Uncertainty and significant figures

[songoku]

Homework Statement

A student makes measurements from which she calculates the speed of sound as 327.66ms^–1. She estimates that her result is accurate to ±3 %. Which of the following gives her result expressed to the appropriate number of significant figures?
A 327.7ms^–1 B 328ms^–1 C 330ms^–1 D 300ms^–1

Homework Equations

significant figures

The Attempt at a Solution

The maximum value = 327.66 x 1.03 = 337.4898
The minimum value = 327.66 x 0.97 = 317.8302

How to proceed and how to take significant figures into account? I know that for multiplication, the answer should follow the least significant figures, but how to apply it here?

The solution is (C) and I don't how to get it.

Thanks

 

[Stonebridge]

There are a number of ways of thinking about this, all of which arrive at the same conclusion.
The 3% uncertainty translates, as you point out, to a value where the uncertainty lies in the 2nd figure. (2nd from left)
337
317
This means that the best of the 4 alternatives is the one expressed to 2 significant figures. Option C.
330 m/s means that you are confident of the first figure (3) and the uncertainty lies in the 2nd figure (3)
328 m/s would mean you are confident in the 1st 2 figures (3 and 2) and the uncertainty is in the 3rd figure. the 8
300 m/s would mean the uncertainty lies in the 1st figure.

 

[songoku]

There are a number of ways of thinking about this, all of which arrive at the same conclusion.
The 3% uncertainty translates, as you point out, to a value where the uncertainty lies in the 2nd figure. (2nd from left)
337
317

How can we know that the uncertainty lies in the 2nd figure?

This means that the best of the 4 alternatives is the one expressed to 2 significant figures. Option C.
330 m/s means that you are confident of the first figure (3) and the uncertainty lies in the 2nd figure (3)
328 m/s would mean you are confident in the 1st 2 figures (3 and 2) and the uncertainty is in the 3rd figure. the 8
300 m/s would mean the uncertainty lies in the 1st figure.

The same question as above. How to determine the position of the uncertainty?

Thanks

 

[PeterO]

Homework Statement

A student makes measurements from which she calculates the speed of sound as 327.66ms^–1. She estimates that her result is accurate to ±3 %. Which of the following gives her result expressed to the appropriate number of significant figures?
A 327.7ms^–1 B 328ms^–1 C 330ms^–1 D 300ms^–1

Homework Equations

significant figures

The Attempt at a Solution

The maximum value = 327.66 x 1.03 = 337.4898
The minimum value = 327.66 x 0.97 = 317.8302

How to proceed and how to take significant figures into account? I know that for multiplication, the answer should follow the least significant figures, but how to apply it here?

The solution is (C) and I don't how to get it.

Thanks

Notice that the extreme answers are 10 either side of the calculated value. That means we know the answer to the nearest 10, so we express the answer rounded off to the tens figure.

 

[songoku]

Notice that the extreme answers are 10 either side of the calculated value. That means we know the answer to the nearest 10, so we express the answer rounded off to the tens figure.

If this is not multiple choice question, can I answer 320?

 

[Dickfore]

In a number, a digit is reliable iff the error is not bigger than 0.5 x positional value of the digit. A positional value is 10^{n - 1}, where n is the position of the digit on the left from the decimal point, and 10^{-n}, where n is the position of the digit to the right of the decimal point.

In your case, the double of the error is 10 < 20 < 100, so n = 3 is reliable, whereas everything to the right of it unreliable. You could have seen this by your writing answer with the upper and lower bounds. You see that the third digit remains unchanged.

Nevertheless, reliable and significant digit are not the same thing.

This article explains it quite well:
http://www.av8n.com/physics/uncertainty.htm#sec-execsum-sigfig

 

[PeterO]

If this is not multiple choice question, can I answer 320?

NO. That answer would indicate you are just leaving off / ignoring any figures after the last figure you deem accurate.

Of these figures:

300, 310, 320, 330, 340, 350

which one is closest to 327?

 

[PeterO]

In a number, a digit is reliable iff the error is not bigger than 0.5 x positional value of the digit. A positional value is 10^{n - 1}, where n is the position of the digit on the left from the decimal point, and 10^{-n}, where n is the position of the digit to the right of the decimal point.

In your case, the double of the error is 10 < 20 < 100, so n = 3 is reliable, whereas everything to the right of it unreliable. You could have seen this by your writing answer with the upper and lower bounds. You see that the third digit remains unchanged.

