The sentry guarding the fort cries out “Alarm, alarm! The indians are attacking us!” The officer asks “How many?” The sentry says “5001” Officer: “WHAT??!!” Sentry: “Well, there’s one in front and about 5000 behind him”.
The concept of uncertainty in calculations involving variable degrees of significant precision is covered in this article, from NIST.
Tris
You are right about this. I was not trying to explicitly and concisely state the rules, but rather to clarify their application in response to a question that implied some familiarity with them. I left what I had hoped would be obvious unsaid: if that was an error in your eyes, then I stand corrected for it.
When all you get as feedback is “No, that’s wrong,” without elaboration, it leaves one guessing as to what the problem is. It sounded like you disagreed with the result - now it seems that the problem was instead with leaving some elements of the reasoning implicit.
An even better method is to explicitly state the error in the measurement, for instance, 3000 ± 0.3 grams. Using sigfigs only allows you to express uncertainties of half of a power of ten of a unit: 3000. is 3000 ± 0.5, and 3000.0 is 3000 ± 0.05, but you can’t specify an error intermediate between the two. Also, there might be times when it’s genuinely reasonable to express your best guess to high precision, even if it is not that precise, expressed as 54321 ± 123 or the equivalent.
Actually, if you follow the rules explicitly, you get the wrong answer. Your implicit assumptions contradict your explicit rules. That, in a nutshell, is the usual problem with determining significant digits–you have to be explicit about the accuracy of your numbers.
I’m not sure why you read my posts as disagreeing with your result. I’d be interested in seeing an explanation of that.
Sorry to have been out of touch for a few days, RM. It’s not really my desire to turn this into a pissing match, although it’s starting to feel like one. 
I think I understand what you’re saying. It seems to me now that the phrase I used in my first response, “count places from the decimal point,” is the sloppy choice of words that’s causing the problem. I knew what I meant, but said it very, very poorly. What I believe we’re both looking for is something along the lines of “The final answer must be rounded to the same precision as the least precise number being added.”
And, as has been discussed, when there are trailing zeros to the left of the decimal point, some convention needs to be adopted to show just how precise these numbers are. Without that, you’re just guessing.
I apologize if I misinterpreted anything you said. I think I formed the impression that you disagreed with something I said when you said “My point is that your rule and smeghead’s rule are wrong.” I hadn’t really formed in my own mind exactly what “my rule” was, hence my confusion.
It sounds to me like we’re in agreement, even if I have trouble articulating my thoughts clearly.
Myself, I could have used a few more words in my posts. I think we are in agreement. See you around brad_d.
Brad clearly understands the concept and reasoning behind Signifigant Figures, but Chronos is closest to the truth: counting SigFig’s is a poor substitute for real error analysis.
-Luckie
Hmmm, it occures to me that a concrete example might be useful to illustrate the inadequacy of the “Signifigant Figures” approach to error analysis.
Supose you have measured 10 weights that have a measured mass of 1,2,3,4,5,6,7,8,9,&10 grams. Each measurement is accurate to the nearest gram. The total mass of these to weights is 55 grams. using “SigFigs” one might think that this total weight is accurate to the nearest gram, but that would be incorrect.
Someone who knows a little more would probably say the uncertainty is ±0.5 for each measurement so the total uncertainty is sqrt(10*(0.5^2))=1.581, but this is still not quite right.
“Accurate to the nearest gram” would imply a uniform distribution. A uniform distribution between -0.5 and + 0.5 has a standard deviation of 1/(2sqrt(3))=0.289, so the total uncertainty is sqrt(10)/(2sqrt(3))=0.913 (Using the Central Limit Theorum to assert that the error distribution will approach a Gaussian). This is a bit larger than what SigFigs would lead you to believe (i.e. that last digit isn’t necessarily correct, there is an ~1.4% chance that the real total is 57 grams or higher).
Just trying to provide some insight into real error analysis.
-Luckie
I’ve glanced at a few of the above posts, and find a majority of them quite interesting but inherently flawed.
