# Error/uncertainty interval goes negative while the value is known to be positive, how to report the interval?

Say we measure the coefficient of friction between two materials to be $0.03 \pm 0.05$ where we have used the formula $\Delta Y = \sqrt{ \left ( \frac{\partial F}{\partial X_1} \right )^2 (\Delta X_1)^2 +...+ \left (\frac{\partial F}{\partial X_N} \right )^2 (\Delta X_N)^2}$ where $Y = F(X_1, ..., X_N)$ in order to compute the error.

If we assume conservation of energy holds, then we know that the coefficient of friction cannot be negative. So I am tempted to think the error of the COF should be reported as $0.03 +0.05/-0.03$.

However when writing this question on PSE I realized what "the error" is (say 1 standard deviation) and thus it should be reported as the formula suggests, namely $\pm 0.05$. Because it does not imply the value could be negative any more than if we had obtained say $0.93 \pm 0.05$. Well maybe more probable indeed, but the point is that in both cases there is some probability for the COF to be negative, since the upper and lower bounds don't mean the value measured is necessarily restrained inside of them.

I'd like to know whether I'm right or off.

Edit: Some people are suggesting me to reduce the uncertainty/error to bypass this "problem" or just to get a better evaluation, by making more and/or better measurements. This is not my goal. Say I measured the quantity above with that particular interval and say I cannot reproduce the experiment for some reason. Note that in reality I don't have to report such a value with such an uncertainty, I am just curious on how to deal with such a case.

• I've run into a similar problem before. For the extremely low friction coefficient that you are reporting, it would be best to take your measurement with more precise equipment. Everyone knows that the friction coefficient can't be negative, and it is also apparent that the stated range of error is so wide compared to your reported value that you can't tell what you have. – David White Mar 16 '17 at 22:03
• I know, but if you read carefully my post, I think that the error bounds cannot ensure the value to be positive even though we know it can't be negative (else conservation of energy is violated). Also note that I am not personally reporting such values, it just occured to me this hypothetical case. – AccidentalBismuthTransform Mar 16 '17 at 22:21
• Maybe the problem is how you calculated uncertainty from your measurements, and perhaps also your experimental technique. – sammy gerbil Mar 16 '17 at 23:04
• @no_choice99, as I stated, I've had to deal with this kind of problem in practice. When I had this issue, I REALLY needed a more precise measurement, but none was available. :-( – David White Mar 17 '17 at 1:37
• @sammygerbil, David White, you are both missing the point. I do not care at all about reducing the error/uncertainty (I even said that this is a hypothetical example, I personally do not have to report such a measurement), I am just interested to know how to deal with such a situation. Say you can't refine the experiment, the question is how do you report the error. – AccidentalBismuthTransform Mar 17 '17 at 7:29

When you report the uncertainty in your measurement you basically state "this measurement could have been obtained with the underlying values of X in this range".

That is not the same as saying "X can have any of these values". If you actually want to give a confidence interval you could say something like "there is 95% confidence that X is in the range [0, y]". But in that case, especially with the numbers you give, you might have to deal with the asymmetry of the situation (the interval is no longer +- 1.96 $\sigma$.)

I am not aware of a uniform convention for this case. When in doubt use words to clarify - compared to the effort of the measurement, writing a few words to communicate unambiguously is well worth it.

• Could you please elaborate slightly more the difference between "this measurement could have been obtained with the underlying values of X in this range" and "X can have any of these values"? I'm extremely grateful for your answer. – AccidentalBismuthTransform Mar 17 '17 at 8:37
• "The measurement could.have been obtained..." is a statistical statement. "The value is between" is a physical interpretation about the measurement and what it possibly tells us about the system being investigated. – Floris Mar 17 '17 at 10:13

The first-order law of propagation of uncertainty that you have reported cannot deal in a coherent way with asymmetric distributions or physical limits like those of your example.

This means that if you want to have a better uncertainty evaluation, taking into account physical limits in a realistic way, you should employ more refined methods of propagation. Just truncating the uncertainty interval is generally considered a poor approach.

In particular, you can consider the Monte Carlo method (see, e.g., this guide) or a Bayesian evaluation. For the latter, refer to a book on Bayesian statistics.

• Could you explain why the 1st-order law of propagation of uncertainty can't deal in a coherent way with the "physical limit" of my example? Also note that I don't want to reduce the uncertainty (my example is hypothetical, I am just caring on how to report the error in that particular example). Ok about truncating the uncertainty interval, that's exactly what I wrote I thought was incorrect. About Monte Carlo or Bayesian evaluation, again, I am not interested to bypass the relatively big uncertainty since this would not deal with the original question. – AccidentalBismuthTransform Mar 17 '17 at 7:34
• @no_choice99 I'll expand on that in the next days. About the " I am just caring on how to report the error in that particular example": before reporting an uncertainty (please avoid the term "error" which means a completely different thing), you should first evaluate it. In your case, the evaluation of the uncertainty through the law of propagation gives an unrealistic result, so you might not want to report it. That's why in certain cases one needs to use more refined models of propagation. Not to "bypass" the "big" uncertainty, but to obtain a realistic, meaningful evaluation. – Massimo Ortolano Mar 17 '17 at 7:45
• @Massimov Ortolano I am not understand why the result is unrealistic? The uncertainty interval just represents 2 sigma (or so) and indeed it implies the COF could be negative but like I said, even if I had the value $0.93 +/-0.05$ the value could also be negative (albeit with a very small probability but still greater than 0) because I am only reporting 2 sigma. So why is the former case unrealistic? – AccidentalBismuthTransform Mar 17 '17 at 7:55

My take on my own question:

The uncertainty should be reported as $\pm 0.05$ in your case, because you used the right formula to compute the uncertainty based on the fact that $Y=F(X_1, ..., X_N)$.

Note that you're not implying that the value of the COF has to lie in the uncertainty interval, because that interval is just worth 2 sigma. So, if you had a COF of say $0.93$ with an uncertainty of $0.05$, even though values near 0 are well outside the uncertainty range, there's still a non zero probability that the COF is actually near 0 even though that probability is in practice indistinguishable from 0.

Now the uncertainty you calculated for such a value of the COF may seem "too big" but that is purely subjective and beside the point of the original question.

One workaround whenever one has a manifestly positive quantity $\mu>0$ is to report the variable $\ln\mu\in \mathbb{R}$ instead, which is then manifestly real, and hence doesn't have the problem with unphysical error intervals. Another is to use lopsided error intervals.