I perform N experiments and have N sets of measurements. Now I know that the instrument I used to make measurements also has an error. Now I want to combine sampling uncertainty and random uncertainty between replicates (i.e. between N replicates). Can I just take root mean square of the uncertainties from replicates and instrument, to get overall uncertainty in the N measurements?
You perform N measurements. From these measurements you get the mean and the standard deviation (SD). If your N measurements are uncorrelated and show a normal distribution, then your statistical uncertainty is uA = SD/sqrt(N).
The instrument "error" can be considered as a systematic uncertainty, which is usually called uB.
The combined uncertainty is then u = sqrt(uA^2 + uB^2).
The first part of the answer of Dirk Piester is completely correct. However, the second part is only partly correct.
First of all the instrument error is not a systematic uncertainty, but it is an type B uncertainty, which is also randomly distributed (while a systematic error isn't)
Secondly, performing the law of uncertainty propagation, the type B uncertainty (the instrument uncertainty) has also to be divided by sqrt(n), because every measurement you made is also uncertain due to the instrument uncertainty.
So the final uncertainty is
u = sqrt(uA^2 + uB^2/N).
Dirk Piester and Jürgen Hartmann thank you, adding another layer of complexity, analysis with standards show that the instrument uncertainty is correlated to measurement values, thus I would need a co-variance value in estimation of uncertainty. In such a case I think I need to modify Piester's expression with some co-variance term, I am unsure on how to bring the co-variance into the equation. You assistance would be much appreciated.
Thank You,
Ah... you have to be very careful with the word “correlation”. Uncertainties can’t be correlated in a strict sense. If you mean that the size of the uncertainty depends on the value, that is something different.
The question is whether the unknown error is correlated. In other words, while we do not know the error, we can say that, for example, two unknown errors are correlated in the sense that if it were possible to know one, we would be able to predict the other. This happens, for example, when we use the same instrument to measure two things that are combined to give a final answer.
I suggest you learn about uncertainty analysis according to the Guide to the Expression of Uncertainty in Measurement (the GUM). That can be downloaded from the BIPM website (search BIPM GUM), but it’s not easy to follow.
My organisation- NPL - has some online training on how to apply the GUM. The introduction to uncertainty analysis is free. It doesn’t cover correlation though. Other courses do - and I have a (free-to-download) textbook on uncertainty analysis for Earth Observation (see my page here) that covers this. Even though it is written for Earth Observation it should be useful (at least early chapters) for your question.
Two comments:
1) I used "uncorrelated" to express that measurement n+1 does not know the result of measurement n, n-1, ...
2) Jürgen Hartmann is right. Instrument errors may have a "statistical" nature. If those have a random impact on each single measurement value then this uncertainty is part of the measurement value distribution in the data set of N and can be treated similar to uA. But it may be sometimes difficult to distinguish those from the pure uA measurement noise. On the other hand, there may be sytematic instrument errors, which do not change from measurement to measurement. Their uncertainties are part of uB, too.
Basically: Never mix errors and uncertainty!
If your instrument has an error, you should determine this and correct for it in each measurement. The uncertainty of this correction should then be combined with other uncertainties in accordance with the methods presented in GUM.
Dear all,
For chemistry field:
https://www.eurachem.org/images/stories/Guides/pdf/Eurachem_CITAC_QAC_2016_EN.pdf
what i would do:
assuming a model of measurement y=x+c being x the reading of the instrument and c its (fixed but unknown) correction. If x is estimated by repeating n measurements and it is assumed a rectangular probability distribution for c (for example) then the uncertainty will be: u(y)=sqr(s^2/n+(dc/sqr(3))^2) where s is the standard deviation and dc represents our doubt (100%) on the correction: c=c0+-dc and c0 is our (pure type B!!) estimation of the correction
Thank you all, I very much appreciate your help.
If in your experiment, the quantity is measured directly using a single measuring instrument and if there are no other influence quantities, the mathematical model of the measurement (the functional relationship between the measurand and the input quantities on which it depends) will only involve the correction c (see the formula presented above by Javier de lucas: y = x + c). Assuming that x and c are independent (uncorrelated), the law of propagation of uncertainty given in GUM (in which the sensitivity coefficients are equal to 1) leads to
u(y) = sqrt [u(xmean)2+u(c)2] (1),
where
- u(y) is the standard uncertainty associated with the measurement result;
- xmean is the arithmetic mean of the N repeated observations, xi, i = 1, 2, ..., N;
- u(xmean) = s(xmean) = sqrt [s(xi)2 / N], where s(xi) is the experimental standard deviation;
- u(c) is a Type B standard uncertainty. For the evaluation of u(c), I see two situations:
1) If the value of c is provided in a calibration certificate of the instrument and its uncertainty is declared as a standard uncertainty, then this value of the standard uncertainty represents u(c) and it is entered into the formula (1) above. If the uncertainty value associated with c is given as an expanded uncertainty U(c) = ku(c), where usually k = 2 (for a normal distribution and a confidence level of 95%), then the value of the standard uncertainty u(c) is the value given for U(c) divided by k.
