We use displacement encoding pulsed field gradient (PEG) nuclear magnetic resonance to measure Fourier components S-q of flow displacement distributions P(xi) with mean displacement (xi) for Newtonian and non-Newtonian flows through rocks and bead packs. Displacement distributions are non-Gaussian; hence, there are finite terms above second order in the cumulant expansion of In(S-q). We describe an algorithm for an optimal self-consistent cumulant analysis of data, which can be used to obtain the first three (central) moments of a non-Gaussian P(xi), with error bars. The analysis is applied to Newtonian and non-Newtonian flows in rocks and beads. Flow with shear-thinning xanthan solution produces a 15.6 +/- 2.3% enhancement of the variance (7 2 of displacement distributions when compared to flow experiments with water. (C) 2007 Elsevier Inc. All rights reserved.