Definition of Uncertainity

I have some confusion regarding Measurement Uncertainty. In some books/articles it is defined wrt true value as "Uncertainty in the average of measurements is the range in which true value is most likely to fall , when there is no bias or systematic component of error is involved in measurement" but if we conduct a very large number of measurements with no systematic error then we would be pretty sure that mean of measurements will be equal to true value (or will have a negligible random Error associated with our result) and uncertainty would make no sense as it is defined previously! Like standard deviation (or Uncertainty) in population mean (no bias included) can't be treated as it previously stated because population mean itself would represent true value.

There is another definition (which is based on confidence level, generally 68% and random measurements not with true value) states that "There is roughly a 68% chance that any measurement of a sample taken at random will be within one standard deviation of the mean. Usually the mean is what we wish to know and each individual measurement almost certainly differs from the true value of the mean by some amount. But there is a 68% chance that any single measurement lies with one standard deviation of this true value of the mean. Thus it is reasonable to say that:". It seems to me Correct to some extensions because it would be applicable to infinite measurements also.

Both the definitions are defined with consideration of no systematic component of error involved but if it is involved what would uncertainty in uncorrected result ( not corrected for systematic error ) represent? Well, second definition will still hold if we measure the further measurements with the same instrument or without correction of systematic component but if someone other would measure measuand with no systematic component or with corrected instruments his 2/3 measurements will not fall in that range of uncertainty. Then in that case what would be the appropriate definition of uncertainty?

And here GUM description for uncertainty came up :-

D.5.1 Whereas the exact values of the contributions to the error of a result of a measurement are unknown and unknowable, the uncertainties associated with the random and systematic effects that give rise to the error can be evaluated. But, even if the evaluated uncertainties are small, there is still no guarantee that the error in the measurement result is small; for in the determination of a correction or in the assessment of incomplete knowledge, a systematic effect may have been overlooked because it is unrecognized. Thus the uncertainty of a result of a measurement is not necessarily an indication of the likelihood that the measurement result is near the value of the measurand; it is simply an estimate of the likelihood of nearness to the best value that is consistent with presently available knowledge.

Above description is like adding an additional point 'systematic error' to first definition which is based on true value and the same confusion arises here that what would uncertainty in population mean (mean of a large number of measurements taken) with no systematic error represent?

Now I ends up here with my point of view on previously stated definitions. And want to ask what would be the correct definition of measurement uncertainty which would be applicable to any case whereas 'systemic error is involved or not' or 'Result belongs a very small or a very large sample of measurements'?

• Please be careful, because there is no such thing as an infinity of measurements. Trying to reason about such infinities frequently leads to confusion, falsehoods, and paradoxes.
– whuber
Commented Sep 8, 2023 at 18:47
• @whuber I meant there a large number of measurements and I know infinite times is what not possible to measure that's why I have wrote there theoretically not practically but I think I should edit it. Commented Sep 8, 2023 at 18:49
• @whuber I have edited the question and I think it is no more a falsehood. Commented Sep 8, 2023 at 19:00

I am posting my answer based on what I learnt from ISO terms and definitions , GUM, a book Taylor : An introduction to error analysis. and a lengthy discussion on physics forums.

What I was lagging with standard deviation and standard error. I didn't know the difference between Standard deviation of the mean (also know as standard error = SD/(N)½) and Standard deviation. Both are deviation but in different sense. Former describes that how close our mean is with respect to conventional true value (population mean) and the later one is based on dispersion of measurements.

Taylor in his book (In chapter 4 and 5) described that Uncertainity reported in measurement must be standard deviation in mean not standard deviation (SD).

Further in a example he tells us we must report a measurement washing off individual indications (measurements) as Mean ± Standard error (Standard deviation in mean).

Now according to formula which is inversely proportional to number of measurements, we would have a negligible standard error associated with mean if we take a large number of measurements (Uncertainity ≈ 0) But that doesn't mean that standard deviation would be negligible!

And as described above ( Uncertainity as SD/(N)½ ) GUM description for uncertainty (Uncertainity is simply an estimate of the likelihood of nearness to the best value that is consistent with presently available knowledge ) would be the best definition for Uncertainity (valid for small or large number of measurements) as sometimes we are not aware with some of bias components of error and in that condition range in mean as Uncertainity, definitely would not show true value but will describe the best estimate that can be taken in those conditions.

There is One another important point that there are same terms(like mean, standard deviation etc.)in mathmatical and physical statistics but they are somewhat different based on in what sense they have used. For example

In Math : Like in a classroom of 100 students if I report a measurement of their performance in test out of 10, we will not use the word Uncertainity in report because each student's mark is exact i.e no error and Uncertainty is associated with that so in that case we would report performance as Average Score ± standard deviation not like Average Score ± standard error and one more point standard deviation shows here that 68 student's marks lie in SD range

In Physics : Suppose I have to measure time period of a pendulum and I take 100 measurements under similar conditions and I know that now unlike to student's test scores these 100 time periods are not correct , they have a random ( or may have systematic component also) error associated with them so I would report time period as Average ± Standard Error and this reporting will not be necessarily reflect to true value (as there may be bias included). Standard deviation here would simply represent that if someone under similar conditions measure time period 68% of his/her readings will fall in SD range.

( In classroom example we are using statistics for different students scores and each one's individual score is known perfectly but in time period calculation we would have only one true value and we are just trying to get close to that by repeating measurements as we don't know about some unknown errors that could get us take away from true value.)

That's all I had to say. Correct me if I am wrong somewhere!

