Is there a rule on how to do the final rounding

[chijioke]

Please help take a look at the following:

The sum of the values to reasonable degree of accuracy is either 3 150.745 or 3 153.755. If you take the mean, you have
$$\frac{ 3150.745 + 3153.755 }{ 2 }= 3152.25$$It is stated the the sum to reasonable degree of accuracy is 3 150 to 3 sig figs. My question is how do I know the number of sig figs to round the value, if am not instructed.


The same question apply here. Is there a rule guiding the value to a reasonable degree to a number of sig figs?

 

[Dr.Peterson]

I believe they are showing you how to decide the appropriate rounding:

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The known digits (highlighted) are 315 (3 digits), so those are the digits to retain. You don't know what the next digit should be. (Though for some purposes it would be appropriate to keep one more digit, that is not the rule they are demonstrating.) Note that the 0 is not considered a significant figure, because they have told you there are only 3.

Similarly,

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Here the 11.975 is so close to 12 that you can consider the digits 12 to be known. I think a good case could be made here for retaining three digits (12.0), but they are being conservative.

Keep in mind that this is a subjective issue; there is no absolutely correct answer.

Was nothing explicitly said in this book to explain their reasoning? Look for the first reference to "reasonable degree of accuracy" and tell us how they described that.

 

[Steven G]

View attachment 34566
The sum of the values to reasonable degree of accuracy is either 3 150.745 or 3 153.755.

That statement is not true. They added up the lower bound of each number and got 3150.745 and they added up the upper bound of each number and got 3153.755.

All this means is that the sum of those numbers is somewhere in between 3150.745 and 3153.755.

In case you do not now what I mean by upper & lower bound, then look at this example.

512+/- 0.5.

The lower bound of 512 +/- 0.5 is 512-0.5=511.5
The upper bound 0f 512 +/- 0.5 is 512+0.5= 512.5

 

[chijioke]

I believe they are showing you how to decide the appropriate rounding:

View attachment 34568​

The known digits (highlighted) are 315 (3 digits), so those are the digits to retain. You don't know what the next digit should be. (Though for some purposes it would be appropriate to keep one more digit, that is not the rule they are demonstrating.) Note that the 0 is not considered a significant figure, because they have told you there are only 3.

Similarly,

View attachment 34569​

Here the 11.975 is so close to 12 that you can consider the digits 12 to be known. I think a good case could be made here for retaining three digits (12.0), but they are being conservative.

Keep in mind that this is a subjective issue; there is no absolutely correct answer.

Was nothing explicitly said in this book to explain their reasoning? Look for the first reference to "reasonable degree of accuracy" and tell us how they described that.

My question is, apart from rounding the value 3152.25 to 3 sig figs, can I choose to round it to round any number of sig figs of my choice? Not necessarily 3 sig figs.

 

[Dr.Peterson]

My question is, apart from rounding the value 3152.25 to 3 sig figs, can I choose to round it to round any number of sig figs of my choice? Not necessarily 3 sig figs.

You can choose anything you want; but it will not necessarily be "a reasonable degree of accuracy" based on the information given, and their standards.

The nominal sum of 2581+512+6+53.25 is 3152.25. Given the sigfigs of the addends, the actual sum might be anywhere from 3150.745 to 3153.755; so we can only be sure of the first three digits of the sum, namely 315_. That is why they recommend rounding to three sigfigs, as 3150. This represents the state of your knowledge better than either calling it 3200 (2 sigfigs) or 3152 (4 sigfigs).

You have been given a "rule", which you asked about. You have been "instructed".

If you choose something else, then you would be asked to justify your choice. Can you? And why do you want to do something other than what you are being taught? That is the question.

 

[Steven G]

My question is, apart from rounding the value 3152.25 to 3 sig figs, can I choose to round it to round any number of sig figs of my choice? Not necessarily 3 sig figs.

When it comes to accuracy, 2 and 2.000 are different numbers!

2 means 2 +/- 0.5, while 2.000 means 2.000 +/- 0.0005

Since no number given in the first problem had 7 sf digits, then I would not use 7 sf in rounding.
You had one number up to 4 sf, so using 3 sf is reasonable.

 

[Dr.Peterson]

I should add that the traditional way of rounding a sum is not by significant figures, but by decimal places. By this approach, we would observe that all four addends are accurate to the units place, but the tenths place is not known for all, so we would round to the units place, giving 3152. Sig figs are irrelevant in addition; they are useful in multiplication.

This is just a rough rule of thumb, and does not take into account all possible variation, as the method being taught does. This is part of the reason I said, "Though for some purposes it would be appropriate to keep one more digit, that is not the rule they are demonstrating."

