Help explaining unrounding concepts

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In summary, when dividing or multiplying a number with more than 2 decimal places it is customary to round to the nearest significant digit. However, when adding/subtracting a number with more than 2 decimal places, we round to the nearest significant digit and take into account the number of significant digits in the input.
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mathnewb99
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We have some people trying to report four places to the right of the decimal with inputs that rounded to 2 and 3 decimal places. Can you please articulate why it is impossible to guarantee four decimal precision? I have been unsuccessful in my attempts and am looking for help from someone with a formal math background. Many thanks in advance.
 
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  • #2
mathnewb99 said:
We have some people trying to report four places to the right of the decimal with inputs that rounded to 2 and 3 decimal places. Can you please articulate why it is impossible to guarantee four decimal precision? I have been unsuccessful in my attempts and am looking for help from someone with a formal math background. Many thanks in advance.

Hi mathnewb99, welcome to MHB!

There are different schools of thought on how many decimals to report.

Mathematically it is correct to report as many decimals as we want since we would assume that the inputs are exact.

In practice we have to take into account that the inputs are not exact but have a measurement and/or rounding error.
It is then common to report as many decimals as is representative for the precision of the final result.
However, we only do that for the final result. Any intermediate result must be reported with a couple more digits to ensure we do not introduce undesired rounding errors in the final result.

So mathematically we can say that 12.3 + 0.456 = 12.756.
In practice it is conventional to assume that the input 12.3 has an error up to 0.05.
In this example the result will also have an error up to 0.05, so it is common to report the result as 12.8, which is rounded to the same number of decimals as the 'worst' input.
That is, unless it is an intermediate result, in which case we would report it as 12.756.
 
  • #3
Thanks for the reply. That makes a lot of sense.

In the example I have the team is dividing a dollar value with 2 decimal places by a decimal that has 3 decimal places. In this case applying "round to the worst precision" would mean the final result should be reported with 2 decimal places to ensure any rounding errors are properly consumed by that final rounding operation. Does my interpretation sound correct to you?
 
  • #4
mathnewb99 said:
Thanks for the reply. That makes a lot of sense.

In the example I have the team is dividing a dollar value with 2 decimal places by a decimal that has 3 decimal places. In this case applying "round to the worst precision" would mean the final result should be reported with 2 decimal places to ensure any rounding errors are properly consumed by that final rounding operation. Does my interpretation sound correct to you?

For multiplication and division it works slightly different than for addition/subtraction.
We have to look at the number of significant digits instead of the number of decimal digits.
Consider 56 / 2.34 = 23.9316.
The input 56 has 2 significant digits and 2.34 has 3 significant digits.
The worst input has 2 significant digits so the final result should also have 2 significant digits.
So if this is the final result, we would report 24, which shows that we have a precision of about +/- 0.5.
 

FAQ: Help explaining unrounding concepts

1. What is unrounding and why is it important in scientific research?

Unrounding is the process of converting a rounded number back to its original, more precise value. This is important in scientific research because rounded numbers may not accurately represent the true value and can lead to errors in calculations and conclusions.

2. How do you determine the number of significant figures when unrounding?

The number of significant figures in an unrounded number is determined by the number of digits that are known with certainty. This includes all non-zero digits and any zeros between non-zero digits. Zeros at the end of a number may or may not be significant depending on the context.

3. Can you give an example of unrounding in a scientific experiment?

Sure, let's say a scientist measures the mass of a substance to be 3.50 grams, but the instrument used only displays two decimal places. In order to unround this value, the scientist would need to determine the number of significant figures, which in this case is three. The unrounded value would then be 3.500 grams.

4. Are there any rules or guidelines for unrounding numbers?

Yes, there are a few rules to keep in mind when unrounding numbers. First, the unrounded value should have the same number of significant figures as the original rounded value. Second, if the last digit of the rounded value is 5, the unrounded value should be rounded up if the preceding digit is odd, and rounded down if the preceding digit is even.

5. Can unrounding lead to more accurate results in scientific calculations?

Yes, unrounding can lead to more accurate results in scientific calculations by providing a more precise value for calculations. However, it is important to keep in mind that unrounding cannot add more information to the original measurement and the accuracy of the final result will still be limited by the accuracy of the original measurement.

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