Uncertainty significant digits

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Discussion Overview

The discussion revolves around the treatment of uncertainty in measurements, specifically how to express values with associated uncertainties in a way that respects significant digits. Participants explore the implications of significant digits in relation to uncertainty and how these concepts should be represented in final answers.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • Some participants propose that the uncertainty should only contain one significant digit, leading to questions about whether to express the final answer as 600 +/- 100 or keep it as 570 +/- 100.
  • Others argue that expressing the uncertainty as 570 +/- 100 suggests a precision that may not align with the significant digits of the value itself, leading to confusion.
  • A participant mentions that using 600 +/- 100 seems more logical, as it aligns the significant digits of the value and the uncertainty.
  • There is a suggestion that a more reasonable uncertainty might be 570 +/- 5, indicating uncertainty in the tens place rather than the hundreds place.
  • Some participants question the grading of an answer that used 600 +/- 100, suggesting that it may have been marked incorrect despite seeming reasonable.
  • Discussion also touches on whether units can exist in exponents, with some participants asserting that they cannot, while others seek clarification on this point.
  • A later reply emphasizes not to mix uncertainties with significant digits, explaining that different expressions imply different ranges of actual values.
  • One participant raises a question about why certain quantities are treated as vectors while others are not, suggesting a potential need for a new thread to explore this topic further.

Areas of Agreement / Disagreement

Participants express differing views on how to handle significant digits and uncertainties, with no consensus reached on the best approach. There is also disagreement regarding the treatment of units in exponents and the classification of quantities as vectors or scalars.

Contextual Notes

Participants note that the historical context of significant digits may have influenced current practices, particularly in relation to older calculation methods like slide rules. There is also mention of potential misunderstandings in grading related to these concepts.

MathewsMD
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When dealing with the intermediate value of example: 570 +/- 100, how do you put this into it's final form? If I'm not mistaken, doesn't the uncertainty only contain 1 significant digit? Doesn't the uncertainty digit have to correspond to the final significant digit on the actual value as well? In that case, is the final answer 600 +/- 100 or is it appropriate to leave as 570 +/- 100 as the final answer?
 
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MathewsMD said:
When dealing with the intermediate value of example: 570 +/- 100, how do you put this into it's final form? If I'm not mistaken, doesn't the uncertainty only contain 1 significant digit? Doesn't the uncertainty digit have to correspond to the final significant digit on the actual value as well? In that case, is the final answer 600 +/- 100 or is it appropriate to leave as 570 +/- 100 as the final answer?
It seems to me that there is some dissonance between 570 (with two sig. digits) and the uncertainty 100 (with one sig. digit). I would go with 600 ##\pm## 100 - that makes more sense to me.

Something more reasonable would be 570 ##\pm## 5. This suggests that there is uncertainty in the tens place (7 digit), but having 570 ##\pm## 100 suggests uncertainty in the hundreds place (5 digit), which makes the 7 digit worthless.
 
Mark44 said:
It seems to me that there is some dissonance between 570 (with two sig. digits) and the uncertainty 100 (with one sig. digit). I would go with 600 ##\pm## 100 - that makes more sense to me.

Something more reasonable would be 570 ##\pm## 5. This suggests that there is uncertainty in the tens place (7 digit), but having 570 ##\pm## 100 suggests uncertainty in the hundreds place (5 digit), which makes the 7 digit worthless.

That's what I was thinking too...i went with 600 and got the mark wrong.

Also, what happens if you have units in the exponent? Just curious, is it possible?
 
MathewsMD said:
That's what I was thinking too...i went with 600 and got the mark wrong.

Also, what happens if you have units in the exponent? Just curious, is it possible?

Yes/no?
 
MathewsMD said:
That's what I was thinking too...i went with 600 and got the mark wrong.
Was this a computer-scored problem or one that was graded by a person? If the latter, I would talk to this person and find out why that answer was considered incorrect.
MathewsMD said:
Also, what happens if you have units in the exponent? Just curious, is it possible?
I don't believe you can have units in the exponent. Any units would have to cancel, leaving an exponent with no units. If I'm wrong in this, please show me an example where things are otherwise.
 
Mark44 said:
Was this a computer-scored problem or one that was graded by a person? If the latter, I would talk to this person and find out why that answer was considered incorrect.

I don't believe you can have units in the exponent. Any units would have to cancel, leaving an exponent with no units. If I'm wrong in this, please show me an example where things are otherwise.

Yep. I actually spoke with the lab coordinator and the explanation I received went along the lines of "You have 570 +/- 100 in the intermediate value, and this shows more precision than 600 +/- 100." Maybe I misinterpreted him, but it didn't really make sense to me since an extra digit is carried in the calculations. I honestly don't care that much about the mark but maybe I'll ask again for another explanation. Although unnecessary marks off aren't fair, I rather just know this for my personal knowledge. I may have done an erroneous calculation and not canceled units properly, but I'll try to find the example I was working on a while ago and post it if I find it. As long as there is no method to actually consider units in exponents in ordinary equations, that should be helpful enough since units most likely do cancel in the aforementioned problem.

My last question is: why are some quantities vectors while others aren't? For example, we can calculate both current and current density, but why do we only consider current density to be a vector and current a scalar quantity? Is it a purely arbitrary convention or is it something more mathematically fundamental? I understand vectors like forces and displacement have directions and magnitudes associated with them, but I don't quite understand why we don't do the same thing for quantities like current? Is it to simplify equations only?
 
MathewsMD said:
When dealing with the intermediate value of example: 570 +/- 100, how do you put this into it's final form? If I'm not mistaken, doesn't the uncertainty only contain 1 significant digit? Doesn't the uncertainty digit have to correspond to the final significant digit on the actual value as well? In that case, is the final answer 600 +/- 100 or is it appropriate to leave as 570 +/- 100 as the final answer?

Don't mix uncertainties and significant digits.

570 +/- 100 is a statement that the actual value lies between 470 and 670 with a certain probability. 600 +/- 100 is a statement that the actual value lies between 500 and 700 with the same probability, and that's a different thing.

It made sense to use significant digits in intermediate values back when people used slide rules for intermediate calculations, so the method of computation itself introduced uncertainties that were directly related to the digit position. But that was then.
 
MathewsMD said:
My last question is: why are some quantities vectors while others aren't? For example, we can calculate both current and current density...

Suggest that you start a new thread for the new question.
 
Nugatory said:
Don't mix uncertainties and significant digits.

570 +/- 100 is a statement that the actual value lies between 470 and 670 with a certain probability. 600 +/- 100 is a statement that the actual value lies between 500 and 700 with the same probability, and that's a different thing.

It made sense to use significant digits in intermediate values back when people used slide rules for intermediate calculations, so the method of computation itself introduced uncertainties that were directly related to the digit position. But that was then.

Hmm..okay so do you mind expanding on which of the two expressions would be more accurate?
 

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