 

[chijioke]

Was nothing explicitly said in this book to explain their reasoning? Look for the first reference to "reasonable degree of accuracy" and tell us how they described that.

This is the reference before the example.

 

[Dr.Peterson]

View attachment 34574
View attachment 34575
This is the reference before the example.

So, the example is intended to clarify what they mean.

I have to say the English is not terribly clear:

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I suppose what they mean by "precision" is the possible error in a number, such as $\pm0.5$. In the example, they added these up to get $\pm1.505$, which implies that the units digit is not certain, which is why they didn't count it as a significant figure in the sum.

It would have been helpful if they explained in words what they are doing in the examples. I suppose they leave that for a teacher to say, which would be something like what I said in my first response.

 

[chijioke]

I should add that the traditional way of rounding a sum is not by significant figures, but by decimal places. By this approach, we would observe that all four addends are accurate to the units place, but the tenths place is not known for all, so we would round to the units place, giving 3152. Sig figs are irrelevant in addition; they are useful in multiplication.

Going with the rule for adding and subtracting numbers, then just as you said the answer would be 3152 to reasonable degree of accuracy. It might interest you to see the exercises they gave under the example.

Now here are the solution.
The solutions to 3, 4,5 and 6 agrees that when adding and subtracting numbers, rounding should be done having number of decimal place in mind not number of sig figs.But solutions 1 and 2 follows different rule which 1 cannot explain. What do you have to say concerning this?

 

[Dr.Peterson]

Going with the rule for adding and subtracting numbers, then just as you said the answer would be 3152 to reasonable degree of accuracy. It might interest you to see the exercises they gave under the example.

I didn't exactly say that the answer should be 3152; by the rule you are apparently being taught, it is 3150. The point was that there are different rules. You follow the rule in use where you are.

The solutions to 3, 4,5 and 6 agrees that when adding and subtracting numbers, rounding should be done having number of decimal place in mind not number of sig figs.But solutions 1 and 2 follows different rule which 1 cannot explain. What do you have to say concerning this?

I think their answer to #1 almost follows their (implied) rule: 34.25±0.005 + 26±0.5 + 10±0.5 = 70.25±1.005, which is between 69.245 and 71.255, so taking their rule strictly, you would say 70 with 1 s.f., as it could be 69, 70, or 71; but that variation is small enough that 2 s.f. would be reasonable. But the answer does agree better with the alternative rule (minimum number of decimal places) that I stated, and which is taught in the pdf you attached.

For #2, they seem to have followed their rule: 126±0.5 + 723±0.5 + 208±0.5 + 366±0.5 = 1423±2, which is between 1421 and 1425, so only the first 3 digits are really known, and they would round to 1420, with 3 s.f. By the rule I learned, we'd call it 1423, though I'd question that.

By the way, they are not using sigfigs for their decisions. They are only stating the number of sigfigs at the end in order to tell you whether to read zeros as significant. They give the answer in terms of decimal places when there are any; otherwise, they give the number of sigfigs rather than, say, telling you that all digits are meaningful, or that it is accurate only to the tens.

For the other problems, they appear to be using the minimum number of decimal places, not their method as I understand it. I definitely wish they had clearly stated what they are doing!

 

[chijioke]

I didn't exactly say that the answer should be 3152; by the rule you are apparently being taught, it is 3150. The point was that there are different rules. You follow the rule in use where you are.


I think their answer to #1 almost follows their (implied) rule: 34.25±0.005 + 26±0.5 + 10±0.5 = 70.25±1.005, which is between 69.245 and 71.255, so taking their rule strictly, you would say 70 with 1 s.f., as it could be 69, 70, or 71; but that variation is small enough that 2 s.f. would be reasonable. But the answer does agree better with the alternative rule (minimum number of decimal places) that I stated, and which is taught in the pdf you attached.

For #2, they seem to have followed their rule: 126±0.5 + 723±0.5 + 208±0.5 + 366±0.5 = 1423±2, which is between 1421 and 1425, so only the first 3 digits are really known, and they would round to 1420, with 3 s.f. By the rule I learned, we'd call it 1423, though I'd question that.

By the way, they are not using sigfigs for their decisions. They are only stating the number of sigfigs at the end in order to tell you whether to read zeros as significant. They give the answer in terms of decimal places when there are any; otherwise, they give the number of sigfigs rather than, say, telling you that all digits are meaningful, or that it is accurate only to the tens.

For the other problems, they appear to be using the minimum number of decimal places, not their method as I understand it. I definitely wish they had clearly stated what they are doing!

Thanks for your reply.