Least Count Error is Random or Systematic?

There are so many articles on internet on Error analysis in measurement but some of them relate least count error with systematic and other with random error and some associate it with both of them. I am so confused that Error in a measurement associated with least count is random or systematic?

Edit :- I think question was not clear about what I want to ask so I am adding an analysis of a measurement. I want to know about error associated with least count of measurement not about Uncertainty due to least count ( ISO guidelines :- Annex F, f.2.2.1 includes it in type B uncertainity) . Let's consider a ideal case in which I have a pen with a known true mass of 12.48gm. When I measure it using a digital weighing machine with a 0.1gm least count, it reads 12.5gm. The uncertainty in measurement due to the least count is ±0.05gm. However, when I compare it to the true value, there's an error of 0.02gm. (In this case we are assuming that there is no calibration or zero error so this error is definitely originated only due to limited resolution of instrument ). Is this error random or systematic?

Reference to 'Least Count Error is Systematic' :-

Reference to 'Least Count Error is Random' :-

Instrument resolution (random) — All instruments have finite precision that limits the ability to resolve small measurement differences. For instance, a meter stick cannot be used to distinguish distances to a precision much better than about half of its smallest scale division (0.5 mm in this case). One of the best ways to obtain more precise measurements is to use a null difference method instead of measuring a quantity directly. Null or balance methods involve using instrumentation to measure the difference between two similar quantities, one of which is known very accurately and is adjustable. The adjustable reference quantity is varied until the difference is reduced to zero. The two quantities are then balanced and the magnitude of the unknown quantity can be found by comparison with a measurement standard. With this method, problems of source instability are eliminated, and the measuring instrument can be very sensitive and does not even need a scale.

Least Count Associated to Both random and systematic:-

• I'm not sure why this is being flagged to move to Mathematics. The last time I checked, mathematicians don't measure things in the lab and they don't have to think about the uncertainties this leads to in their results. Sep 3 at 14:05

Error in a measurement associated with least count is random or systematic?

All of the references that use the terms “systematic error” or “random error” are outdated. These terms have been deprecated since 1995. The current (for almost 30 years) approach was largely developed in order to resolve the difficulties that you are asking asking.

The definitive reference is the BIPM’s Guide to Uncertainty in Measurement:

https://www.iso.org/sites/JCGM/GUM/JCGM100/C045315e-html/C045315e.html?csnumber=50461

https://www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf

See sections 2.3 and 3 for details. Uncertainties are not classified as random or systematic. They are classified as to whether or not they are evaluated by the statistical analysis of a series of observations.

Under this categorization, least count error is clearly not evaluated by the statistical analysis of a series of observations. Techniques for estimating this type of uncertainty are listed in the guide in section 4. I like treating it as a uniform probability with a width, $$w$$, of 1 count. The standard uncertainty, $$u$$, is then $$u^2=w^2/12$$

• First and foremost, I want to express my gratitude for dedicating your time to provide such a valuable response and reference. However, I have a query regarding Section 3.3.2, specifically point f, which mentions that finite instrument resolution or discrimination threshold (often associated with the least count error) is considered a component of uncertainty. I'm hoping you could shed some light on why instrument resolution is linked to uncertainty rather than error, as I'm finding it challenging to grasp the distinction between uncertainty and error. Sep 3 at 15:34
• @GovindPrajapat see section 2 for an explanation of these terms. The uncertainty is the the dispersion of the values that could reasonably be attributed to the measurand. The error is the difference between the measurement and the true value. Error is unknowable because we never know the true value. You might use error for a theoretical calculation where you can theorize some true value. But we must use uncertainty for real measurements.
– Dale
Sep 3 at 16:44
• @GovindPrajapat no. Uncertainty and error are different concepts. Error is fundamentally unmeasurable. Uncertainty is the measurable dispersion. The dispersion need not be random.
– Dale
Sep 3 at 22:21
• The Guide to Uncertainty in Measurement (GUM) is a great resource for everyone and essential for metrologists, and I completely agree with the last sentence of your answer, but how measurements are categorized is not the same in all research fields. For example "statistical" and "systematic" uncertainties are alive and well in particle physics, just look at these arXiv abstracts. Sep 4 at 1:39
• @GovindPrajapat as I stated at the end of my answer, the least count uncertainty is a “not statistically evaluated” uncertainty. Some people call that a systematic uncertainty. It is not an error of any kind
– Dale
Sep 4 at 13:56

The terminology and methods of uncertainty analysis vary in different research fields, and as @Dale has noted in his excellent answer, "statistical" and "systematic" are no longer used by metrologists and others who follow the Guide to Uncertainty in Measurement (GUM), but they are still useful (and used) in fields such as experimental particle physics, (e.g. see these arXiv abstracts). Other fields, e.g. statistics, may use still other ways to categorize uncertainty.

If you do use "statistical" and "systematic" categorization of uncertainties, then roughly speaking, a statistical uncertainty is one that gets smaller as the square-root of the amount of primary data, and a systematic uncertainty is one that does not.

In a well designed measurement system, we want the precision (e.g. the least count) to be smaller than the accuracy (i.e. the total uncertainty) so it does not contribute significantly to the final uncertainty. In that case, the least count reading uncertainty would usually be considered as a non-Gaussian statistical uncertainty. It could, however, also contribute a systematic uncertainty if you don't know whether the instrument is determined by (unbiased) rounding or (biased) truncation.

If the least count is the dominant uncertainty, e.g. for some bizarre reason the readout is only 3 digits when the intrinsic device accuracy is 6 digits, then the least count would be a systematic uncertainty because when you made repeated measurements you'd get the same least count digit 99.9% of the time. (I have never seen an instrument that badly designed, but I have used instruments that careful calibration showed they were clearly more accurate than their displayed precision.)

"Error" and "uncertainly" are often confusingly used interchangeably. If you are actually asking about systematic error and not systematic uncertainty, where "error" is the difference between the measured value and the (unknowable) true value, then I don't think there is a single correct answer to your question since different people use different terminology. I'd consider rounding error a systematic effect, since given a value with 0.01 precision, I can systematically tell you the error due to rounding to 0.1 precision, just as I could tell you the error if the measurement device is known to systematically read 1% high or has a -0.2 offset.

• Thank you for taking your time to answer. But I want to know about error not uncertainty. Let's consider a ideal case in which I have a pen with a known true mass of 12.48gm. When I measure it using a digital weighing machine with a 0.1gm least count, it reads 12.5gm. The uncertainty in measurement due to the least count is ±0.05gm. However, when I compare it to the true value, there's an error of 0.02gm. (which is definitely originated due to limited resolution of instrument as in this case we are assuming that there is no calibration or zero error). Is this error random or systematic? Sep 4 at 15:26
• @Govind-Prajapat I added a paragraph to my answer in response to your comment. Sep 4 at 21:10