I Error propagation and symmetric errors

AI Thread Summary
The discussion centers on the interpretation of symmetric errors in error propagation, particularly in the context of calculating kinetic energy from velocity. The formula derived suggests that if velocity has symmetric errors, kinetic energy should also reflect this symmetry. However, examples show that when calculating specific values, the uncertainties in velocity do not remain symmetric, raising questions about the relationship between symmetric errors and their propagation. Clarity is emphasized between relative uncertainties and absolute uncertainties, as symmetric absolute uncertainties do not guarantee symmetric relative uncertainties. Proper notation is also highlighted to avoid confusion between uncertainties and derivatives.
Malamala
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Hello! I am a bit confused about how to interpret symmetric error when doing error propagation. For example, if I have ##E = \frac{mv^2}{2}##, and I do error propagation I get ##\frac{dE}{E} = 2\frac{dv}{v}##. Which implies that if I have v being normally distributed, and hence having a symmetric error (by this I mean that the upper and lower limit of the uncertainty interval in the value is the same), the same is true for E, as that formula is predicting just one ##dE## and people would quote (at least in most of the papers I read) it as ##E \pm dE##. However if I have say, ##E = 1000## eV, ##dE = 100## eV, ##m = 100## amu and I calculate the associated speed, I get, for ##E = 1000## eV, ##v = 43926## m/s, for ##E = 1100## eV, ##v = 46070## m/s and for ##E = 900## eV, ##v = 41672## m/s and as you can see, the associated uncertainties on v are not symmetric. So how should I think of the symmetric uncertainty given when doing error propagation, as when plugging in the numbers this is not the case. Doesn't it mean that if I have a symmetric (gaussian) error on ##v## the error on ##E## will not be symmetric? And is this not the case most of the time, except when there is a linear relationship between the variables? Thank you!
 
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First, you need to decide if you are going to analyze and report relative uncertainties or uncertainties. You are confusing things by mixing them. ##\sigma_v## is the uncertainty in ##v## and ##\frac{\sigma_v}{v}## is the relative uncertainty in ##v##. If one is symmetric then the other is usually not, so you need to be clear about what you are saying is symmetric.

By the way, I would not use ##dv## for an uncertainty, I would use ##\sigma_v##. Your notation risks lots of confusion with derivatives.
 

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