How Many Sig Figs Given Error Range?

In summary, the conversation discusses the appropriate number of digits to include when writing a result with a given estimated uncertainty. The question is considered ambiguous and flawed, but a possible solution is to include three digits based on the first digit of the result being a 1.
  • #1
Cosmophile
111
2

Homework Statement


"A calculator displays a result as 1.3250780 x10^7 kg. The estimated uncertainty in the result is +/- 2%. How many digits should be included when the result is written?"

Homework Equations


I'm not sure that there are any relevant equations here, aside from (maybe)

Actual result = Measured result + (Measured result)*(+/- 0.02)

The Attempt at a Solution


1.3250780x10^7 = 13250780

13250780*(0.02) = 265016 (rounding to nearest 1)
13240780 + 265016 =
13515795

Where I have bolded the numbers that do not change when the possible error is added to the measured result. Because I am only sure of these two numbers, does that mean that my number of significant figures should be 2?
 
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  • #2
Hi Cosmo,

Cosmophile said:
I'm not sure that there are any relevant equations here
Perhaps not equations, but some rules could be mentioned.

An interesting question. Note that even the 2% is an estimate -- and the question doesn't reveal if that estimate is a rough estimate from say, squinting an eye at the scale from which the thirteen million is read off, or if it's an accurate estimate from calculating the standard deviation of a set of a hundred observations. On top of that, the question can be considered ambiguous: it's not clear if 'the result' can include reporting the error or not.

In the former case the answer is easy: (13.2 +/- 0.3) (13.3 +/- 0.3) kilotonne should both be considered reasonable, so three digits.
In the latter case things become really difficult: the tacit assumption is that the error is plus or minus one half of the last digit (i.e. 13 means between 12.5 and 13.5). Then two digits makes one lose accuracy unnecessarily. However, 13.3 would then mean 'in the range [13.25, 13.35]' and that's way too accurate.

My short reply would be: three digits. Main argument: the first digit of the result is a 1. Can you unravel the logic in that ?
 
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Likes Cosmophile
  • #3
The question is certainly (hah) flawed, in my opinion. My physics class is being done entirely online, and it is questions like this which cause me to become irritated with the course. Especially when it is my performance in this class which will go on a transcript, and not my ability to solve more "physical" problems, like those posed in Kleppner & Kolenkow.

Woe is me, /plays world's smallest violin.

I've already submitted the assignment (I went with two significant figures, so whoops!) but I appreciate the write-up! I'll give it a moe diligent read when I'm finished with my other assignments. Thank you!
 

1. How do you determine the number of significant figures given an error range?

The number of significant figures in a measurement is determined by the precision of the measuring instrument. To determine the number of sig figs given an error range, count the number of digits in the error range that are known with certainty. This will give you the number of significant figures in the measurement.

2. What is considered a significant figure in an error range?

A significant figure in an error range is any digit that is known with certainty. This includes all non-zero digits and any zeros between non-zero digits. However, trailing zeros after a decimal point are also considered significant figures in an error range.

3. Can there be more significant figures in an error range than in the actual measurement?

Yes, it is possible to have more significant figures in an error range than in the actual measurement. This can happen when the precision of the measuring instrument is greater than the precision of the measurement itself. In this case, the error range will have more digits, but the extra digits beyond the significant figures of the measurement should be considered insignificant.

4. How do you round a measurement to the appropriate number of significant figures when given an error range?

To round a measurement to the appropriate number of significant figures when given an error range, use the following rule: if the first digit after the significant figures is 5 or higher, round up the last significant figure. If the first digit after the significant figures is 4 or lower, keep the last significant figure the same. This rule applies to both addition/subtraction and multiplication/division calculations.

5. Is it possible for a measurement to have an infinite number of significant figures?

No, all measurements have a finite number of significant figures. This is because there is always some level of uncertainty in any measurement, and therefore, there will always be a limit to the number of digits that can be known with certainty.

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