# Multiple choice on uncertainty

1. A student measures the density of a metal and calculates answer to be 8543.2kgm^-1 . The result is accurate to +- 1.1%
Which of the following states answer to an appropriete number of sig figs?
A 8.543x10^3
B 8.5x10^3
C 9x10^3
D 8.54x10^3

2. Relative Uncertainty = Absolute Uncertainty/ Measured Value

3. First i got the abosute uncertainty by: 8543.2x0.0011 to get +-93.9752
in order to compair the initial value and the initial+-absolut uncertainty values in order to compair them and determain to how many sf they are identical.
I got 8637 and 8449 and thus i couldnt get the answer.

kuruman
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This is a round off question. You can write the student's answer as 8.5432×103. What is 93.9752 in the same powers of 10? Write it down and then see if you can determine how many figures from the student's answer you can drop as insignificant.

This is a round off question. You can write the student's answer as 8.5432×103. What is 93.9752 in the same powers of 10? Write it down and then see if you can determine how many figures from the student's answer you can drop as insignificant.
Would that be 0.093975x10^3 ? I am not sure if i quite get the concept.

kuruman
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Would that be 0.093975x10^3 ? I am not sure if i quite get the concept.
The concept is simple, you do not report more significant figures than it makes sense. First of all look at your uncertainty, 0.093975×103. It's an uncertainty in the measurement of 8.5432×103. Suppose you reported the value (8.5432±0.093975)×103. Does it make sense to report so many sig figs in the uncertainty? If not, how many sig figs should you report in the uncertainty? Then we will worry about the number itself.

The concept is simple, you do not report more significant figures than it makes sense. First of all look at your uncertainty, 0.093975×103. It's an uncertainty in the measurement of 8.5432×103. Suppose you reported the value (8.5432±0.093975)×103. Does it make sense to report so many sig figs in the uncertainty? If not, how many sig figs should you report in the uncertainty? Then we will worry about the number itself.
So we report to one dp as the uncertainty effects the value starting from the 2nd dp?

kuruman
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So we report to one dp as the uncertainty effects the value starting from the 2nd dp?
Yes, you round off the uncertainty to 1 sig fig. So now you have (8.5432±0.09)×103. Is the last "2" significant? If the uncertainty is ±0.09, would you be able to tell the difference between 8.5432 and 8.5433 or 8.5431? Do you see where this is going?

Yes, you round off the uncertainty to 1 sig fig. So now you have (8.5432±0.09)×103. Is the last "2" significant? If the uncertainty is ±0.09, would you be able to tell the difference between 8.5432 and 8.5433 or 8.5431? Do you see where this is going?
Yes, now i got it. Thank you so much!

haruspex
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The relationship between significant figures and error ranges is something I have never been sure of.
In this question, if we quote ##8.5\times 10^3## as the answer, does that imply 8450 to 8550? The original ±1.1% includes values out of that range. To encompass the whole of it would lead to ##9\times 10^3##, but that seems like overkill.

haruspex
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I would write (8.54±0.09)×103.
Sure, but the requirement in this question is to write it as a single number, no ±. I.e. choose between B and C.

kuruman
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Sure, but the requirement in this question is to write it as a single number, no ±. I.e. choose between B and C.
Why limit the choices between (B) and (C)? I would choose (D) with the understanding that it would be followed by ±0.09 for a complete answer.
As you know, if one writes 8.54±0.09, one implies that about 68% of the measurements would be between 8.63 and 8.45. If one writes 8.5±0.1, then one implies that about 68% of the measurements would be between 8.6 and 8.4. In this case 100×0.1/8.5 = 1.2% to 2 sig figs. So I think we are in a fuzzy area between answering (B) or (D). Error analysis is not an exact science. One needs to be aware of its implicit fuzziness when writing multiple choice questions.

haruspex
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I would choose (D) with the understanding that it would be followed by ±0.09 for a complete answer.
The question as set does not give you the option of adding a range. I think it is clear that we are being asked to represent the precision purely in terms of the number of significant digits. D is definitely out.
As you know, if one writes 8.54±0.09, one implies that about 68% of the measurements would be between 8.63 and 8.45.
Not necessarily. Many lab readings are based on a graduated scale, e.g. we might measure a distance to the nearest mm. The reading is ±0.0005m with a uniform distribution. The Gaussian interpretation emerges, typically, when a number of sources of uncertainty are combined.

• Tom.G
Tom.G
I found this from NIST ('National Institute of Standards and Technology' in the USA), but after reading it found it non-helpful.
https://www.nist.gov/information-technology-laboratory/sed/topic-areas/measurement-uncertainty

The International standard is GUM (Guide to the Expression of Uncertainty in Measurement) from BIPM (Bureau International des Poids et Mesures). At 134 pages, I haven't read it.
http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf

Cheers,
Tom

kimbyd
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I found this from NIST ('National Institute of Standards and Technology' in the USA), but after reading it found it non-helpful.
https://www.nist.gov/information-technology-laboratory/sed/topic-areas/measurement-uncertainty

The International standard is GUM (Guide to the Expression of Uncertainty in Measurement) from BIPM (Bureau International des Poids et Mesures). At 134 pages, I haven't read it.
http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf

Cheers,
Tom
This quote from the above PDF matches with what I would expect:

"7.2.6 The numerical values of the estimate y and its standard uncertainty uc(y) or expanded uncertainty U should not be given with an excessive number of digits. It usually suffices to quote uc(y) and U [as well as the standard uncertainties u(xi ) of the input estimates xi ] to at most two significant digits, although in some cases it may be necessary to retain additional digits to avoid round-off errors in subsequent calculations."

