# How to evaluate an uncertainty involving an experimental correlation?

## Main Question or Discussion Point

hello,
Thank you to understand that this is not a homework!

I have to evaluate the uncertainty of a measurement but the baseline is also with uncertainty (correlation). I would like to have some ideas or suggestions on the best way to evaluate the uncertainty. Here is the problem:

We have an aqueous solution for which, the density D is temperature T and concentration X dependent:
D = a + bX + cX² + T (d + eX)
This expression is an experimental correlation (from the literature) whose accuracy is 2%.
T: temperature; absolute uncertainty = 0.5°C

We measure our solution temperature T and density D. We solve the above correlation equation in order to find the concentration X. WHAT IS THE UNCERTAINTY OF THE OBTAINED VALUE X?

Is it correct to consider that the uncertainty of “variable” D is 2%?

Thank you.

## Answers and Replies

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Stephen Tashi
In the context of fitting equations and making measurements, I know of no standard definition for the term "uncertainty". The answer to this problem will depend on what you and the authors of the equation mean by "uncertainty".

The authors of the equation might mean that all the true values of the D's lie within plus or minus 2% of the the value predicted by the equation. Or they might mean that the errors in prediction are normally distributed with a standard deviation of about 2% of the predicted value. Or maybe the 2% describes two standard deviations, etc.

If you can find out what "uncertainty" means or state your preferred assumption about what it means, then the problem might become well posed.

We measure our solution temperature T and density D. We solve the above correlation equation in order to find the concentration X.
Must we also consider possible errors in you measurement of T ?

Thank you for the answer. Actually, I did not "care" about because I usually think "uncertainty" and nothing more. But now, I can find, you are right, the authors stated "standard deviation".

Temperature sensor: as suggested in my first post, the accuracy of our thermocouple is 0.5°C and has to be taken into account in the uncertainty calculation.
Thank you.

chiro
You can fit models like this with standard programs but there are usually assumptions about the residuals of the errors (which are typically assumed to be normal, and all errors are assumed to be independent against all the other errors).

You can do all of this kind of thing in most common statistical packages including a very nice free one called R. After you check the assumptions and take into account the nature and context of your data (and see if all of this seems OK for the following analysis) then you can generate estimators for the parameters including the point estimates at the standard error which can be used to generate intervals and also to help describe the "uncertainty" of the model.

Stephen Tashi
I'll guess about the hocus-pocus that a software package would do::

Let $D = f(X,T) = a + bX + X^2 + T(d + eX)$

Use the linear approximation
$D + \triangle D = D + \triangle X \frac{\partial f}{\partial X} + \triangle T \frac{\partial f}{\partial T}$

$\triangle D = \triangle X \frac{\partial f}{\partial X} + \triangle T \frac{\partial f}{\partial T}$

$\triangle D = \triangle X (b + Te + 2X) + \triangle T (d + eX)$

So $\triangle X = \frac{1}{b + Te + 2X} ( \triangle D - (d + eX)\triangle T )$

Let $A(X,T) =\frac{1}{b + Te + 2X}$ and $B(X) = d + eX$

$\triangle X = A(X,T) \triangle D - B(X) \triangle T$

Consider $\triangle D$ and $\triangle T$ to be independent random variables representing measurement errors and let $D,T,X$ be the unknown true values of the variables

$\sigma^2 _{\triangle X} = A^2(X,T) \sigma^2_{\triangle D} + B^2(X) \sigma^2_{\triangle T}$

The "accuracy" of thermocouple is another ambiguous term. Taking "accuracy" to mean "standard deviation", $\sigma_{\triangle T} = 0.5$.

$\sigma^2_{\triangle X} = A^2(X,T) (.02 D)^2 + B^2(.05 )^2$

This would be helpful if we knew $X,D,T$. However, we only know the measured values $X + \triangle X, D + \triangle D, T + \triangle T$. I speculate that a software package might substitute the values measured values for $X,T,D$ in the above equation. To justify that, we'd have to examine whether the right hand side of the equation is sensitive to small changes in $X,T,D$. (You can do some numerical tests to find out.)

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How you approach this problem depends on how serious the analysis must be. Are you simply looking for a number to put in a report that nobody is likely to scrutinize? Or is some critical theoretical or financial decision going to be made that depends on your estimate of $\sigma_{\triangle X}$ ?

Thank you both for your responses. Yes, I will try to well understand the calculations of Stephen Tashi. There is no critical decision that is going to be taken but I as a scientist, we usually need to evaluate the uncertainties of our measurements in order to evaluate the significance of our results. For example if the two "different" concentrations values are obtaines, let say for example 55.2 wt% and 56 wt% and the precision of the measurement is 1.5 wt%, we could hardly say that there is a difference between the values.
chiro: I do not know actully if R could help me because the aim is not to fit a model. We measure only T and D and we calculate X using the correlation. We then need to evaluate the uncertainty. I am sorry that my vocabulary is not so correct: "uncertainty", "precision", "accuracy". We are so used to used them interchangeably.
thank you.