Let's say that $x$ is some statistic that represents your measurement of something in nature, such as (say) the sample average of human height in a country.

Suppose that your measurement of $100$ randomly chosen objects (e.g. people) you find out that $x=179.0234234524312123545990088$.

Obviously many of these decimal places are probably highly likely to be due to noise, and will probably change drastically if I re-sample again. One may use eyeballing, and follow his feelings to choose (say) only 4 decimal places, thus writing down $179.0234$.

My question is: Is there any principled approach to decide how many decimal places should be chosen in order write down the number $179.0234234524312123545990088$ for various uses?

For example, should we write $x$ as $179.02342$? $179.0234$? $179.023$? $179.02$? $179.0$? Or maybe just $179$ if the statistic is too unstable to render its fractions meaningful?

So far I have been using my guts/feelings/eyeballing to decide what to display for a final answer. Usually I just choose $4$ decimal places. So usually I would write down $179.0234$.

But the problem is that my approach so far is not principled, and is rather based on guesses/feelings/eyeballing.

I wonder if there is any principled approach that can settle this debate down once and for all? For example, how many significant places should be used for computation and how many for display? There have been prior answers for display of results, for example, Number of significant figures to put in a table? and Why don't we use significant digits? but I do not feel that these addresse the issue of propagation of error during calculations.

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    $\begingroup$ @ josliber There's a difference between ow many decimal places to show in a final answer and how many to carry at intermediate steps. $\endgroup$ Dec 5 '16 at 4:00
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    $\begingroup$ See also: Why don't we use significant digits? $\endgroup$ Dec 5 '16 at 15:50
  • $\begingroup$ Might I suggest to just modify the question to ask something new about the use of significant digits for calculations versus their presentation. $\endgroup$
    – Carl
    Dec 6 '16 at 4:47
  • $\begingroup$ @Carl OK. You can edit the question. I will then approve (if my approval is needed). $\endgroup$
    – caveman
    Dec 6 '16 at 5:09
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    $\begingroup$ @whuber Modified question to include novel material, old questions referenced. I am still waiting for an application of the Caveman Information Criterion ;) $\endgroup$
    – Carl
    Dec 6 '16 at 5:31

When I record data for analysis, I use all of the available data and do not truncate, indeed I zero pad if I do not have enough digits. That is because to do otherwise is 1) detrimental and 2) unnecessary. Our habits concerning significant figures are mostly history from the slide rule era and ignore issues like ties, and the deleterious effect that has on calculations. Moreover, the number of significant figures $N$ drops drastically when certain processing occurs. For example, for regression alone, you should consider yourself lucky to get $\frac{N}{2}$ significant figures out of it. Thus the error propagation of data manipulation can take 50 significant figures and reduce it to 5 without half trying. Moreover, it you started with 5 significant figures your final answer can have -6 significant figures accuracy. Moreover, I check significant figures as I proceed to do calculations, just so that I know what the final precision is.

Moreover, I have another very good reason. I use long numbers as fingerprints that allow me to know if two processes (e.g., think different computer languages) really are giving the same answers and that allows for debugging that might otherwise not even occur. Sufficient warning, if you do not do as I do, you will not see what is going on.

Then, when I am finished, I give a 'human result', the number of significant figures of which represents 1) how accurate and precise the answer is and 2) the need that people think they have for those significant figures. This I usually present with error calculations included. However, the actual results I usually store to 20 places. Just in case I want to do further processing.

  • $\begingroup$ How do you perform the error calculations when writing the numbers for people? I agree that storing all of them is great, but when you wish to present the numbers to boss, I think it's a good idea to show him only the reliable numbers that are not likely to be due to some fluke. $\endgroup$
    – caveman
    Dec 6 '16 at 4:27
  • $\begingroup$ Lots of ways. Here is one example. Current atomic clocks are accurate to 1 part in $10^{13}$. $\endgroup$
    – Carl
    Dec 6 '16 at 4:36
  • $\begingroup$ I see that the $1$ in $10^{13}$ accuracy measure is found by those who design current atomic clocks. However, how to identify the accuracy level of my statistic $x$? $\endgroup$
    – caveman
    Dec 6 '16 at 5:08
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    $\begingroup$ By doing the tedious work of carrying propagation of error along with your primary calculations. For example for ridge regression and what ever you are doing, the errors carry forward. $\endgroup$
    – Carl
    Dec 6 '16 at 5:47

The suitable number of significant figures always depends on the purpose for which the number is to be used and the context. That means that there probably cannot be an entirely principled argument regarding the appropriate number. In general, you can round a number until you start to care about the least significant figure. However, don't round values used in calculations quite that far so that there is no effect of the rounding of interim values on the final value.

Consider the case where there is a steep relationship between your x=179.0... and another variable of interest. In such a case the difference between 179.0 and 179.02 might be important and so you should not round it away. However, if the slope of the relationship is shallow then a distinction between 179 and 180 might be of little importance.


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