# Definition of Uncertainty

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 take infinitely times measurements (theoretically) with no systematic error then we would be pretty sure that the mean of measurements will be equal to the true value and uncertainty would make no sense as it is defined previously!

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, the 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 theoretically infinitely times measurements) with no systematic error represent?

Now I end 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 'systematic error is involved or not' or 'measurements are finite or infinite'?

Edit:- In context of infinite measurements, 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. I wanted to know that for example what does population standard deviation in population mean shows when no bias component is involved ? Because then we can safely assume population mean to true value and don't need to show a standard deviation if it is related to only true value not the random measurements taken. ( population mean ISO 3534-2: 1.2.1 )

• If you don*t know anything of a systematic error, you can mot say a thing about the real error. Imaging you have a scale which is 10% of the real value, but you don't know about it. son your mean value just says "up to systematic error. Second to use the second definition you have to assume that the errors are statistical, which may often not be true. the 68% are ok for measurements of the rate of decay of isotopes , since there you know its statistical. Talking of infinit many measurements makes no sense. Sep 8 at 17:18

but if we take infinite measurements ( theoretically ) with no systematic error then we would be pretty sure that mean of measurements will be equal to true value and uncertainty would make no sense as it is defined previously!

It is not possible to take infinite, or infinitely many, measurements, so this is not a valid complaint.

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.

This statement in the GUM description is explicitly stating that we do not include systematic biases inside the definition of uncertainty, because we might never know if yet another unknown systematic bias is being missed. You are simply reading it wrong if you think GUM is asking for systematic biases to be included in this evaluation.

Actually, what is really happening is that you lack of comprehensive understanding of what it is we are trying to do. It is not very helpful to focus upon measurement uncertainty; we should first seek to construct a framework to understand the basic uncertainty types out there.

The most easiest to understand of them all is random uncertainties. The most common mental model of a type of random uncertainty is when a variable of interest is the sum of a huge number of tiny fluctuations, washing out the details of how each single tiny fluctuations are, and appearing as if it is just one single broad distribution. If this distribution is symmetric, then it is very likely to be Gaußian in character, or at least close enough such that using Gaußians to model them is not vastly wrong. In which case, we may write down just one single uncertainty estimate. This is the source of all the appearances of 68%, for it is the Gaußian's single standard deviation confidence interval.

It is must better to think of uncertainty estimates as describing probability distributions, like the Gaußians as mentioned above, because a lot of the time, what we really want to do with these uncertainty estimates, is to propagate them to another quantity of interest. Say, we obtained an estimate of the two lengths of a rectangle, and we want to figure out the area of the rectangle, along with the uncertainties on what it would be. It will be helpful if we can run computer simulations to sample probability distributions from the length estimates, and generate directly the probability distribution of the area from there. Modelling them with Gaußians automatically discourages the error estimates from being simultaneously extreme, favouring instead for area estimates close to the mean values. It is much more important to be able to accurately model complicated combinations with probabilistically sensible distributions, than to attempt to capture absolutely maximal errors (impossible when the distributions have infinitely long tails anyway), say. It enables an entire industry of future uses of quantities if we abide to the conception that we want to use these uncertainty estimates as probability distributions.

If the uncertainties are definitely not symmetric, then we may obviously not use Gaußians. A common approach is to give the upper and lower uncertainty estimates, and some other probability distribution will be good for that, leaving the choice up to the modelling user.

If there are systematic biases, it is typical for its values and uncertainty estimates to be stated separately from the reporting of the mean and uncertainties. This happens whether or not the mean value has already subtracted the systematic bias or not, with explicit notes telling the user whether this subtraction is done or not. It can even happen for each type of systematic bias identified; they do not have to be combined into one big systematic bias before application, and in fact, they are usually considered separately, again for ease of modelling.

Measurement uncertainty actually also comprises quantisation errors, amongst other possibilities. Quantisation errors are neither the usual random uncertainty nor a systematic bias. They are modelled separately and is definitely never modelled via a Gaußian. These cannot be fixed by measuring infinitely many times either, which is thus a refutation of your first misconception.

• "It is not possible to take infinite, or infinitely many, measurements, so this is not a valid complaint." 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. I wanted to know that for example what does standard deviation in population mean shows when no bias component is involved ? Because then we can safely assume population mean to true value and don't need to show a standard deviation if it is related to only true value not the random measurements taken. (iso:3534:-2: 1.2.1) Sep 9 at 3:20
• "You are simply reading it wrong if you think GUM is asking for systematic biases to be included in this evaluation." I appreciate your thoughts for reading my mind. I was actually reading it as we must include systematic errors in Uncertainity but now it's clear we shouldn't. Sep 9 at 3:22
• I think you are having yet another misconception, then. Even if, by repeated measurements to reduce the random uncertainty on a fixed variable, you can reduce the variance of the sample mean (not population mean, really; you are estimating the population mean), that just means that you have a good grasp of what this fixed variable is. However, when you want to use it later, you still have to figure out what a single measurement's variance is, and that will be large. i.e. you state the estimated population mean with small variance, and then state the large variance expected for a single pick. Sep 9 at 9:07
• Hey, I have post an answer would you like to cross verify it and correct if it's wrong somewhere. Sep 10 at 2:02

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 he reported measurement as Mean ± Standard deviation in mean in a example. 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!

but if we take infinite measurements ( theoretically ) with no systematic error then we would be pretty sure that mean of measurements will be equal to true value and uncertainty would make no sense as it is defined previously!

Two problems here. The first is the presumption that we can take an infinite number of measurements. We cannot.

But, let's stay with this hypothesis for a minute. If we can make an infinite number of measurements even when the system has a fixed systematic bias error, then an exact truth is that the mean of the distribution is the true value after correcting exactly for any (fixed) systematic bias.

But ... !!!!

Even when we can make an infinite number of measurements, we never will be able to measure the same value every time (to infinite number of times). Why not? Because real measurement devices have finite precision. No one has created a measurement device that is infinitely precise. In simple terms, the measurement uncertainty of any device is half the distance between the tick marks on its measurement scale.

With this in mind, even if the system absolutely does not change at all over the time period when you measure it, what you measure now and what you will measure later has a real, finite chance of being two different values due solely to the limit for the measurement precision of the device you use to make the measurement. Hence, a distribution of measured numbers arises out of the infinite number of measurements made even on a system that itself has undergone absolutely no change down to the infinitesimal degree on the value that you measure.

In short, the only way to eliminate an uncertainty distribution entirely from measurements on a system parameter is to do an infinite number of measurements with an infinitely precise measuring device on a system that has absolutely no random variation to it at all points in time when any one measurement is made. Alternatively said, if you have a device that reports to infinite number of significant digits (no measurement uncertainty), and if you know your system does not change in any way over time (no random fluctuations at all), only then you can confidently make one measurement and report it. Otherwise, you have a distribution curve at hand.