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Is there a well founded rule for the number of significant figures to publish?

Here are some specific examples / questions:

  • Is there any way to relate the number of significant figures to the coefficient of variation? For example, if the estimate is 12.3 and the CV is 50%, does that mean that the information represented by '.3' approaches zero?

  • If a confidence interval has a range of orders of magnitude, should they still have the same number of significant figures, e.g.:

    12.3 (1.2, 123.4) vs 12 (1.2, 120)

  • Should the number of significant figures in an error estimate be the same or less than the number of significant figures in a mean?

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  • $\begingroup$ If you can, don't use a table :) A graphic is, IMO, almost always easier to read than a table (he obvious exception being if you don't have many numbers). Journals and their reviewers don't always agree, unfortunately.... $\endgroup$ – JMS Mar 24 '11 at 20:26
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    $\begingroup$ @JMS Good point, but Tables are useful for summarizing detailed characteristics of statistical units (cross-classified by a factor interest, e.g. clinical diagnosis or whatever), with variables of different types (continuous, nominal, and ordinal), and other results derived from statistical modeling per se (confusion matrix, regression coef. etc.) that won't fit into Figures (or not always if you think of Gelman's approach for showing reg. coef. as dotcharts). We need both; the question is when do we really need a Figure instead of a Table, IMO. $\endgroup$ – chl Mar 24 '11 at 22:37
  • $\begingroup$ @chi Fair. I did say almost always :). Things like big n-way tables are impossible to (completely) reproduce graphically. It depends on the forum I'd say. Tables have the benefit of being complete, sure, but does your reader actually absorb all that extra information? If there are too many parameters to fit in a graph, I'd contend that a table is often at least difficult to read. However, I do think complete results should be accessible (online, appendix, etc) if for nothing else but reproducibility. In that case I'd also like data & code though! Wandered OT, sorry.. $\endgroup$ – JMS Mar 25 '11 at 3:57
  • $\begingroup$ Also I think regression coefficients and confusion (correlation, covariance, ...) matrices are usually better suited to a graphical display, dotplots or similar for the former and heatmaps or graphs for the latter. $\endgroup$ – JMS Mar 25 '11 at 4:00
  • $\begingroup$ @JMS I agree with your point, but in this case there is a figure limit, some other cases there are figure charges. Also, in this case if readers glance over the table and focus on the figures that are presented, then they won't waste time trying to figure out the point of an esoteric figure. But I fully support reproducibility, and while I am at it, I could (if I get around to it) add a visualization of the table to the code that is attached. $\endgroup$ – David LeBauer Mar 25 '11 at 5:40
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I doubt there's a universal rule so I'm not going to make any up. I can share these thoughts and the reasons behind them:

  • When summaries reflect the data themselves--max, min, order statistics, etc.--use the same number of significant figures used to record the data in the first place. This provides a consistent representation throughout the document concerning the precision of the data.

  • When summaries have higher precision than the data, write the values in a way that reflects that extra precision. For instance, a mean of $n$ values has $\sqrt{n}$ times the precision of the individual values: roughly, include one extra significant figure for $3 \le n \le 30$, two for $30 \lt n \le 300$, etc. (This is rounding on a log-10 scale, obviously.)

    -Note that the CV does not provide useful information in this regard.

    -Some estimates can be obtained with great precision. They don't have to be rounded to match something else. For instance, the mean of 1,000,000 integers might be 10.977 with a standard error of 0.00301. My decision to write the mean to three decimal places (and 4-5 sig figs) was based on the order of magnitude of the SE, which indicates the last digit is partially reliable. The decision to write the SE to three sig figs (five decimal places) is more arbitrary: two sig figs would work; one probably would not; four sig figs would also work and be consistent with the 4-5 sig figs in the mean; more than four sig figs would be overkill. (One could estimate the standard error of the SE itself in terms of the fourth moment of the data, and use that to determine an appropriate amount of rounding, but most of us don't go to such trouble...)

  • Signal the reader when you are doing substantial rounding. Be especially careful when the report is discussing the statistical test itself. The reason is that people may use your work to check their own calculations. Sometimes even a slight difference can reveal an error. You don't want to cause trouble because you rounded 123 to 120 and someone else, checking the work, obtains 123 and suspects one of you has erred.

  • Be consistent. You might lose some readers if you list a value as 123 at one point and later reference it as 120.

  • Don't be ridiculous. (I automatically suspect incompetence when I encounter reports that give statistical results to 15 sig figs when the data have only two sig figs, for instance.)

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    $\begingroup$ My very big +1 because it is really a lot of good advices. In the same vein, I like to show to students that it is really pointless to summarize data gathered from surveys (or votes) as % with a lot of decimals without considering sample size (which impacts standard error). $\endgroup$ – chl Mar 24 '11 at 22:46
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I'd suggest 12 (1.2, 123.4). Omit the .3 since it's nearly meaningless, but many people when they see (1.2, 120) will assume that the last '0' in 120 is significant.

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  • $\begingroup$ Why do you suggest to omit a decimal for the statistic of interest if you agree to show them in the CIs (i.e., if it's meaningless for 12, why does it make sense for 123.4)? $\endgroup$ – chl Mar 24 '11 at 22:50
  • $\begingroup$ @chl: it doesn't make much sense, but omitting it might be misleading. If I put in 123.4, someone like you will see the extra digits and just disregard them, no harm done. If I put in 120, many readers will think this is accurate to 3 digits - bad. $\endgroup$ – AVB Mar 24 '11 at 23:59
  • $\begingroup$ still not clear why you recommend 123.4 instead of 123 (why omit .3 but not .4 in the example?) $\endgroup$ – David LeBauer Mar 26 '11 at 23:24

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