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Mathematical models of Neolithic spread use dispersal histograms to estimate some of the parameters necessary to obtain quantitative spread rates that can be compared to those inferred from the archaeological record. However, it has been never determined if dispersal histograms are a reasonable approximation to the complete distribution of dispersal distances. Indeed, it is unknown if long-distance dispersal events are important in Neolithic spread, similarly to what happens in many ecological invasions. In this paper, we first exemplify the possible importance of long-distance dispersal by using a detailed histogram for a modern, industrialized population. We show that using such an histogram yields substantially faster spread rates than those of Neolithic waves of advance, and that this is due to the existence of long-distance dispersal events (of several hundred kilometers). Next we address the question of whether such a behavior is also observed in pre-industrial populations. For this purpose, we use a complete set of dispersal distances for the individuals of a pre-industrial population for the first time, and we find that long-distance dispersal events (i.e., of several hundred kilometers) are absent. We also show that, using this complete set of dispersal distances, the spread rates predicted by a mathematical model are consistent with those of the Neolithic, both in continental Europe and in Scandinavia. Moreover we observe, quite surprisingly, that computing histograms (even with only 4 bins) from the complete set of individual distances introduces negligible changes in the results. We argue that these results (the absence of long-distance dispersal events, the agreement with the archaeological record, and the validity of the histogram approach) imply that the propagation of Neolithic waves of advance can be described using a sound mathematical approach, which also yields reliable estimates on the relative importance of demic and cultural diffusion. This is applied to several case studies (Europe, Scandinavia and some specific ceramic cultures in Neolithic Europe). • We apply a pre-industrial census to estimate dispersal distances between birthplaces of parent and their children. • Dispersal histograms (even with only 4 bins) are a reasonable approximation to the complete set of dispersal distances. • We analyze the spread of the Neolithic in Europe, Scandinavia and some specific ceramic cultures in Neolithic Europe. [ABSTRACT FROM AUTHOR]