Spatial spread is stochastic, and small numbers at invasion fronts amplify the importance of stochasticity, yet the spread of an invader is typically modeled deterministically. Moreover, variance in the dynamics of invasive spread is an important yet especially neglected aspect of biological invasions: if an invasion were repeated many times, how different would it be each time? Both for predictions and model fitting the stochastic aspect is essential. We present stochastic models for the spread of a sexually reproducing species in discrete time and space. From stochastic processes at the level of individuals, we derive spatially explicit stochastic models for the population level. We include contributions from demographic stochasticity, demographic heterogeneity, and environmental stochasticity. Our stochastic models predict the variance in population size at different locations along the advancing invasion wave and the variance in the rate of spread. Significantly, our models provide direct calculations of the likelihood and so are ideal for fitting to data. We fitted the models to data from thirty replicated invasions of the flour beetle, Tribolium castaneum, under controlled conditions in a laboratory experiment. We show that by including only demographic stochasticity, the variance in the spread rate is dramatically underestimated, by 15 times in our experiment. We show that to adequately model the spatial spread of animal populations it is necessary also to include a stochastic sex ratio, as well as variation in fecundity and dispersal between individuals.