Uncertainty of weighted average

[Mehran90]

Hello,

I have a bunch of reflectances measured, each with a specific uncertainty at different wavelengths. I am trying to do a weighted average, knowing the weights for each measured reflectance. My question is what is the uncertainty of the weighted average?

In other words,
F_ave = Sigma(w_i*F_i) / Sigma(w_i)
if each F_i has a measured uncertainty of s_i, what would be the uncertainty of F_ave?

Thank you for your help

 

[katxt]

Work through the variances. The variance of the sum = sum of the variances
S(Fbar)=sqrt(sig(Si^2.Wi^2))/sig(Wi)
I think I have enough brackets