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