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I am currently having a disagreement with a coauthor on a paper over the matter of significant figures.

He would like to specify (for example) that a certain organization has an environmental footprint of 7,622 gha. Environmental footprinting rests on a tonne of assumptions and is not a precise methodology, so I don't think we have anything like 4 significant figures of precision in our calculations - more like 2 at most. So I would like to publish that figure as 7,600 gha.

I have pointed out that doing otherwise implies a level of precision we don't have. Currently we are publishing an executive summary prior to peer reviewed journal paper, so I also said it would be embarrassing if the precision issues were picked up at peer review stage and we thus ended up publishing two different sets of figures. He responds that he has never had a problem with publishing more accurate figures in previous journal publications.

The coauthor does not have a quantitative background so has limited understanding this issue, however he is more senior.

So two questions

  • does this issue even matter?
  • what would you do about it?

UPDATE

It looks like you all agree with me that this matters. Good, I'm not insane :)

As we are currently writing an executive summary/press release for the general public, there is no place for scientific notation or estimates of error - they will make it harder to read and likely put off some people.

Can someone link me to a good, polite and authoritative rant on why this matters that I can show to my colleague? The trouble with almost all material I have seen on significant figures, is that it doesn't discuss rounding off digits left of the decimal point. i.e. it talks about turning 7.36654 into 7.4 but not 736654 into 740000. I suspect my colleague draws an arbitrary line in his head at the decimal point, being unaware that the position of the point is just a function of the units used. And I doubt they would be open to that sort of argument.

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Did your coauthor give a reason why he does not want to use the 7600 figure? Why does he even care? –  ff524 Jun 9 at 13:43
    
No, and my next step might be to ask him that question, but I want to stop and think first! –  Sideshow Bob Jun 9 at 14:22
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It may be worth showing this question to your colleague (send him the link or view it side-by-side) as evidence that you are not crazy in this endeavor. While this question is not a primary source, it shows that at least 13 people in the first 2 hours since posting (who have 15 or more rep on this site) have identified that this issue, which specifically pertains to your current work, is worth discussing. The highest voted answer also supports your argument. –  NauticalMile Jun 9 at 15:48
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If you want authoritative references, look for journal style manuals. This issue isn't mentioned that often because, at least in technical disciplines, it is tacitly understood. Some examples, however, are the AMA style guide and the J. Food Comp. Anal. guide for authors. –  episanty Jun 9 at 16:47
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To me, 7,600 gha is no less precise than 7,622 gha. How about 7,6 kgha? –  Eckhard Jun 9 at 22:55
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5 Answers 5

up vote 9 down vote accepted

Just to add to the answers with some academic citations, style guides for Tables often list this point. Although they often give examples of extraneous fractional digits (as these tend to be the most egregious examples of where they are illogical), the logic behind the rounding extends wherever the significant digits are location.

For instance, Feinberg & Wainer (2011) suggest that at maximum only 3 digits should be displayed in the table, as humans have a hard time making comparisons among any more digits (and they strongly suggest two or less). So here is advice extending beyond just decimals. (In Table 3 they give an example of rounding to the billions.)

They also appeal to the trivial amount of error introduced by such rounding. In your example the error would be 22/7622 ~ 0.003 - I think 0.3 percent is alright for a wide variety of circumstances.

To justify reporting all 4 digits from a statistical perspective the standard error of the estimate would need to be less than 0.5. If the error is more than 50 you really shouldn't report any more than the hundreds.

I provide more citations in my blog post for the stats site; Some notes on making effective tables, although not all are applicable to this question.

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Just because you have the additional digits, doesn't mean you should report them.

Accuracy and precision are extremely important concepts in the reporting of scientific data. Excessive precision, as you suggest, implies too much reliability in the accuracy of scientific data. For instance, someone turned in a report to me saying that a particular parameter was known to be:

E = 1.3405987423423 GPa

If this were to be published in a paper, it could be rejected out of hand for incompetence, because the parameter E is often known only to one or two decimal places, let alone four.

In general, you need to make your reporting consistent with the assumptions and the accuracy of the measurement techniques. One way to figure this out is to note the accuracy of your measurement—what is the expected error in the measurement? If the error in your measurement technique is, for instance, plus or minus 1 global hectare, then by all means report it to the nearest hectare. If it's plus or minus 1000, though, reporting to the nearest hectare will look quite silly.

How do you deal with this? (I've actually had to do something like this before. Similar argument, but not exactly the same.) In this particular instance, engaging in a little hyperbole might be useful. Instead of reporting it to four significant digits, show him a version of the text that lists the data to the full precision you have available—something clearly ridiculous that reports things down to six or seven decimal places, if possible. Then, when he argues that what you're doing doesn't make sense, you can argue "what's the harm in reporting extra digits?" That should be enough to have your coauthor "see the light."

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Alas I don't think that tactic would work :/ –  Sideshow Bob Jun 9 at 14:48
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Protip: "The bigfloat package is a Python wrapper for the MPFR library for arbitrary precision floating-point reliable arithmetic." :-) –  mmh Jun 9 at 16:49
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Ideally you should give an indication of the uncertainty to go with your value e.g. 7600 ± 100 (or whatever) even if the uncertainty is only a rough estimate.