• At a glance I don't see anything terribly incorrect, but it looks like you are missing some important distinctions among the different forms and sources of uncertainty, often grouped into "epistemic" and "aleatory" uncertainty. The latter concerns actual variation, which should be further divided (although perhaps somewhat arbitrarily in some circumstances) into various "components of variance," some of which can be attributed to the measurement process.
– whuber
Commented Sep 10, 2023 at 14:34

Your question and answer indicate some confusion in terms and concepts, which I'll try to clear up first.

The acronym "GUM"

My expertise in measurement uncertainty is restricted to ISO/IEC Guide 98 Uncertainty of measurement. That Guide has multiple parts and supplements. The acronym "GUM" usually means Part 3, i.e. ISO/IEC Guide 98-3:2008 Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement (GUM:1995) (pdf at bsigroup.com). I suggest using unqualified "GUM" to refer to only the whole of that document, e.g. not Part 6: Developing and using measurement models and not Annex D of the GUM.

I also suggest using the ISO Online Browsing Platform to check for ISO recognised definitions.

Population parameters vs. measurands

From GUM

3.1.1 The objective of a measurement (B.2.5) is to determine the value (B.2.2) of the measurand (B.2.9), that is, the value of the particular quantity (B.2.1, Note 1) to be measured. A measurement therefore begins with an appropriate specification of the measurand, the method of measurement (B.2.7), and the measurement procedure (B.2.8).

NOTE The term “true value” (see Annex D) is not used in this Guide for the reasons given in D.3.5; the terms “value of a measurand” (or of a quantity) and “true value of a measurand” (or of a quantity) are viewed as equivalent.

3.1.2 In general, the result of a measurement (B.2.11) is only an approximation or estimate (C.2.26) of the value of the measurand and thus is complete only when accompanied by a statement of the uncertainty (B.2.18) of that estimate.

<snip>

B.2.9 measurand particular quantity subject to measurement

EXAMPLE Vapour pressure of a given sample of water at 20 °C.

NOTE The specification of a measurand may require statements about quantities such as time, temperature and pressure.

In passing, I note that GUM:Annex D - to which you refer as "ISO terms and definitions" and "GUM description for uncertainty" - is really explanation of why GUM eschews reference to "true value" and "error", and doesn't really help define measurement uncertainty.

Note that the objective of a measurement is not to estimate a population parameter. Considering the two examples in your answer:

In Math: Usually the student's mark is of no interest except as an indication of that particular student's ability (the measurand) and as such it is certainly subject to error. Less commonly the objective may be to estimate the average ability of a group, but even then the average mark is only an estimate of the measurand.

In Physics: Due to unavoidable changes in conditions there will be differences in the (unknowable) true periods of different oscillations. You seem to be suggesting that the measurand is some essentially hypothetical average of an infinite number of oscillations.

OK, on to your question "what would be the correct definition of measurement uncertainty which would be applicable to any case whereas 'systemic error is involved or not' or 'Result belongs a very small or a very large sample of measurements'?"

You don't mention the definition of measurement uncertainty in GUM which seems appropriate to all the circumstances you cite:

2.2.3 The formal definition of the term “uncertainty of measurement” developed for use in this Guide and in the VIM [6] (VIM:1993, definition 3.9) is as follows:

uncertainty (of measurement)
parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand

NOTE 1 The parameter may be, for example, a standard deviation (or a given multiple of it), or the half-width of an interval having a stated level of confidence.

NOTE 2 Uncertainty of measurement comprises, in general, many components. Some of these components may be evaluated from the statistical distribution of the results of series of measurements and can be characterized by experimental standard deviations. The other components, which also can be characterized by standard deviations, are evaluated from assumed probability distributions based on experience or other information.

NOTE 3 It is understood that the result of the measurement is the best estimate of the value of the measurand, and that all components of uncertainty, including those arising from systematic effects, such as components associated with corrections and reference standards, contribute to the dispersion.

Your final 'Result belongs a very small or a very large sample of measurements' suffers from imprecision in language. Using ISO terminology the "measurement" is the final result for which the uncertainty is required - "a ... sample of measurements" doesn't make much sense. If you mean "the (final) measurement value is calculated from a ... number of other quantities" then the number and nature of those quantities is part of "the method of measurement (B.2.7), and the measurement procedure (B.2.8)" in GUM 3.1.1 quoted above.

The relationship between those other quantities and the measurement value is the subject of the "measurement model" covered in "Guide to the expression of uncertainty in measurement — Part 6: Developing and using measurement models" which your answer refers to as "GUM". In my experience the considerations in Part 6 are rarely used in practice.

Hope this helps.

• Could you summaries which terms I have understood wrong? Commented Sep 11, 2023 at 16:50
• I hoped my answer did exactly that. You'll have to make your question more precise if you need further clarification. Commented Sep 11, 2023 at 18:21
• You said that "Your question and answer indicate some confusion in terms and concepts, which I'll try to clear up first." But in your long answer I could find which terms were you referring and had some confusion like that is uncertain or anything else. Commented Sep 12, 2023 at 7:50
• @GovindPrajapat You confuse the GUM with other documents. You confuse discussion and descriptions of measurement uncertainty (GUM:Annex D, Part 6) with it's definition (GUM:2.2.3). You confuse measurements (marks, times) with measurands (student ability, unclear property of a pendulum). You do not consider the GUM definition of measurement uncertainty. I deduce English is not your first language, it may help to read the GUM in your preferred language. Commented Sep 12, 2023 at 10:43