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.:
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
(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,