I work at a university-run R&D institute. I have been very active in calibration, metrology, and traceability issues for a number of years. WE SELL DATA FOR A LIVING, so I feel somewhat qualified to comment on the question at-hand.
The reason everyone is having so much trouble determining “the number of significant digits to display in the final answer” is because the original problem is missing some VERY crucial information. The absence of this information renders any calculation to be virtually meaningless, and any attempt to calculate the final “number of significant digits” is ultimately a futile endeavor. In the real world, THE NUMBER OF SIGNIFICANT DIGITS IN THE FINAL ANSWER CANNOT BE DETERMINED STRICTLY BASED ON THE NUMBER OF SIGNIFICANT DIGITS IN EACH VARIABLE. If you TRIED to do it you would be taken out back and beaten.
So what information is missing?
Answer: Measurement uncertainty. EVERY measurement has two components: 1) the measured value, 2) the measurement uncertainty. The latter is COMPLETELY missing from the original problem, and therefore it is impossible to compute ANYTHING of value, including “significant digits.”
So let’s get specific: The original problem said:
“Water with a mass of 35.4g is added to an empty flask with a mass of 87.432g. The mass of the flask with the water is 146.72g after a rubber stopper is added. Express the mass of the stopper to the correct number of sig. fig.”
First of all, with the exception of high school math teachers, NO ONE in the real world tries to directly calculate “the number of significant digits” strictly based on the number of significant digits of other variables, because it would be WRONG. (Again: We sell data for a living, so you’ve got to believe me on this.) While you certainly need a minimal amount of resolution in your final answer, the value of the final measurement uncertainty is orders of magnitude more important.
This is what would happen in REAL LIFE if you were trying to determine the mass of the stopper (and subsequently sell the data to a paying customer):
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You would measure the mass of the water with a NIST-traceable balance. You would record ALL displayed digits.
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You would determine the measurement uncertainty of the water’s mass. This can be found on the balance’s calibration certificate.
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You would measure the mass of the empty flask with a NIST-traceable balance. You would record ALL displayed digits.
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You would determine the measurement uncertainty of the flask’s mass. This can be found on the balance’s calibration certificate.
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You would measure the mass of the water + flask + stopper. You would record ALL displayed digits.
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You would determine the measurement uncertainty of the mass of the water + flask + stopper. This can be found on the balance’s calibration certificate.
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You would compute the mass of the stopper using simple addition and subtraction. For the time being, you would carry out a reasonable number of significant digits (be conservative; carry out more than you think may be necessary).
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VERY, VERY, VERY IMPORTANT: You would use an appropriate, statistically-derived, and industry-accepted mathematical function to compute the uncertainty of the stopper’s mass value determined in the previous step. The mathematical function will have as inputs the uncertainties determined in steps 2, 4, and 6, along with a confidence factor.
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Finally, THE CALCULATED UNCERTAINTY OF THE STOPPER’S MASS VALUE WILL DICTATE THE MINIMUM NUMBER OF SIGNIFICANT DIGITS TO DISPLAY.
For example, if the uncertainty calculated in step 8 was +/- 0.1 grams, it would be perfectly acceptable to tell your customer the mass of the stopper is 23.888 +/- 0.1 grams. It would also be acceptable to tell your customer it is 23.89 +/- 0.1 grams, or possibly 23.9 +/- 0.1 grams. But it would NOT be acceptable to tell your customer it is 24 +/- 0.1 grams.
Another example: if the uncertainty calculated in step 8 was +/- 0.05 grams, it would be perfectly acceptable to tell your customer the mass of the stopper is 23.888 +/- 0.05 grams. It would also be acceptable to tell your customer it is 23.89 +/- 0.05. But it probably would NOT be acceptable to tell your customer it is 23.9 +/- 0.05 grams.
Again, THE NUMBER OF SIGNIFICANT DIGITS IN THE FINAL ANSWER CANNOT BE DETERMINED STRICTLY BASED ON THE NUMBER OF SIGNIFICANT DIGITS IN EACH VARIABLE.
