# Propagation of uncertainty with some constants

• I
• happyparticle
In summary, the OP wants to find the uncertainty in a function given a value and a function that calculates the propagation of uncertainty. However, he is not sure how to do it. He suggests using a simplified version of Differential Calculus that uses ω and δω2.
happyparticle
TL;DR Summary
How to calculate the propagation of uncertainty with some constants
Hi,

I have a value ##100 \pm 0.1## and a function ##\frac{V}{E} = \frac{1}{\sqrt{1 + (\omega r c)^2}}## and I would like to find the uncertainty.
Where r = 1000 and c = ##5 \cdot 10^{-8}## are constants.
However, I'm not sure to understand how.

Here's what I think and did.
Since I multiply the uncertainty by a constant.

##\sigma= (1000 \cdot 5 \cdot 10^{-8}) \cdot 0.1 = ##

and then for the power
##\sigma= \frac{2 (5\cdot 10^{-6}) \cdot (\sqrt{\omega r c)}}{\omega r c}##

Where I'm using this formula ##\frac{\sigma_f}{f} = \frac{n \sigma_a}{a}##

Say ##Q \equiv V/E##, then ##\sigma(Q) = (\partial Q/\partial \omega) \sigma(\omega) = -(\omega r^2 c^2 / [1+(\omega rc)^2]^{3/2}) \sigma(\omega)##

EpselonZero said:
Summary:: How to calculate the propagation of uncertainty with some constants

Hi,

I have a value ##100 \pm 0.1## and a function ##\frac{V}{E} = \frac{1}{\sqrt{1 + (\omega r c)^2}}## and I would like to find the uncertainty.
What variable is the value that you have? And what uncertainty do you want to find?

Dale said:
What variable is the value that you have? And what uncertainty do you want to find?
The value is 100 and since the value has an uncertainty If I use this value in a formula I have to take into account the propagation of this uncertainty.

EpselonZero said:
The value is 100
The value of what is 100? I see three possible variables that could be 100, but you just keep saying that “the value” is 100 without any hint which value you are talking about.

I am done here. You have wasted people’s time here twice. I could have already answered your question if you had bothered to be clear either the first time or when I asked for clarification. I am not going to waste my time any more.

100 is just an arbitrary variable, it could be 1000 it could be 10. I gave an example to show exactly what I meant. I could use any other function with 1 variable and some constant. It could be ##\omega##, r or c. That is why I put those 3 together. However, since I wrote that r = 1000 and c = 5E-8, I don't know what you mean?

I don't understand why you tell me this is not clear and that I waste people's time. I thought it was obvious that I was talking about ##\omega##, I mean I didn't even think about if it was clear or not, I just wrote as it, since r and c has their values, I really thought it was obvious, my bad if it wasn't.

Maybe I thought It was clear because I don't understand this concept really well, which is possible and that's why I'm asking this question.

Last edited by a moderator:
If the OP had only made it clearer by explicitly choosing one of ω, r or c then there would be no problem here. It was probably 'too obvious' for him.
It does show, however, how the algebraic approach make things much clearer (albeit a bit less friendly, initially). Just expanding the equation with ω replaced by ω+δω would show what's going on. Ignore terms higher than δω2 and you're there.

@EpselonZero I wrote down the answer in post #2, does it make sense?

ergospherical said:
@EpselonZero I wrote down the answer in post #2, does it make sense?
Yes - except that it's very minimalist for the OP. It's actually harder, conceptually than your two line answer implies. I suspect that answer just appeared as a bit of a blur and he passed over it to encounter getting his backside kicked about bad presentation .
@EpselonZero The OP needed help digging himself out of the quandary.
I'd suggest that
1. Numbers should be avoided; stick with all the symbols until the end.
2. Add the uncertainty δω to ω and insert (ω+δω) where you had ω. Expand the resulting expression and ignore terms with δω2 and then reach for your calculator.
I know that's the basis of Differential Calculus but there's no harm in being explicit when explaining things to an elementary question.

ergospherical

## 1. What is the concept of propagation of uncertainty?

The propagation of uncertainty refers to the process of estimating the uncertainty in the result of a calculation or measurement that involves multiple variables with their own uncertainties. It allows scientists to determine the overall uncertainty in a final result by taking into account the uncertainties of each individual variable.

## 2. How is uncertainty propagated with constants?

With constants, the uncertainty is propagated using a mathematical formula that takes into account the uncertainties of each constant and how they affect the overall uncertainty of the final result. This formula is known as the law of propagation of uncertainty or the law of error propagation.

## 3. What is the importance of considering constants in the propagation of uncertainty?

Constants play a crucial role in the propagation of uncertainty as they can significantly affect the overall uncertainty of a result. By taking into account the uncertainties of constants, scientists can obtain a more accurate and reliable estimate of the uncertainty in their final result.

## 4. How do you calculate the uncertainty of a result with constants?

The uncertainty of a result with constants can be calculated using the law of propagation of uncertainty, which involves taking the partial derivatives of the result with respect to each variable and multiplying them by their corresponding uncertainties. These values are then squared, summed, and the square root is taken to obtain the overall uncertainty of the result.

## 5. Can the uncertainty of a result be reduced by manipulating constants?

No, the uncertainty of a result cannot be reduced by manipulating constants. The uncertainty of a result is determined by the uncertainties of the variables involved and cannot be changed by manipulating constants. However, by improving the accuracy of the constants, the overall uncertainty of the result can be reduced.

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