Species abundance distributions are an essential tool in describing the biodiversity of ecological communities. We now know that their shape changes as a function of the size of area sampled. Here we analyze the scaling properties of species abundance distributions by using the moments of the logarithmically transformed number of individuals. We find that the moments as a function of area size are well fitted by power laws and we use this pattern to estimate the species abundance distribution for areas larger than those sampled. To reconstruct the species abundance distribution from its moments, we use discrete Tchebichef polynomials. We exemplify the method with data on tree and shrub species from a 50 ha plot of tropical rain forest on Barro Colorado Island, Panama. We test the method within the 50 ha plot, and then we extrapolate the species abundance distribution for areas up to 5 km2. Our results project that for areas above 50 ha the species abundance distributions have a bimodal shape with a local maximum occurring for the singleton classes and that this maximum increases with sampled area size.