# How do you state the uncertainty of an instrument?

Here is the resource that I was reading in order to understand the uncertainty at an introductory level: http://study.kis.net.ua/pluginfile.php/58076/mod_resource/content/1/Errors%20%20Uncertainty%20Notes.pdf

I have two questions about it. It says: 'Sometimes you hear that uncertainties should generally be reported as ½ the least count; this is technically correct. But since they should be reported with the same # of decimal places as the instrument, in practice this amounts to stating them as ± the least count.'

By that logic, the uncertainty of a meter rule can be stated both as +/-1mm and 1mm. This seems wrong. What am I missing here?

My attempts to decipher what this phrase means have been futile: 'If the number of decimal places in your uncertainty does not match the measurement, round your uncertainty up so that it does. In most cases, you do this anyway – only when you are reporting a value beyond the least count would you not do this. Decimal uncertainties always have 1 significant figure.'

My problem is not knowing any examples of these two statement. May those who answer please provide examples?

Significant figures is a method for trying to keep track of the bounds of certainty that result from limited precision inputs (a rather crude and not always consistent method, but relatively easy).

Making explicit error estimates and following a set of rules for carrying those through calculations is a different (and largely better) means of keeping track of the bounds of certainty.

You don't want to be applying both at the same time.

If you are a making explicit error estimates and propagating those errors then keep one to two extra figures (relative what the sigfigs rules would tell you) as you work. Then when you get to the end of the calculation you can re-establish the sigfigs methodology by writing the number so that only one digit of error-estimate is needed.

What I have written above, is me prescription for 'how to do it right'. You should be aware of the old saying "right way, wrong way, navy way." (or "army way" or "[just-about-any-institution] way" at the end). It means that sometime you have to follow a rigid prescription that isn't quite right because it is what is expected of you.

In the classroom you may need to do it your instructor's way even if that causes trouble.

• Hello, thank you very much, but my point is that I don't understand what to follow. I don't know what this source is saying. – Mathematician Apr 29 '17 at 17:54
• @dmckee, as a high school physics teacher, I applaud your ending comment. The methods chosen by instructors have a context associated with them, and mixing recommended methods can occasionally lead to unintended inconsistencies. – David White Apr 29 '17 at 17:57
• @DavidWhite Certainly. I teach—at times—our gen ed physical science course, our 'non-majors' intro to physics and our for intro for physics majors, engineers, mathematicians and the like, so I get to struggle with the question of how much and what kind of uncertainty handling to do with different groups of students. It's a tough call every time and I don't want to sound like I am criticizing the use of sigfigs as a first pass at this idea. – dmckee --- ex-moderator kitten Apr 29 '17 at 18:02
• I have no teacher. I am self studying this, so explaining this resource is my only way to understand. – Mathematician Apr 29 '17 at 18:31