We present comprehensive derivations for the statistical models and methods for the use of pulsed gradient spin echo (PGSE) NMR to characterize the molecular weight distribution of polymers via the well-known scaling law relating diffusion coefficients and molecular weights. We cover the lognormal and gamma distribution models and linear combinations of these distributions. Although the focus is on methodology, we illustrate the use experimentally with three polystyrene samples, comparing the NMR results to gel permeation chromatography (GPC) measurements, test the accuracy and noise-sensitivity on simulated data, and provide code for implementation.