we report the average value of N measurements, the uncertainty we should associate with this average value is the standard deviation of the mean, often called the standard error (SE)."
ahiijny
CC BY-SA 4.0
2019-04-14T00:42:00.890Z
[NIST](https://physics.nist.gov/cgi-bin/cuu/Info/Constants/definitions.html): "The standard uncertainty u(y) of a measurement result y is the estimated standard deviation of y."
ahiijny
CC BY-SA 4.0
2019-04-14T08:42:56.840Z
@uhoh Indeed, those were my guesses.
pglpm
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2019-04-14T08:43:31.943Z
@ahiijny Thank you. I've added some references to the *Guide to the Expression of Uncertainty in Measurement*.
pglpm
CC BY-SA 4.0
2019-04-14T08:48:10.710Z
Hi @pglpm if you are providing the information from *e Guide to the Expression of Uncertainty in Measurement* as part of an answer, you should post it as a second answer to your question, which is perfectly fine. Only include it in your question post if it is a necessary part of your question. It certainly looks interesting!
uhoh
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2019-04-14T09:10:59.460Z
@uhoh Thank you. I don't think it provides an answer to my question, because I couldn't find any guidelines about superscript-subscript notation in that guide. But I've added a reference to it in my question because it was my starting point. Cheers!
pglpm
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2019-04-14T12:45:48.447Z
@pela That's very useful, thank you. What do you mean by "central value"?
pglpm
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2019-04-14T20:05:21.073Z
@pglpm I mean the "$x$" in $x_{-\sigma_{-}}^{+\sigma_{+}}$, which may then be the mode, mean, or median, and where $\sigma_{-}$ and $\sigma_{+}$ then are the lower and upper errors, respectively. Note that if you want to add to numbers with asymmetric uncertainties, you can't simply add lower and upper errors in quadrature. That is, $x_{-\sigma_{x,-}}^{+\sigma_{x,+}} + y_{-\sigma_{y,-}}^{+\sigma_{y,+}} \ne (x+y)_{\sqrt{\sigma_{x,-}^2 + \sigma_{y,-}^2}}^{\sqrt{\sigma_{x,+}^2 + \sigma_{y,+}^2}}$
pela
CC BY-SA 4.0
2019-06-21T13:26:11.667Z
@ahijny What your sources say is the rule is irrelevant to this question since there is only one value of the standard deviation of a distribution whereas the question here asks what do (different) upper and lower error bars mean. Upper and lower error bars are quoted in cases where the probability distribution is asymmetric and convey information beyond merely what the standard deviation is.
ProfRob
CC BY-SA 4.0
2019-04-13T13:43:26.913Z
2019-04-13T14:21:07.633Z
2019-04-13T14:21:07.633Z
ProfRob
<p>While reading publications in astrophysics I've encountered this kind of notation (fictitious example):
<span class="math-container">$$M = 6.1^{+0.6}_{-0.1} \times 10^{30} \textrm{ kg}.$$</span></p>
<p>Several examples are <a href="https://doi.org/10.3847/2041-8213/ab0ec7" rel="nofollow noreferrer">here, p. 2</a>.</p>
<p>I <em>assume</em> that the superscript and subscript refer to the right and left uncertainty of the main magnitude. I have three questions:</p>
<ol>
<li><p>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?</p>
</li>
<li><p>Can someone provide early (or the first) examples of this particular <em>superscript-subscript</em> notation? I only knew the &quot;<span class="math-container">$x \pm \triangle x$</span>&quot; notation.</p>
</li>
<li><p>Can someone provide a written reference – textbook, manual, article, or similar – where this notation is explained?</p>
</li>
</ol>
<p>I checked <a href="https://astronomy.stackexchange.com/q/8841/26647">this question</a>, but it didn't really help me. I also checked the <a href="https://www.bipm.org/en/publications/guides/gum.html" rel="nofollow noreferrer"><em>Guide to the Expression of Uncertainty in Measurement</em></a>, published by the <em>Joint Committee for Guides in Metrology</em> (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 &quot;coverage intervals&quot; and &quot;confidence intervals&quot; (see <a href="https://www.bipm.org/utils/common/documents/jcgm/JCGM_101_2008_E.pdf" rel="nofollow noreferrer">JCGM 101:2008</a> definitions 3.12–3.16). But I can't find anything about the superscript-subscript notation there.</p>
<p>Cheers!</p>
<p><strong>Late addendum:</strong> the interpretation of the accepted answer (median and left &amp; right 16th percentile) is valid for the Notes under Table 3, p. 10, and subsequent tables of <a href="https://doi.org/10.3847/2041-8213/ab1141" rel="nofollow noreferrer">Paper VI</a>. However, the interpretation is &quot;median [...] with stated uncertainties corresponding to the 25th and 75th percentiles&quot; in Table 14, p. 44, of <a href="https://doi.org/10.3847/2041-8213/ab0e85" rel="nofollow noreferrer">Paper IV</a>.</p>