Basically it's saying to retain the two most significant digits of the error, and include the value up to those same two digits.

So for the above example, since the error is +/- 94, it would make sense to report it as "8543 +/- 94". In the notation given in the PDF, the notation would be 8543(94).

haruspex
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report it as "8543 +/- 94"
You are missing the point of this thread. We do not have the luxury of specifying +/-. We are asked to express the uncertainty purely in terms of the number of digits, as in options A to D.

• Tom.G
Tom.G
I would choose "B", 8.5×103, using the following rationale.

Given 8543.2 ±94 indicates that the true value of digits in the "94" position are unknown; that is why they are called "uncertain."
Logically, this indicates that there is no information available for those positions in the original number.
With no information available, the original number would be 85??.?
Ergo, B 8.5×103

Cheers,
Tom

kimbyd
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I would choose "B", 8.5×103, using the following rationale.

Given 8543.2 ±94 indicates that the true value of digits in the "94" position are unknown; that is why they are called "uncertain."
Logically, this indicates that there is no information available for those positions in the original number.
With no information available, the original number would be 85??.?
Ergo, B 8.5×103

Cheers,
Tom
I agree with this statement.

kuruman
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I would choose "B", 8.5×103, using the following rationale.

Given 8543.2 ±94 indicates that the true value of digits in the "94" position are unknown; that is why they are called "uncertain."
Logically, this indicates that there is no information available for those positions in the original number.
With no information available, the original number would be 85??.?
Ergo, B 8.5×103

Cheers,
Tom
I see your argument and I accept it. However, suppose that the question offers the same choices (A) - (D) but specifies instead a 0.2% accuracy. Then one gets 8543.2 ± 17. According to the argument the answer would still be (B) with nothing to reflect the more than 5-fold improvement of accuracy. Answer (B) then implies a range of accuracy values.

I guess that's why the ± is needed.

mfb
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Logically, this indicates that there is no information available for those positions in the original number.
Assuming a Gaussian uncertainty, there is some information available about the third digit (and a tiny bit about the fourth). 8.54 is more likely than 8.59 (by a factor ~1.15).

"You have to give the number without an uncertainty" is bad practice. If you cannot choose, such as in this multiple choice question, 8.5 sounds like the most reasonable answer.

Staff Emeritus
It's a multiple choice question. We're not allowed to make new choices even if we like them better.

8.5 x 103 implies that the true answer is in about a 1% range around 8.5; otherwise we would write 8.4 or 8.6. The given uncertainty is 1.1%. That is enough information to provide the best answer.

Orodruin
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Assuming a Gaussian uncertainty, there is some information available about the third digit (and a tiny bit about the fourth). 8.54 is more likely than 8.59 (by a factor ~1.15).

"You have to give the number without an uncertainty" is bad practice. If you cannot choose, such as in this multiple choice question, 8.5 sounds like the most reasonable answer.
I agree with this. I would also never round an error down, regardless of what "conventional" rounding states, and you should add the decimals you are throwing away from the central value to the error before rounding the error. All of this to be conservative with your measurement and avoid undercoverage.

Now, it is not reasonable to ask high-school students to quote ##\pm## in anything they do (bachelor students hardly know how), and hence we are left with some more or less arbitrary rules for how many significant digits to quote and it is not always clear where the line should be drawn.

I agree with @mfb, I would say 8.5 is the most reasonable choice. If I was allowed to quote errors I would quote ##8.54 \pm 0.10##. The interpretation of this in terms of Gaussian confidence intervals, the confidence intervals have appropriate coverage. An argument can also be made for ##8.543 \pm 0.095##. It is a matter of how precise you feel you can be on the errors.

haruspex
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8.5 x 103 implies that the true answer is in about a 1% range around 8.5; otherwise we would write 8.4 or 8.6. The given uncertainty is 1.1%. That is enough information to provide the best answer.
I quite agree that 8.5 x 103 is the most reasonable answer, but I don't think it is quite as straightforward as you suggest.

The "1% range around 8.5" so implied would be 0.6% either side, i.e. 8.45 to 8.55. This is a distinct tightening, more so if taking the Gaussian interpretation of the original ±1.1%.

On top of that, we have the context that the student is claiming a precision for a result. Writing it in such a way as to reduce the range at all might be regarded as inappropriate.

As several have posted, this is why stating a range is far preferable.

kimbyd
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I see your argument and I accept it. However, suppose that the question offers the same choices (A) - (D) but specifies instead a 0.2% accuracy. Then one gets 8543.2 ± 17. According to the argument the answer would still be (B) with nothing to reflect the more than 5-fold improvement of accuracy. Answer (B) then implies a range of accuracy values.

I guess that's why the ± is needed.
For this I'd suggest a cutoff at 3.1: Error values above 3.1 do not include the error digit. Errors below 3.1 include the error digit.

Why? 3.1 is the (rounded) square root of 10. This is similar to order-of-magnitude rounding. Since the effect of selecting the number of significant figures is the equivalent of order-of-magnitude rounding on the error, it seems like a reasonable choice here.

So, for example:
$$8543 \pm 40 \rightarrow 8.5 \times 10^3$$
$$8543 \pm 13 \rightarrow 8.54 \times 10^3$$

But really, include the error if at all possible.

vela
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