Nevertheless, reliable and significant digit are not the same thing.

This article explains it quite well:
http://www.av8n.com/physics/uncertainty.htm#sec-execsum-sigfig

Hold on: the third digit remained unchanged, but the second digit did change !

 

[Dickfore]

Hold on: the third digit remained unchanged, but the second digit did change !

so, what's your point?

 

[PeterO]

so, what's your point?

I had not read your positional descriptions correctly

 

[songoku]

In a number, a digit is reliable iff the error is not bigger than 0.5 x positional value of the digit. A positional value is 10^{n - 1}, where n is the position of the digit on the left from the decimal point, and 10^{-n}, where n is the position of the digit to the right of the decimal point.

In your case, the double of the error is 10 < 20 < 100, so n = 3 is reliable, whereas everything to the right of it unreliable. You could have seen this by your writing answer with the upper and lower bounds. You see that the third digit remains unchanged.

Nevertheless, reliable and significant digit are not the same thing.

This article explains it quite well:
http://www.av8n.com/physics/uncertainty.htm#sec-execsum-sigfig

Sorry I don't find about positional value in the link and I also don't understand about positional value you said.

The error should not bigger than 0.5 x positional value of the digit. I think the error in the question is 10 since the upper and lower bounds are more or less differ by 10 from the actual value. So positional value of the digit must be bigger than 20

You take n = 3 because 20 < 100 = 10^n-1, where n = 3. Then what does n = 3 mean? n is the position of the digit on the left from the decimal point, but I can't interpret what to do with n = 3

NO. That answer would indicate you are just leaving off / ignoring any figures after the last figure you deem accurate.

Of these figures:

300, 310, 320, 330, 340, 350

which one is closest to 327?

What is the meaning of "any figures after the last figure you deem accurate"? Which 'last figure' are you talking about?

Thanks

 

[PeterO]

What is the meaning of "any figures after the last figure you deem accurate"? Which 'last figure' are you talking about?

Thanks

I agree with you that the uncertainty is ±10, therefore the "units" digit is unnecessary.


The number 327 can be rounded off to 3.3 x 10² sometimes written as 330

or as some people write, 3.2 x 10² sometimes written as 320.

This second answer means the all digits after the "tens" were ignored - which in this case gives and error.

I see lots of students who merely write down the first few digits that are displayed on their calculator, paying no attention to the next digit to see if the answer should be rounded up. They know the final answer should be expressed to 2 significant figures, but instead of rounding to 2 figures, they merely leave off everything except the first 2 figures.
I have even seen students express an answer like 32.97 as 32

When you suggested 320 as a possible answer, you were looking like one of those people who just drops off the unnecessary digits, rather than rounding correctly.

 

[songoku]

I agree with you that the uncertainty is ±10, therefore the "units" digit is unnecessary.


The number 327 can be rounded off to 3.3 x 10² sometimes written as 330

or as some people write, 3.2 x 10² sometimes written as 320.

This second answer means the all digits after the "tens" were ignored - which in this case gives and error.

I see lots of students who merely write down the first few digits that are displayed on their calculator, paying no attention to the next digit to see if the answer should be rounded up. They know the final answer should be expressed to 2 significant figures, but instead of rounding to 2 figures, they merely leave off everything except the first 2 figures.
I have even seen students express an answer like 32.97 as 32

When you suggested 320 as a possible answer, you were looking like one of those people who just drops off the unnecessary digits, rather than rounding correctly.

Ok, so first we find the error (uncertainty), which is ± 10, to decide where we should round the number. In this case, it is the "tens". The number will consist of "hundreds" and "tens", which means that we should express the final answer in 2 significant figures.

Am I getting the correct concept here? Thanks

 

[PeterO]

Ok, so first we find the error (uncertainty), which is ± 10, to decide where we should round the number. In this case, it is the "tens". The number will consist of "hundreds" and "tens", which means that we should express the final answer in 2 significant figures.

Am I getting the correct concept here? Thanks

That is correct - 2 significant figures. AND rounded off, not truncated. don't just leave off unwanted digits, round off correctly
That was why 320 was wrong. 320 is just 327 with the 7 ignored. 330 is 327 rounded off.

 

[songoku]

That is correct - 2 significant figures. AND rounded off, not truncated. don't just leave off unwanted digits, round off correctly
That was why 320 was wrong. 320 is just 327 with the 7 ignored. 330 is 327 rounded off.

ok thanks a lot for your help :)