CC BY-SA 4.0
2019-04-13T18:56:11.937Z
Related Stats SE questions: [Usage of Plus-minus sign](https://stats.stackexchange.com/q/160474), [Is it correct to use plus or minus symbol before standard deviation?](https://stats.stackexchange.com/q/147469), [Can I specify the Standard Error of the Mean (SEM) with the plus minus sign?](https://stats.stackexchange.com/q/234720), [How should the error-term in reporting results be properly written to make sense?](https://stats.stackexchange.com/q/185655), [When providing plus or minus of a mean value, do I use the standard deviation or the variance?](https://stats.stackexchange.com/q/250242)
ahiijny
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2019-04-13T18:56:25.763Z
^ nothing very helpful in those questions either.
ahiijny
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2021-05-06T20:13:57.040Z
can this simply mean like +/- error where two values of error are possible like + 0.6 or - 0.1 so value lies between 6 x 10^30 and 6.7 x 10^30
Raghavendra Singh
<astrophysics><magnitude><notation>
Superscript-subscript uncertainty notation
pglpm
# Superscript-subscript uncertainty notation

(Asked by: username_0 on 2019-04-13)

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](https://doi.org/10.3847/2041-8213/ab0ec7).

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](https://astronomy.stackexchange.com/q/8841/26647), but it didn't really help me. I also checked the [*Guide to the Expression of Uncertainty in Measurement*](https://www.bipm.org/en/publications/guides/gum.html), 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](https://www.bipm.org/utils/common/documents/jcgm/JCGM_101_2008_E.pdf) definitions 3.12–3.16). But I can't find anything about the superscript-subscript notation there.

Cheers!

**Late addendum:** the interpretation of the accepted answer (median and left & right 16th percentile) is valid for the Notes under Table 3, p. 10, and subsequent tables of [Paper VI](https://doi.org/10.3847/2041-8213/ab1141). However, the interpretation is "median [...] with stated uncertainties corresponding to the 25th and 75th percentiles" in Table 14, p. 44, of [Paper IV](https://doi.org/10.3847/2041-8213/ab0e85).

**Tags:** astrophysics, magnitude, notation
> **Comments:**
> **username_1**: Related Stats SE questions: [Usage of Plus-minus sign](https://stats.stackexchange.com/q/160474), [Is it correct to use plus or minus symbol before standard deviation?](https://stats.stackexchange.com/q/147469), [Can I specify the Standard Error of the Mean (SEM) with the plus minus sign?](https://stats.stackexchange.com/q/234720), [How should the error-term in reporting results be properly written to make sense?](https://stats.stackexchange.com/q/185655), [When providing plus or minus of a mean value, do I use the standard deviation or the variance?](https://stats.stackexchange.com/q/250242)  
> **username_1**: ^ nothing very helpful in those questions either.  
> **username_2**: can this simply mean like +/- error where two values of error are possible like + 0.6 or - 0.1 so value lies between 6 x 10^30 and 6.7 x 10^30  

---

## 1 answers

### Answer by username_3 on 2019-04-13. Score: 7 | Accepted answer

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.

