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Final zeros to the right of the decimal point are considered significant. What do those zeros indicate and why are they significant? For example, in 2.000 there are four significant figures.

Melanie Shebel
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AksaK
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    They are treated just the same as any other numbers; the uncertainty is in the next place and assumed to be $\pm 0.0005 $. – porphyrin May 15 '17 at 07:32
  • @porphyrin But unlike leading zeros to the right of decimal and final zero after a digit (1-9), they are considered as significant. I want to know if these zeros are placeholders then why zeros in .0001 and 10 are not considered significant? – AksaK May 15 '17 at 07:59
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    $10, 1000, 12340$ etc. are integers so are absolute numbers. $0.001$ can be written as $10^{-3}$ or $1/1000$ so the zeros are necessary to indicate the magnitude of the number. – porphyrin May 15 '17 at 08:04
  • @porphyrin Then what does zeros in 2.000 indicate.I am confused – AksaK May 15 '17 at 08:11
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    it means that this is a real number not an integer and that its values lies somewhere between $1.9995$ and $2.0005$ but that we don't know exactly where between these two numbers and as a result of our ignorance we write the best guess which is $2.000$ . – porphyrin May 15 '17 at 08:17
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    Try not to answer a question in the comment session guys. – Jeppe Nielsen May 15 '17 at 08:26
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    Related question on Mathematics.SE. – Apoorv Potnis May 15 '17 at 11:25
  • I chose $\pm 0.0005$ but having checked in some textbook it seems that $\pm 0.001$ would be better, i.e. twice the value I gave. The example given below by @DavePhD deals with a different case where the error is already known and some rounding is then done in the standard way. – porphyrin May 15 '17 at 14:17

3 Answers3

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"$2.000$" does not mean $2.000 \pm 0.0005$.

"$2.000$" does not mean the interval [1.9995,2.0005].

"2.000" means that there is an unspecified amount of uncertainty in at least the last, and possibly the last two digits.

See the NIST Good Laboratory Practice for Rounding Expanded Uncertainties and Calibration Values:

Example 5

The correction for a weight is computed to be 285.41 mg and the uncertainty is 33.4875 mg. First, round the uncertainty to two significant figures, that is 33 mg. Then, round the correction to the same number of decimal places as the uncertainty statement, that is, 285 mg

So in other words, for people who follow the NIST standard of stating uncertainty to two significant digits:

2.000 can mean anything from 1.990-2.010 to 1.901-2.099

Others only express uncertainty using one significant digit, in which case:

2.000 can mean anything from 1.999-2.001 to 1.991-2.009

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DavePhD
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  • This might be the irrelevant question but I have to ask. In above example 5 you rounded off the number 33.4875 to two significant figures as 33 but my teacher taught me that if you want round off a number to some desired sig fig you must do it in different steps like 33.4875 = 33.488 = 33.49 = 33.5 = 34. So which is correct one? – AksaK May 15 '17 at 11:38
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    @AksaK I didn't write the example, I'm just quoting from NIST. NIST is correct. Teacher is wrong. "33.4875" is closer to "33" than "34" so you round to "33". – DavePhD May 15 '17 at 11:42
  • Though I didn't understand NIST but I understand rest of your answer. – AksaK May 15 '17 at 11:49
  • @AksaK In physics publications it's more common to see two uncertain digits reported. For high school or undergraduate science, it's more common to see just one. I was told in college to use one digit, unless the digit would be "1", in which case to use two digits. – DavePhD May 15 '17 at 11:54
  • Please note that my answer is dealing with precision of measurement, not uncertainty! The two are different! In terms of precision 2.000 definitely means that the measurement is in the interval [1.9995,2.0005[ as you would otherwise state a different measurement value! The uncertainty of a measurement is only relevant when you do more than one measurement! – Jeppe Nielsen May 15 '17 at 13:30
  • @JeppeNielsen I disagree, because, taking the NIST example, the measurement is 285.41 mg. The measurement interval is 285.405 to 285.415 mg. The reported value is 285 mg. – DavePhD May 15 '17 at 13:37
  • Perhaps you should use international standards instead of national ones then: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3387884/ – Jeppe Nielsen May 15 '17 at 14:07
  • The example the OP has given is different to yours in that no error is given so there has to be some assessment as to what that might be. I chose $0.0005$ but having checked in some textbook sit seems that $0.001$ would be better, i.e. twice the value I gave. – porphyrin May 15 '17 at 14:15
  • And the international method of stating the above measurement would then be 285.41(+- 0.005)mg. – Jeppe Nielsen May 15 '17 at 14:18
  • @JeppeNielsen NIST follows this international standard: http://www.bipm.org/en/publications/guides/gum.html – DavePhD May 15 '17 at 14:27
  • Dave, if you measure something with 5 significant digits, then you either round it to 4 significant digits or report 5 digits including the uncertainty. You do not throw away a perfectly good digit of precision eg. the reported value in the NIST example should be 285.4 mg or 285.41 (+-0.005) for maximum precision and clarity. – Jeppe Nielsen May 15 '17 at 14:54
  • @JeppeNielsen In that example, the value is measured to 5 digits, but not all 5 are "significant". The uncertainty in the value is over 10%. I would throw away at least 2 digits like NIST does, or maybe even 3 digits and say $290 \pm 30$ – DavePhD May 15 '17 at 15:32
  • And that is were our disagreement is. You are concerned about the uncertainty of the mean of the measurements. I am concerned about the precision of each individual measurement. I will naturally agree that each measurement when compared to the mean may be different, but if your measurement device is capable of five significant digits for each individual measurement then you should report that as a measurement. The uncertainty and systematic/random errors should be determined after the measurements have been performed. – Jeppe Nielsen May 17 '17 at 13:46
  • @JeppeNielsen No, what I'm saying is not limited to a mean. "A Type B evaluation of standard uncertainty is usually based on scientific judgment using all the relevant information available, which may include – previous measurement data, – experience with, or general knowledge of, the behavior and property of relevant materials and instruments, – manufacturer’s specifications, – data provided in calibration and other reports, and – uncertainties assigned to reference data taken from handbooks." https://www.nist.gov/sites/default/files/documents/2017/05/09/tn1297s.pdf – DavePhD May 17 '17 at 13:49
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The zeros to the right of the decimal point denotes the expected precision of a measurement.

Thus a value of 2.0 indicates that the the measurement falls in the interval [1.95,2.05[. The value 2.00 correspond to the interval [1.995,2.005[ and 2.000 corrospond to the interval [1.9995,2.0005[.

Therefore the amount of zeros are significant for indicating the precision of a measurement.

Jeppe Nielsen
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I think it's worth noting that the number of 0s is not guaranteed to indicate significance/precision, as it's possible the author did not follow standard conventions. For instance, they might have reported the number of digits that some measurement device displays, which may or may not correlate with the overall measurement precision in the context of the particular experiment. When in doubt, look for a discussion of the context of the measurement and an analysis of possible errors; if that's not present, consider asking the author to confirm whether they did, in fact, mean for the number of digits to reflect the precision.

Dan Bryant
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