# How can one use the % uncertainty in order to calculate the maximum or minimum values a value? [closed]

Here's the exam paper which provides the scenario that lead me to this question. (Question 1 is what I'm focusing on here.) http://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Physics/2013/Exam%20materials/WPH06_01_que_20160201.pdf

For starters, am I crazy? Or is there a difference between a +/- uncertainty and an uncertainty that is not +/-. Part (b)(iii) shows no discrimination. It states 0.6% or 1.1% as the uncertainty.

Moreover, here is why that's important in the part that really confuses me: part (d). How would I calculate the maximum and minimum values possible from the uncertainty provided? The value that is talked about is 1.16, and it has a 4% uncertainty. I used 1.16(1.04) in order to calculate the maximum value. Is this right? I would guess not based upon the fact that this the 4% uncertainty isn't +/-. If yes, then based upon a random phrase that I've seen somewhere in a stats book, 'the confidence interval moves around the mean', if this weird 'confidence interval' concept applies here then I may be right, but can someone please explain?

Finally, back to part (b)(iii), if the answer for part (d) is indeed 1.16(1.04), then would the minimum value for the volume of the rubber bung be 1.52(1-0.0057) or 1.52(1-0.0104) because I have to multiply the 0.57% uncertainty by 2 (since its +/-)?

Honestly thanks to everyone on this site for their continuous help and very valuable time. Someone please help me. I have a final exam coming up and uncertainties are stupidely difficult because of the many discrepancies and what seems to be a requirement of a strong background of knowledge of statistics, which I haven't time to develop. I understand high rep users are busy, but someone please help me.

## closed as unclear what you're asking by ZeroTheHero, Kyle Kanos, John Rennie, Yashas, Jon CusterApr 30 '17 at 18:18

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• If I understand the question correctly, here are some suggestions for improving it. A better question title might be: “How should I interpret an uncertainty given as a percentage without a +/- sign?” A better start to the question body might be: “A question on a practice exam tells me that the uncertainty of a quantity is 4%. Elsewhere in the answers for the exam, uncertainties that differ by a factor of two are both considered acceptable. I am now unsure if a 4% uncertainty means ±4% or ±2%.” In general the question could be more succinct, and the last paragraph is unnecessary. – David Bailey May 1 '17 at 3:43

How uncertainties are presented is defined by convention, and there is no convention universally used by everyone in all situations, so if there is any chance of ambiguity it is important to clearly state the convention you are using. If I were to read that the uncertainty on a physics quantity $x$ is 4%, then like you I would assume that it means that the value is expected to be in the range 0.96$x$ to 1.04$x$. According to the marking scheme for question parts (a)(i) and (b)(iii), however, the examiners appear to accept either this “half-range” convention or an alternate “full-range” convention that a “4% uncertainty” means that the value is expected to be in the range 0.98$x$ to 1.02$x$.

Different conventions are common and can be confusing. In statistics I would assume “±4%” means that the standard deviation of the expected distribution of possible values is 4%, but in metrology it might mean the standard deviation is 2%. In particle physics I would assume “±4%” means that that there is a 16% chance that the value is larger than 1.04$x$ and a 16% chance that it is smaller than 0.96$x$. In engineering “±4%” might be a tolerance meaning that the item being measured should be rejected if the measurement is outside the range 0.96$x$ to 1.04$x$.

A student answering an exam question should just be clear about any assumptions made and any conventions used, but for part (d) the convention doesn’t really matter. The numbers the examiners have provided give a difference of 27% in density between the two rubbers, which for any reasonable interpretation of uncertainty is much too large for the rubbers to be the same. This large difference may be intentional, exactly so that students do not worry that the answer depends on any unspecified assumptions or conventions.

Here is an example of how one might succinctly answer (d):

• "The measured densities of the rubber bung and band are 1.52 g/cm$^3$ ± 0.6% and 1.16 g/cm$^3$ ± 4% respectively. The minimum likely density for the bung (1.52 - 0.6%×1.52 = 1.51 g/cm$^3$) is much, much larger and not consistent with the maximum likely density of the band (1.16 + 4%×1.16 = 1.21 g/cm$^3$), so they do not appear to be made from the same type of rubber."

One could expand the answer noting that one has conservatively assumed that the "4%" is a half-range uncertainty, and that if "4%" is the full-range uncertainty, then the two rubber densities are even more inconsistent.

• While you're explaining conventions, may you please explain how the 'percentage difference' method that is shown works? I have exams coming up, but I promise to edit the question when I get the time, such that it better reflects what I need. – Mathematician May 1 '17 at 15:49