Tuesday, 2 June 2015

statistics - Why is propagation of uncertainties quadratic rather than linear?




1) Up until now, during practical work sessions, I always used these formulas for uncertainty propagation:


if C=A+B or C=AB ΔC=ΔA+ΔB

if C=AB or C=AB ΔCC=ΔAA+ΔBB
if C=Am ΔCC=|m|ΔAA


These formulas are derived from the expression for the differential of a function of multiple variables:


if C=f(A,B) dC=CAdA+CBdBΔC=CAΔA+CBΔB


and I think that makes sense because the value C is just a function of two variables that happen to be measurements.


2) But this morning, I was told that this is wrong and I should actually use these instead:


if C=A+B or C=AB (ΔC)2=(ΔA)2+(ΔB)2

if C=AB or C=AB (ΔCC)2=(ΔAA)2+(ΔBB)2


which are apparently derived from a general formula:


(ΔC)2=(Cx1Δx1)2+(Cx2Δx2)2+...


or



ΔC=(Cx1Δx1)2+(Cx2Δx2)2+...


3) So:



  • Why is this formula better?

  • Where does it come from?

  • What does it actually represent? (Do I recognize the shape of a norm in that last formula?)

  • What is wrong with the formula of the differential?


I'm asking a lot, but the propagation of uncertainty always confused the ship out of me.




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