[chanelo]

When the number on the scale shifts between, say, 8.9955 and 8.9956 without settling on one or the other, which do I use?

[Borek]

Whichever you want. 10^-5 error is not the serious problem in typical analytical applications.

[curiouscat]

8.995 ?

[Borek]

Or 8.996, depending on the rounding method. But that would mean losing accuracy, I don't see a reason to ignore the last digit.

[Arkcon]

This is exactly where significant figures comes in, taught to students early in their career, then forgotten, basically because Wikipedia say so.  You report 8.995 or 8.996 as appropriate for your rounding rules.  You haven't surrendered accuracy, you're admitting that the instrument, although designed for more accuracy, isn't giving you that accuracy -- either that day, or because it needs maintenance, or the sample is volatile, or there are heating drafts, or static or whatever.  Or, you know, poor user technique, that can also be the case.  Still, you don't always get the maximum manufacturer touted accuracy from any instrument.

[curiouscat]

This is exactly where significant figures comes in, taught to students early in their career, then forgotten, basically because Wikipedia say so.  You report 8.995 or 8.996 as appropriate for your rounding rules.  You haven't surrendered accuracy, you're admitting that the instrument, although designed for more accuracy, isn't giving you that accuracy -- either that day, or because it needs maintenance, or the sample is volatile, or there are heating drafts, or static or whatever.  Or, you know, poor user technique, that can also be the case.  Still, you don't always get the maximum manufacturer touted accuracy from any instrument.

Exactly this. If your reading bounces between 8.995 and 8.996 8.9955 and 8.9956 it is in principle wrong  (in practice, probably irrelevant) to report either of those results because the fact is that you just don't know.

Reporting that last fluctuating digit gives a false impression of accuracy that you don't really have.

[Borek]

You've lost me. 8.9955 - even with the last digit unsure - is much more accurate than 8.995. I don't see why you suggest to drop the last digit completely.

[curiouscat]

You've lost me. 8.9955 - even with the last digit unsure - is much more accurate than 8.995. I don't see why you suggest to drop the last digit completely.

Maybe I'm wrong; but the way I always thought about this is if you write 8.995 (or 8.996 ; depends on your rounding convention) you imply you don't know what the last digit is; could be 8.9951, 8.9952 etc.

Writing 8.9955 implies you have confidence that the measurement is indeed 8.9955 (and not 8.9951 nor 8.9959 etc.)

Note that I'm reccomending reporting 8.995  and not 8.9950.

I admit I'm not a 100%  sure.

[Arkcon]

Because you're lying?  Consider this, I've determined, using the analytical balance I have here, for the same sample (the O.P. sent it to me via TCP/IP) the actual mass is 8.995572 g.  Aren't I awesome?

No, I'm not.  You know I'm lying, you know the outcome described is false.  There is, IMHO, no distinction between reporting accuracy you're not sure of, reporting accuracy that isn't possible, or even reporting a measurement you didn't make.

One time, the maintenance crew came into the lab, there was something wrong with some part of the facility, and they wanted to compare their handheld pH meter or check it, or something.  Its hard to know what they wanted ...

Anyway, we compared their grimy handheld pH meter, with our validated, weekly maintained pH probe.  I carefully preformed a two-point flanking calibration with NIST-traceable buffers, cracking open a fresh bottle to be extra sure.  Our pH meter gave a different result than his, and he blew up at us -- his was obviously better -- it cost $80 and gave the answer to 3 decimal places, and ours stopped at 2.  He just arbitrarily decided that more decimal places = better.  And that's not true.

[curiouscat]

Reminds me of this joke:


"A museum guide says a dinosaur skeleton is 100,000,030 years old, because a staff-scientist told him it was 100 million years old when the guide had started working there 30 years ago"

[Borek]

There is nothing strange with a good analytical balance reporting results with four decimal points and being accurate, that's what their specification usually says (either 0.0001g or 0.1mg - which is the same). I would not go further than ±0.1 mg, but correctly maintained balance should keep that accuracy. Hence I would read 8.9955 g as something like 8.9955(1). I would not care much about whether the last digit is 5 or 6, but I would not throw it away either.

If you are ready to arbitrarily dismiss the result why stop at 8.995 and not say it is just 9 g (with one significant digit)?

Maybe I'm wrong; but the way I always thought about this is if you write 8.995 (or 8.996 ; depends on your rounding convention) you imply you don't know what the last digit is; could be 8.9951, 8.9952 etc.

Writing 8.9955 implies you have confidence that the measurement is indeed 8.9955 (and not 8.9951 nor 8.9959 etc.)

Actually it doesn't matter what the last digit is, but what is the interval in which you expect the correct result. You can write 8.9955±0.0001 (which reflects expected accuracy of the balance) - doesn't mean you know the last digit is 5, but it definitely means you know the result is not 8.9951 (although it can be 8.9954).

Most likely the real expected accuracy of the balance is not 0.0001 g, but something looking much more random - say 0.000678 g - which is rounded down to 0.0001 g, so the ±0.0001 g is just an artifact of the decimal number system we use.

[curiouscat]

Most likely the real expected accuracy of the balance is not 0.0001 g, but something looking much more random - say 0.000678 g - which is rounded down to 0.0001 g, so the ±0.0001 g is just an artifact of the decimal number system we use.

I think that'd be a very irresponsible thing to do if you were the balance manufacturer. A balance with a "real expected accuracy " of 0.000678 g ought to be advertized as accurate to ±0.001 g and never as ±0.0001 g

Actually it doesn't matter what the last digit is, but what is the interval in which you expect the correct result. You can write 8.9955±0.0001 (which reflects expected accuracy of the balance) - doesn't mean you know the last digit is 5, but it definitely means you know the result is not 8.9951 (although it can be 8.9954).

To me the ± notation is the only way to do this rigorously.

In the absence of that the implicit understanding I assume is that 8.9955 means 8.9955±0.00005

I prefer 8.995 because I think it conveys 8.995±0.0005

If you are ready to arbitrarily dismiss the result why stop at 8.995 and not say it is just 9 g (with one significant digit)?

Because that'd likely be taken to mean 9 g ±0.5

[Borek]

I think that'd be a very irresponsible thing to do if you were the balance manufacturer. A balance with a "real expected accuracy " of 0.000678 g ought to be advertized as accurate to ±0.001 g and never as ±0.0001 g

Sorry about it, my mistake - I ate a zero. What I meant was 0.0000678 g reported as 0.0001 g.

To me the ± notation is the only way to do this rigorously.

In the absence of that the implicit understanding I assume is that 8.9955 means 8.9955±0.00005

I prefer 8.995 because I think it conveys 8.995±0.0005

You are making error margins much wider than necessary this way, and for no reason. To some extent I agree with you that ± notation is the only way to do it correctly - but I feel like the way you suggest to use it is wrong. As it was already said, ± 0.0001 is just a simplification. Real values look differently - like 8.3144621(75) or 8.3144621±0.0000075. In this context notion of "accurate digit" makes no sense at all.

If you are ready to arbitrarily dismiss the result why stop at 8.995 and not say it is just 9 g (with one significant digit)?

Because that'd likely be taken to mean 9 g ±0.5

Yes, that would mean it - but you already told us that you think 8.9955±0.0001 means 8.995±0.0005. If you have no problems with widening your error margin 5 times for no apparent reason, why is it a problem to widen it another 1000 times?