> **Comments:**
> **username_0**: Thank you! Do you happen to have a reference to a manual, or journal guidelines, or similar, about this?  
> **username_0**: There must be a reason why they choose asymmetric increments (skew distribution) and an algorithm to calculate them.  
> **username_3**: @pglpm No, I don't. But in an astrophysics journal, if nothing else was defined, then this is what I would assume.  
> **username_0**: Mmm wait, are you *assuming* that that's their meaning, or do you *know* that that's their meaning? :)  
> **username_3**: @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.  
> **username_0**: 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...  
> **username_4**: @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.  
> **username_0**: @pela I understand that there is no convention. Yet, Rob Jeffries says: "in an astrophysics journal, if nothing else was defined, then this is what I would assume". Well: why? how does he know? Someone must have told him or he must have read it somewhere. What I now presume, is that there must be many astrophysics articles out there where the authors *explicitly* say to be using the 16-50-84% quantiles, so Rob is just summarizing his statistical knowledge. It'd be nice to know if this was the source of his knowledge, or some other experience or text.  
> **username_5**: @pglpm I'm just guessing that the ways one might know without having seen an explicit source can include 1) written enough papers personally to see how journals treat notation, 2) read enough papers where the authors explain their method of determining asymmetric limits to know that journals are flexible as long as it's carefully explained in the manuscript. So "Someone must have told him or he must have read it somewhere." may not be exactly the case.  
> **username_3**: The strongest convention is that the lower and upper bounds define a 68% confidence interval. I would assume this is standard in all physical sciences and would expect to see an explicit statement if otherwise. After that, it really doesn't matter that much what central value is quoted.  
> **username_1**: 1 SD is how I learned it in high school, and it does seem to be the convention. Here's [one source](https://www.webassign.net/question_assets/unccolphysmechl1/measurements/manual.html) that says it directly: "When we report the average value of N measurements, the uncertainty we should associate with this average value is the standard deviation of the mean, often called the standard error (SE)."  
> **username_1**: [NIST](https://physics.nist.gov/cgi-bin/cuu/Info/Constants/definitions.html): "The standard uncertainty u(y) of a measurement result y is the estimated standard deviation of y."  
> **username_0**: @uhoh Indeed, those were my guesses.  
> **username_0**: @ahiijny Thank you. I've added some references to the *Guide to the Expression of Uncertainty in Measurement*.  
> **username_5**: Hi @pglpm if you are providing the information from *e Guide to the Expression of Uncertainty in Measurement* as part of an answer, you should post it as a second answer to your question, which is perfectly fine. Only include it in your question post if it is a necessary part of your question. It certainly looks interesting!  
> **username_0**: @uhoh Thank you. I don't think it provides an answer to my question, because I couldn't find any guidelines about superscript-subscript notation in that guide. But I've added a reference to it in my question because it was my starting point. Cheers!  
> **username_0**: @pela That's very useful, thank you. What do you mean by "central value"?  
> **username_4**: @pglpm I mean the "$x$" in $x_{-\sigma_{-}}^{+\sigma_{+}}$, which may then be the mode, mean, or median, and where $\sigma_{-}$ and $\sigma_{+}$ then are the lower and upper errors, respectively. Note that if you want to add to numbers with asymmetric uncertainties, you can't simply add lower and upper errors in quadrature. That is, $x_{-\sigma_{x,-}}^{+\sigma_{x,+}} + y_{-\sigma_{y,-}}^{+\sigma_{y,+}} \ne (x+y)_{\sqrt{\sigma_{x,-}^2 + \sigma_{y,-}^2}}^{\sqrt{\sigma_{x,+}^2 + \sigma_{y,+}^2}}$  
> **username_3**: @ahijny What your sources say is the rule is irrelevant to this question since there is only one value of the standard deviation of a distribution whereas the question here asks what do (different) upper and lower error bars mean. Upper and lower error bars are quoted in cases where the probability distribution is asymmetric and convey information beyond merely what the standard deviation is.  

---

