Help Understanding Uncertainties

[Ali0086]

I'm having a tough time understanding uncertainties and the concept of how to find the absolute and relative uncertainties of functions. I understand how to find the relative and absolute uncertainties in problems like 40±2 m (The absolute uncertainty is 2m and the relative is 0.05 if I'm not mistaken). But I don't remotely understand how to apply this concept to functions. I'm sorry I don't have an exact question I'm having difficulty with so I'll have to just create some of these questions.

For example:
-Find the absolute and relative uncertainty of the sine of 30±0.03 deg.
-Find the absolute and relative uncertainty of log(10.7±0.3)
-Find the absolute and relative uncertainty of ln(6.2±0.4)

I just want to understand the concept of how uncertainties work and how to approach uncertainties regarding functions such as sin,cos, and ln. Any help is greatly appreciated.

 

[happysmiles36]

This is what I think it means.

The absolute uncertainty of sine of 30±0.03 deg is 0.03
Relative would be 0.03/sine of 30

In this case, (sine of 30) is like the 40 in your example.

For log(10.7±0.3) because it is in brackets, your absolute uncertainty would be 0.3 and relative would be 0.3/10.7

For ln(6.2±0.4) again, absolute uncertainty would be 0.4 and relative would be 0.4/6.2

Because these 2 are in brackets, your uncertainty is already with your measured value.



Found this, may help
http://www.capphysics.ca/PhysLab/Phys114115/App%20A%20-%20uncertainties/appA%20propLogs.htm

Hope that was helpful and correct

 

[The Learner]

The way I see it, you're confused with how uncertainties carry through when there are functions. And from what I understand, you would just carry them through the function in a sense.

Let's say you wanted to calculate circumference and to do this you needed to measure r. Imagine your measurement of r gave 40±2cm. So what's our uncertainty in the circumference? The function here is f(x) = 2*pi*r so you calculate circumference to be about 251. But your measurement of r was uncertain so how much could you be off by? You can plug into your function the biggest overestimate (42cm) and then plug in the smallest underestimate (38cm) and then you can get your uncertainty (here you can plug in 2 and you'll get the same thing, but I'm not sure if that holds true in general, I'd say try it out and play with it a bit).

The idea is that the uncertainty of your measurement carries through to the quantity you're calculating. How much that uncertainty matters depends on the specific function and you want to capture the biggest uncertainty represented by this (I don't believe it's always the max and min values of the measurement that give the biggest uncertainties so I don't want to say just plug the uncertainties into the function).

Hope that helps.

 

[composyte]

To understand uncertainties for functions requires a little bit of knowledge in calculus and partial differential equations. Assuming you have this knowledge I will try to help you understand...A general formula for the uncertainty of a function of several variables (you can use this formula for the specific case of one variable as well) is as follows...

Suppose that x,..., z are measured with uncertainties δx,...,δz and the measured values are used to compute the function q(x,...,z). If the uncertainties in x,...,z are independent and random (ie. the errors are not caused by defective equipment) then the uncertainty in q is

δq=[itex]\sqrt{(\frac{∂q}{∂x}δx)^{2}+...+(\frac{∂q}{∂z}δz)^{2}}[/itex]


now I am sure you can find a proof for this formula if it would please you. However just thinking about how it is reasonable might help as well.

First it seems fair to assume they would be added in quadrature, considering that this is the case when always adding uncertainties... The only difference between this formula and the ones found without functions is the partial derivatives.

However, this seems to make reasonable sense as well because you are taking the uncertainty in the value and multiplying it by how greatly the rate of change the function has with respect to that uncertainty. If the function depends greatly on one value over another, than clearly the functions uncertainty should account for that difference. Hope this helps! let me know if something is still unclear as I may be able to provide further explanation..

 

[Ali0086]

To understand uncertainties for functions requires a little bit of knowledge in calculus and partial differential equations. Assuming you have this knowledge I will try to help you understand...A general formula for the uncertainty of a function of several variables (you can use this formula for the specific case of one variable as well) is as follows...

Suppose that x,..., z are measured with uncertainties δx,...,δz and the measured values are used to compute the function q(x,...,z). If the uncertainties in x,...,z are independent and random (ie. the errors are not caused by defective equipment) then the uncertainty in q is

δq=[itex]\sqrt{(\frac{∂q}{∂x}δx)^{2}+...+(\frac{∂q}{∂z}δz)^{2}}[/itex]


now I am sure you can find a proof for this formula if it would please you. However just thinking about how it is reasonable might help as well.

First it seems fair to assume they would be added in quadrature, considering that this is the case when always adding uncertainties... The only difference between this formula and the ones found without functions is the partial derivatives.

However, this seems to make reasonable sense as well because you are taking the uncertainty in the value and multiplying it by how greatly the rate of change the function has with respect to that uncertainty. If the function depends greatly on one value over another, than clearly the functions uncertainty should account for that difference. Hope this helps! let me know if something is still unclear as I may be able to provide further explanation..

Thanks for the well thought out explanation. Unfortunately, I'm in first year calc for university so I don't think I can follow due to no knowledge of partial diff.equations. Anyway to dumb it down so I could understand even better? The other two explanations have helped a bit so I guess I have a working understanding for now.

 

[Ali0086]

The way I see it, you're confused with how uncertainties carry through when there are functions. And from what I understand, you would just carry them through the function in a sense.

Let's say you wanted to calculate circumference and to do this you needed to measure r. Imagine your measurement of r gave 40±2cm. So what's our uncertainty in the circumference? The function here is f(x) = 2*pi*r so you calculate circumference to be about 251. But your measurement of r was uncertain so how much could you be off by? You can plug into your function the biggest overestimate (42cm) and then plug in the smallest underestimate (38cm) and then you can get your uncertainty (here you can plug in 2 and you'll get the same thing, but I'm not sure if that holds true in general, I'd say try it out and play with it a bit).

The idea is that the uncertainty of your measurement carries through to the quantity you're calculating. How much that uncertainty matters depends on the specific function and you want to capture the biggest uncertainty represented by this (I don't believe it's always the max and min values of the measurement that give the biggest uncertainties so I don't want to say just plug the uncertainties into the function).

Hope that helps.

Thanks a lot! It seems to be a bit more clear now.