sig figs

[charmanderp]

Only on ATARNotes could the matter of significant figures catalyse such an argument.

[ecvkcuf]

Here is the 'simplified' derivation of S.F.. I'm not sure about your background, but I assume you have relatively strong mathematics which is why you are asking about this. There isn't any point for me to keep discussing this with you, S.F. in my opinion is quite useless once you are outside of high-school (some undergraduate work still use it, but many use better methods).

1. Why S.F. is not a good method.
Suppose we want to deliver a volume of 90 mL with a 10.00 mL pipette. The implicit error of the 10.00mL pipette is mL. Since we need to do this 9 times, the final result should be . The final answer should only be accurate to 3 S.F. given the relative uncertainty of the initial measurement.
However, by the rules of S.F., we have
This is one of the reasons why we don't use S.F. for proper treatment of uncertainties.

2. Derivation of S.F. laws
Suppose we have a quantity , then its significance can be approximated by (this is a rough approximate of what the sf actually does. Doing the following calculations become a lot more messy if we use the proper definition, as the proper definition involves a discontinuous function)

i. Multiplication and division by an exact quantity



ii. Multiplication of two quantities with uncertainties

Ignoring the second order uncertainties, we get

Here, two things can happen. Firstly, , therefore where and , so
Alternatively, , implying , thus , thus , in other words, we take the sf of the quantity with fewer sf.
Thus, we arrive at the final rule

iii. Taking the reciprocal

The significance is


iv. Division of two quantities with uncertainties
Combining ii. and iii., we arrive at

v. Addition and Subtraction

Applying the scale argument, we see that absolute errors are important here, not relative errors. S.F. is not used here.

vi. Logarithms

It can be shown for reasonably sized values of , the digit-significance (i.e. the decimal place) of the absolute error is , since is small for small , we have thus shown that , that is, the significance of the error (i.e. the decimal place) is equal to the number of sf in x.

vii. Exponentials
A similar argument to vi. can be constructed to show that , that is, the decimal place of the initial error is equal to the number of sf in the exponential.

viii. Other transcedentals

For range of values of that are of interest, if , then .
This really depends on what kind of function you have. If , then , which goes to and at certain x values.
On the other hand, the power functions will always behave nicely for reasonable values of n. (Which makes sense in terms of rules ii, iii and iv)


The following are methods I use.

3. Method of relative errors
This is more popular at lower level undergraduate studies.

Suppose we have a bunch of quantities and so on. Define the relative error to be

Rule of addition: Use absolute uncertainties
Rule of multiplication:
Rule of transcendentals:

4. Method of standard deviations
This is the most popular method for treating uncertainties of physical quantities, as everything is almost always normally distributed. Since the normal distribution is a nice function, the rules for propagation is exact. For some non-linear functions, the errors need to be relatively small.

Suppose we have a bunch of quantities with standard deviations and so on.

Rule of addition: Use absolute uncertainties
Rule of multiplication:
Rule of transcendentals: , can be generalised to multivariable functions using partial derivatives and the covariance matrix.


I'm out.

[yawho]

Suppose we want to deliver a volume of 90 mL with a 10.00 mL pipette.  90 mL is the true physical value?

It's not about what we want to do, it's about how a measurement compares with the physical world.

Let's for instance look at the pipette. Let's presume we have a pipette which has a physical volume of exactly . The manufacturer tests the pipette, and find its volume to be , so the measurement range is . Since this braces the actual physical value, it is a good measurement.

We then accurately deliver 9 aliquots using this pipette. The total volume of the aliquots is .

If we use standard methods to treat the uncertainty, our estimate for the volume delivered will be , the measurement range is thus 89.955mL to 90.045mL. This is also a good measurement, as the measurement range braces the true physical value.

If we were to use s.f. only, then our estimate for the volume delivered will be , implying that we are confident the last digit is 0. However, the true physical value rounds to 90.01 mL. So in fact, s.f. has been shown to be inaccurate for this case.

Your explanation was based on sf(cx)~sf(x) , so you are saying the number of sig figs of the sum of repeated measurements equals the number of sig. figs of a single measurement.
Please explain how did you arrive at this conclusion? I could not find any authority on this.

'the general rule that uncertainties scale linearly with multiplcation' Did you mean the |uncertainty| (i.e. |+/- ...|) of the sum equals the sum of the |uncertainties| of the single measurements? I totally agree if that is the case.

[thushan]

Suppose we want to deliver a volume of 90 mL with a 10.00 mL pipette.  90 mL is the true physical value?

It's not about what we want to do, it's about how a measurement compares with the physical world.

Let's for instance look at the pipette. Let's presume we have a pipette which has a physical volume of exactly . The manufacturer tests the pipette, and find its volume to be , so the measurement range is . Since this braces the actual physical value, it is a good measurement.

We then accurately deliver 9 aliquots using this pipette. The total volume of the aliquots is .

If we use standard methods to treat the uncertainty, our estimate for the volume delivered will be , the measurement range is thus 89.955mL to 90.045mL. This is also a good measurement, as the measurement range braces the true physical value.

If we were to use s.f. only, then our estimate for the volume delivered will be , implying that we are confident the last digit is 0. However, the true physical value rounds to 90.01 mL. So in fact, s.f. has been shown to be inaccurate for this case.

Your explanation was based on sf(cx)~sf(x) , so you are saying the number of sig figs of the sum of repeated measurements equals the number of sig. figs of a single measurement.
Please explain how did you arrive at this conclusion? I could not find any authority on this.

'the general rule that uncertainties scale linearly with multiplcation' Did you mean the |uncertainty| (i.e. |+/- ...|) of the sum equals the sum of the |uncertainties| of the single measurements? I totally agree if that is the case.

Yawho, I've already sent you a polite PM telling you to be a little more tactful in the way you write. Mao is one of the most respected members on AN and his expertise on Chemistry knows no bounds (I should know, I've worked with him personally). You are being very rude to him in the way that you talk - you are coming across as arrogant and rude - and you are, quite frankly, wasting Mao's time when he could be doing more useful things. Think about how much bother you are giving other people with your smart-ass comments (I've looked at Mao's arguments and I understand them and they are quite watertight) before posting again on this forum (and other threads I've noticed you've done the same thing). Cut it out otherwise we will take action.

You might be thinking that we are being tight, but you'll understand where we are coming from if you take it from our perspective.

Locked.