I didn’t watch this thread for a while, but I’ll just make a quick comment to RM Mentock. It appears everything’s smoothed out, but I’ll mention that I would never have posted what I said without checking first. Smeghead’s set of rules clearly does state ‘least precise’. Brad_d’s didn’t (as has been further elaborated by him). Your question was put to ‘you guys’, and I lumped the two together since the method was mostly the same; that was my error. What I should have pointed out, as I think Garfield did, was that Smeghead provided the complete answer.
Crafter_Man, I seem to have misplaced my NIST-traceable balance. could I borrow yours for a few days?
I think you should rephrase “NO ONE in the real world tries to directly calculate ‘the number of significant digits’ strictly based on …” with “No one who’s livelihood depends on it tries to …”.
There’s a lot of situations where incompletely-specified measurements simply have to be dealt with. The usual method of significant figures, while inherently flawed, is a workable approximation for such situations.
This does not mean that correct error analysis should not be done; it just means that erring when analyzing errors is not an erroneous thing to do; one should always keep in mind the limitations of one’s method.
That way everyone can come to see the error of one’s weighs.
I’d have to disagree with you there, PJ. (And what does “incompletely-specified measurements” mean?)
If you’re doing ANY kind of physical measurement for a commercial, industrial, or government customer you MUST use a traceable instrument. If it’s not traceable the measurement is utterly meaningless and you’re ripping the customer off. In fact, I think anyone who knowingly takes an untraceable physical measurement and sells the data to a customer, or uses the data for a report to be given to a customer, etc., that person should be charged with fraud.
There really is no excuse for not calibrating your instruments. Virtually any instrument can be calibrated, and most can be made traceable to NIST if certain guidelines are followed. And once it is calibrated the total measurement uncertainty can be calculated or estimated with minimal effort. (The measurement uncertainty can then be used to determine the number of significant digits to be published.)
I can’t say you said anything I disagree with in principle, Crafter. BUT, there are uses for sig figs. The one I keep harping on is chemistry. It’s not a matter of cheating your customers, it’s just a matter of keeping track of how accurate your measurements are. There are times when you need to know exactly how imprecise you’re being in chemistry, but there are plenty of times when sig figs are plenty close enough.
Crafter, by ‘incompletely-specified’ I meant measurements that don’t have all the proper qualifiers, those you have no idea where they came from but have to work with anyway. The OP’s question seems a more typical situation (23.5 g, 140.62 ml ).
I agree with you completely that if you are being relied upon for correct data, you must produce it correctly. However, it can be quite common in situations where the measurement was not considered critical for people to forego the precise error analysis. Such measurements are not utterly meaningless, as you claim. They simply lack precision, which is a pretty far cry from losing all of their meaning.
panama jack
“I would rather have a vague but correct answer than one which was precisely detailed yet failed to understand the problem” - from a preface to a book which I can’t recall at the moment.
Look, here’s my point: Math teachers and some professors love to talk about sig. digits, and how many digits should be displayed in the final answer based on the # of sig. digits in intermediate measurements, blaa blaa blaa. They grill these concepts into students’ heads. Not surprisingly, these students will try to determine measurement uncertainty strictly using the “sig. digits” approach when they work in the real world. And the result of this? The space shuttle blows up.
So I’m hear to tell ya: Your math teachers were DEAD WRONG. Your professors were DEAD WRONG. Do yourself a favor and purge this crap from your brain. Using the “significant digits” technique all by itself, you cannot even ESTIMATE how many significant digits the final answer can have, let alone figure out what the measurement uncertainty is; your answers will be wildly off. In fact, if you cannot calculate the measurement uncertainty, for whatever reason, it would be better to allow the final answer to simply have an “abundance” of sig. digits than try to “calculate how many there should be” based on the number of sig. digits in your intermediate measurements.
I think math teachers and professors do a grave disservice to students by perpetuating this myth. But is anyone blowing the whistle? Nope. Perhaps a few more blown-up space shuttles will convince them. (And I’m being serious: the o-rings failed due to a measurement miscalculation.)
I’m sorry if I seem so overboard on this. But I run into this all the time here at work. I plan on drinking some good microbrew tonight; maybe that will help me calm down.