2) If the instrument does not have a calibration certificate and the value of c is taken from the literature or from the manufacturer's specifications where also the maximum error of the instrument, emax, is indicated, then we can assume that the distribution of the values of c (c = -e) is symmetrical rectangular and the uncertainty u(c) is calculated with the equation u(c) = sqrt [(emax 2)/3].
Regarding the correlation between x and c: I think they could be correlated only if they were measured simultaneously in the same experiment.
There may be a known error that needs correcting. And I agree with Kaj - the GUM says that known errors must be corrected before doing uncertainty analysis.
After correcting for all known errors you have a measurement that probably relies on combining several input quantities through a measurement equation- this might be explicit: you measure several things and combine them in an equation, or implicit: the instrument has some equation behind how it functions.
Each term in the equation will be subject to one or more effects that will be a source of uncertainty. Uncertainty is a measure of the spread (standard deviation) of the residual unknown error for each effect.
Metrologists can learn to avoid the word error - I think that is a mistake, because if we don’t use the word error we struggle to talk meaningfully about error correlation. And I think error correlation is badly understood and confused by other correlation - for example in satellite Earth Observation the value in one pixel of an image may be correlated with the value in another pixel because of geophysics- for example ocean currents. When geophysicists calculate zonal averages - averaging pixels over a square km for example - these correlations are maintained by the average and that’s fine.
However, deep below we have an error correlation that must be taken into account. A thermal IR sensor calibrates itself against onboard references once per scanline. That means that the unknown error in that calibration- the specific draw from the distribution described by the uncertainty- will be the same for all observed pixels on that scanline. This error correlation must be taken into account in zonal averaging.
So back to the original question. When you say the instrument has an error - do you mean a known error that you should correct or a residual unknown error that is a draw from a probability distribution described by an uncertainty? If the former - correct it.
If the latter,you may have a calibration certificate for the instrument providing an uncertainty associated with that calibration. You’d be right that that uncertainty provides an unknown error that is then common to all N of your observations with that instrument- and that uncertainty will not reduce by having multiple measurements.
In addition your N measurements are not identical because of noise. The uncertainty associated with that can be added in quadrature - that will work out (my book shows the working) as the uncertainty associated with noise divided by the square root of the number of measurements. If you add that in quadrature (sqrt sum squares) with the uncertainty associated with the calibration (assuming they are in the same units - go to the introductory courses on uncertainty analysis to get this), you’ll have an estimate of the uncertainty associated with that measurement.
Note further if N is sufficiently large (~20 or more), the best estimate of the uncertainty associated with noise is the standard deviation of the measurements. If N is small, you probably need to do an initial measurement with a very large N to work out what the uncertainty associated with noise is. The standard deviation of the large-N measurements is then the uncertainty and the later measurements when you only do say m measurements should divide this uncertainty by sqrt m
About vocabulary of metrology:
https://www.bipm.org/en/publications/guides/vim.html
A good summary for GUM Method:
METROLOGY - IN SHORT
https://www.euramet.org/get/?tx_stag_base%5Bfile%5D=3743&tx_stag_base%5Baction%5D=downloadRaw&tx_stag_base%5Bcontroller%5D=Base
If we do not know the error of the device, then we need to divide the accuracy class into a root of three (as Javier de lucas said). Then it must be squared and added to squared standard deviation under the root.
If we know the systematic error of the device, then we need to correct the measurement result. But the question is: was the correction estimated for similar real conditions? Do the loading effect, temperature shift or drift between two calibrations occur?
If you are sure the correction was estimated properly then you ought to combine the standard deviation and the uncertainty of measurements in determining the correction (as Kaj Heydorn said).
Thank you all, @emma wooliams, so in terms of the insrument, I prepared samples whose measurement values are know. These known measurements were then analyzed by the instrument. The know measurement samples analyzed had values in a range that represented possible values of that elements in the set of N could have. So I was thinking that I could fit the errors from insrument to the values of known samples I measured. Then use that fitting equation to estimate the instrument error for elements in set N. Then I can combine the instrument error from my fitting expression that relates error to actual values with the random error obtained from replication in quadrature. To be more complete I can also add the error from the fitting expression in quadrature to the above mentioned two errors.
