Mechanical Engineering
• 0

# propagation of uncertainties

## Question

Hi,

I am writing my master thesis and stumbled upon a problem while trying to calculate the uncertainty of my results. I generally used the following formula, valid for independent variables:

sf = ((df/dx)^2*sx^2+(df/dy)^2*sy^2+(df/dz)^2*sz^2+...)^(1/2)

With sf being the standard deviation of the function f and sx being the standard deviation of the variable x. This works well in most cases. But in some cases, I have data captured by a sensor e.g. a thermocouple. This sensor sends an analog signal to a data acquisition module, which converts the data into a digital signal. This digital signal is then sent to the computer. Now both the sensor and the module have an uncertainty and I don't know how to calculate the overall uncertainty of the data. I have found a source (bachelor thesis), which calculates these errors with e.g.:

sT=(sTC^2+sM^2)^(1/2)

with sT: standard deviation of the measured temperature, sTC: standard deviation of the thermocouple, sM: standard deviation of the data acqusition module.

But I don't see how this is correct. The author claims that this formula is what you get when you apply the general forumla that I mentioned to this specific case. But this formula resolves if you have two variables which are simply added. In that case though, there are not two variables added. Instead, mathematically, you could say the temperature function is

T=modulesignal(thermocuplesignal)

(of course this neglects the conversion from voltage to degrees C)

But ideally the signals would be the same so basically simply modulesignal = thermocouplesignal, but of course with an inaccuracy.

So the general formula in my opinion cannot be used in this case because the signal from the module has to be regarded as a variable (since if you treat it just as a function like f in the general formula, it cannot have it's own inaccuracy), but it is dependent on another variable: the signal of the thermocouple.

So I think I need another approach to calculate the overall inaccuracy of the temperature. But when I look for methods that work with dependent variables I only found complicated formulas for which also more infromation like covariance would be needed. But I think in this very simple case it should be possible to calculate the inaccuracy in a very easy way.

Does anyone have an idea how to do this? of course if I misunderstood something and wrote some rubbish here I am also thankful if someone could point out the mistakes of my thought process.

Thanks and cheers,

## Recommended Posts

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You are quite right. The true uncertainty of the process you're describing is much more complicated than it may seem from reading the literature. I have a particular interest in this subject, work for a company that is intimately concerned with such things, and am currently collaborating with a group retired professors to adequately address this gap in the literature. Not only does the sensor itself present multiple uncertainties, at the very least both random and systematic, but the sampling process also contains uncertainty. So does the analog-to-digital conversion. Thermocouples have greater uncertainty than platinum RTDs, due to consistency and sampling. It is very difficult to accurately measure small DC voltages. I once set up a test to prove this to a colleague, using everything from a cheap analog multimeter from Radio Shack to a digital one costing many thousands of dollars--all connected to a single nominal 1.5V C size Duracell battery. There is also the matter of how long do you sample a moving target? Most systems of practical interest vary over time. There are far too many people who think that, if you sample long enough, you will know a quantity with certainty; but this is not the case. There are also far too many people who think you're supposed to divide the uncertainty or the standard deviation by sqrt(n), which magically makes all the uncertainty vanish. This is just ignorance and wishful thinking. There are glaring errors in prestigious test codes, including ASME PTC-19.1 (Test Uncertainty) and even ISO JCGM 100 (Evaluation of measurement data--Guide to the expression of uncertainty in measurement), which can be demonstrated with actual data and also Monte Carlo simulations. In recent years, NIST has begun using different words, including "repeatability," which means, "We keep getting the same number, but we have no idea if it's right." History is littered with examples of people who were absolutely sure beyond a shadow of a doubt that they were obtaining accurate measurements of something we now know doesn't exist or isn't what they thought it was. I encourage you to diligently pursue this matter and consider all of the complicating circumstances you can think of in doing so. I don't mean to be overly negative. Just because we may not know something to the level of precision that we might like, doesn't mean that what we do know is worthless or that attention to detail is without reward.

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