Error propagation with averages and standard deviation

[rano]

Ok thank you for the clarification. I got mixed up because a few posts back the errors were additive rather than subtractive (but I see now the variables were switched). So we would report mean ± σ_X from your equation, where σ_X is the error propagated standard deviation of the population.

This analysis has been very helpful; I've searched through 3 textbooks and numerous websites but this has been the only discussion I have seen on the subject.

 

[viraltux]

Ok thank you for the clarification. I got mixed up because a few posts back the errors were additive rather than subtractive (but I see now the variables were switched). So we would report mean ± σ_X from your equation, where σ_X is the error propagated standard deviation of the population.

This analysis has been very helpful; I've searched through 3 textbooks and numerous websites but this has been the only discussion I have seen on the subject.

where σ_X is the [STRIKE]error propagated[/STRIKE] standard deviation of the population. (Remember X are the real values, the error has been accounted for in σ_X)

Good Luck!

 

[viraltux]

So we would report mean ± σ_X

Oops, no, no, no, hold it right there. you want to report the estimation of E(Y) with \bar{Y} ?

σ_X is the s.d. for the population taking into account the errors (removing them), if this is what you want then we already discussed how you get that, but if you want to report \bar{Y} then you cannot get rid of the error and you need to report \bar{Y}±σ_Y.

Think about this example, imaging the concentration of sugar is 1 in the three samples, when we measure it we will get that the errors account exactly for the the machine and human error, that means σ_X=0, but you cannot get rid of the errors if you are interested in the estimation of E(Y), you cannot tell \bar{Y} ± 0.

In this example you could say all three have the same value because σ_X=0, but you don't know that value, so whatever that value is will be within an error ±σ_Y which is additive, now I see why you were insisting the additive model, anyway, you need to simply report:
\bar{Y} ± \sigma_Y

which already accounts for all the human errors and machine ones. At the end it was just that, go figure, ha!

 

[Joel Levitt]

It depends on which question you are asking. Are you asking what is the average weight of the three rocks (error = 2.4g) or are you asking what is the probability of finding a rock with a specified weight (depends on the distribution that applies, we know that there are almost uncountably many tiny rocks on Earth and only one as big as the Eurasian continental mass, unless you count the planet itself as a rock)?

Applied math is a tool for answering many questions, but only after the questions have been carefully posed.