Dear Sankaranarayanan Ayyakudi Ravichandran,
I think the following MOOC [ The University of Tartu: estimation of measurement uncertainty, specifically related to chemical analysis (analytical chemistry) ] material will be useful for you:
https://sisu.ut.ee/measurement/uncertainty
Dear Sankaranarayanan Ayyakudi Ravichandran,
From the latest reply as I understand it you have N measurements (say N = 22 for the sake of argument) and you have some physical measurements that you are sure of. Let the physical measurements be yP(i) for i = 1,2,...,N. In addition you also have some instrument measurements as well and let them be yI(i) for i = 1, 2, ..., N. Based on this information it seems that you are defining the error as eI(i) = [ yI(i) - yP(i) ] and then you have the data as:
True Physical | Instrument Measurement | Instrument Error
yP(1) | yI(1) | eI(1)
...
yP(N) | yI(N) | eI(N)
The problem as I see it is that you are not taking into account the corresponding physical uncertainty u(yP) and the corresponding instrument uncertainty u(yI). To be a bit more tangible you might have some information of the physical accuracy, say for example u(yP) = +/- 1.2% for the sake of argument, and in addition suppose the instrument has a a fixed digital resolution.
If you do not take into account the corresponding uncertainties u(yP) and u(yI) then technically you cannot really say anything about the corresponding uncertainty of the error. Note that uncertainty is a quantitative statistical indication using for example a probability density function, and that error is more of a qualitative indicator of the difference between the measured and true value. An error without an uncertainty is not really useful.
Also note that the GUM (Guide to Uncertainty of Measurement) used by metrologists (measurement scientists) assumes that systematic errors (basically gross errors) has already been accounted for and mainly focuses on aleatric uncertainties (basically random errors).
Kind regards,
Vishal Ramnath
We tried to solve the problem before it was rigorously defined. The information provided later has produced a number of clarifications (even if they are not complete enough and some conditions of measurement still need to be guessed). Conclusion: solving the problem requires a different approach.
The proposed problem seems to be the calibration of a measuring instrument by comparing it with known (but different) sample values. By denoting by xi (i = 1, 2, ..., N) the N known values of the samples, and by zi (i = 1, 2, ..., N) the values of the N samples measured with the instrument, the correction
ci = xi - zi (1)
can be estimated for each (xi, zi) pair.
To calculate the uncertainty associated with each correction, hereafter denoted by u(ci), it is sufficient to combine the uncertainties of xi and zi. The uncertainty of xi, u(xi), is probably negligible (no information is provided about it). The uncertainty of zi, u(zi), is obtained from the dispersion of the independent repeated observations zi,k (k = 1, 2, ..., M) made to determine a single zi value from the series of the N; thus
u(zi ) =s(zi,k,mean) = sqrt [s(zi,k)2/ M] (2)
where zi = zi,k,mean = (z1 + z2 + ... +zM)/M.
If u(xi) is not negligible, it can be combined with u(zi). If the experimental conditions are indeed those deduced here, then xi and zi are not correlated (even if they refer to the same quantity), because they are obtained in different experiments. (As has been said, other uncertainties may be added, as appropriate: uncertainties associated with the instrument's finite resolution, environmental temperature influence, non-linearity, etc.)
As a result,
u(ci) = sqrt [u(xi)2 + u(zi)2]. (3)
However, from the information provided we can also deduced that the issue of interest is the evaluation of the uncertainties associated with the corrections that should be applied to other zi values (than those measured in the experiment). In other words, a curve is fitted to the measured corrections and the uncertainties of the intermediate values must be evaluated. If the problem is really this, the uncertainty of a predicted correction is calculated on the basis of the parameters of the chosen fitting curve, curve which is the new measurement model: a first/second/third etc. degree polynomial. For example, for a linear curve, the equation is written as
cj = azj + b (4),
where the parameters a and b are determined on the basis of the N pairs of values, (ci, zi). As a result, the uncertainty of the cj correction to be applied to the instrument for another value zj of interest will result by combining - using the law of propagation of uncertainty for equation (4) which is the mathematical model - the uncertainties associated with all known values ci, that were the basis for calculating the two input quantities in the model (4): the parameters a and b.
By means of the N pairs (ci, zi) on which they depend, a and b are correlated, and their covariance can be readily determined using the equation given in GUM (see 5.2, equation (17)). There is, of course, also an uncertainty of fit, that can be evaluated on the basis of the differences between the N values ci measured and the N values cj determined for the same zi.
The reply by Sonia Gaita is correct in terms of the general approach.
As stated other uncertainty contributions should be added in quadrature unless otherwise specified if they make a significant contribution to the measured values z_i (i=1, 2,...,N).
The form of Eq(4) could be in principle anything as it doesn't have to be a straight line. Under these circumstances a chi-squared merit function could be used to obtain the values of the parameters a, b, c, d, ... for the particular form of Eq(4). The uncertainties of the parameters u(a), u(b), u(c), u(d), ... would also have to be specified to fully characterize the uncertainties of the corrections for values of non-observed measured values. But apart from that refinement the mathematical formulation of the original problem posed is basically complete.
Again, for chemistry...
https://www.eurachem.org/index.php/publications/guides/quam
and,
https://www.eurachem.org/index.php/publications/mnu-rdlst/128-rdl-uncert
Vishal Ramnath: Can you please suggest a reliable source which estimates both the physical uncertainty u(yP) and the corresponding instrument uncertainty u(yI) that you have mentioned in your first answer?
Kind regards,
Soham
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