The problem is that your proposed solution doesn't really give any indication of your accuracy is 7600 the true value, have you rounded to 2s.f or 3? On the other hand the other solution suggests too great an accuracy which is also wrong.

If you do state an uncertainty the actual value you put matters slightly less. Ideally you should still give the answer to similar levels as that in your uncertainty, as you suggested, but my putting more your not actually creating any ambiguity.

As for whether it matters I suspect in lots of fields people don't really care about this sort of thing. But it does matter so they should care and so should you.

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hmm, anyone know why my latex doesn't work here? I think its correct. –  nivag Jun 9 at 14:02
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Not all SE sites support the same level of markup. We generally don't need LaTeX type markup, so we don't have it. You can test things out in our sandbox. –  StrongBad Jun 9 at 14:11
    
Unfortunately stating uncertainties isn't appropriate for the audience of the document we're writing at the moment (see my update). –  Sideshow Bob Jun 9 at 14:35
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@SideshowBob: Would your audience and data permit to make a general statement about the accuracy, e.g., writing at an appropriate place: “The relative accuracy of environmental footprints reported here varies between 0.1 % and 2 %?” (Which would then be an additional argument against your coauthors, as you would now be reporting digits which are not reliable according to yourself.) –  Wrzlprmft Jun 9 at 14:40
    
Alas no, we're really removing all unnecessary figures from the press release - I doubt that would pass editing (and would take quite a while to calculate in any case). –  Sideshow Bob Jun 9 at 14:50
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This will be an addition to aeismail's answer with which I agree. When reporting numbers like 7600 the trailing zeros are not significant but there is no way to know it. Therefore, writing the number as 7.6 kgha or 7.6 x 10^3 gha directly points to the significant numbers. A key point of publishing information is to provide the reader with an accurate picture for the results and providing quantitative information in its proper context is key. Using appropriate notation to present numbers is therefore important so rather than removing digits that are not deemed significant, it is better to present the data in a different notation by either using established SI-prefixes or scientific notation.

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See my update. Scientific notation is not appropriate for the audience of this document unfortunately. –  Sideshow Bob Jun 9 at 14:34
    
@SideshowBob Does gha stand for giga-hectare? You could write 7.6 pha? –  gerrit Jun 9 at 14:54
    
no, global hectare. –  Sideshow Bob Jun 9 at 14:58
    
I would avoid scientific notation with this particular example. It ends up adding more digits, e.g. 4 digits in 7622 compared to 5 (numeric) digits in 7.6 x 10^3. This will likely make the table harder to read, especially as 7.6 x 10^3 and 7.6 x 10^4 will not stand out very much in right aligned data like writing out the entire estimates would. –  Andy W Jun 9 at 16:47
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If you're specifically interested in the case of writing for a general audience, I think this problem can be usefully resolved by thinking about your choice of units. I have very little sense of what a global hectare actually is, or how to visualize it. So can you express these quantities in terms of units/objects that might be more accessible to your readers? See this and this for a discussion. Presumably, once you've moved to non-scientific units the arguments for precision fall away...

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"I have very little sense of what a global hectare actually is". Good point. I hadn't heard of a "global hectare" until now, I read the wikipedia link to your answer...and it turns out to be the most complicated and confusing "unit of measurement" I've ever seen. After reading this, part of what the OP seems to be saying is that it is dubious to measure anything in global hectares to four significant figures. For a press release, maybe it is best not to use "gha" at all: if STEM-my academics don't know what that is, I wonder how many people will? –  Pete L. Clark Jun 9 at 18:41
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I regret the scare quotes around unit of measurement in my last comment. I don't want to come off as dismissive of this particular area of science. But as a STEM-ist, I have a couple of objections (i): the "dimensions" of a global hectare are (eco-biological productivity)/(hectare). I have never seen a unit named for the dimensional quantity in its denominator: that seems almost calculated to confuse.... –  Pete L. Clark Jun 9 at 20:27
    
And (ii) more precisely a global hectare is defined to be the earth's total eco-biological productivity divided by the number of "productive" hectares. Okay: great idea to try to quantify this, however roughly. But as a "unit": to how many significant figures can we really calculate the numerator and the denominator? If two researchers independently calculated these quantities, to how many significant figures would their answers agree? I would find a reliable computation to three significant figures extremely impressive. –  Pete L. Clark Jun 9 at 20:29
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@Sideshow Bob: I'm not sure that I agree that the dimension of "global hectare" is "area": that would seem to make a "global hecatare" the same as a "hectare", which is not the desired intent. But given that didn't know what a gha was yesterday, I am more than willing to believe that I am missing some subtlety. It's no big deal, but I'm curious: could you point me to some academic source for the definition of gha? –  Pete L. Clark Jun 9 at 23:03
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@Max: Hmm. A global hectare was the most confusing unit of measurement I had ever encountered....until I learned about disability adjusted life years. At first glance -- so please forgive my lack of expertise -- I am not sure that a daly is what I would call a "unit of measure" at all. The wikipedia page (not the ultimate authority, but usually pretty solid...) calls it a "metric", which is not necessarily exactly the same. I had never met a unit before with definition of the form "X+Y". I wonder if there is any SE site where I might ask fundamental questions about dimensional analysis? –  Pete L. Clark Jun 10 at 15:46
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