Recent studies have shown that in the presence of noise, both fronts propagating into a metastable state and so-called pushed fronts propagating into an unstable state, exhibit diffusive wandering about the average position. In this paper, we derive an expression for the effective diffusion coefficient of such fronts, which was motivated before on the basis of a multiple scale ansatz. Our systematic derivation is based on the decomposition of the fluctuating front into a suitably positioned average profile plus fluctuating eigenmodes of the stability operator. While the fluctuations of the front position in this particular decomposition are a Wiener process on all time scales, the fluctuations about the time-averaged front profile relax exponentially.