Animal movement models allow ecologists to study processes that operate over a wide range of scales. In order to study them, continuous movements of animals are translated into discrete data points, and then modelled as discrete models. This discretization can bias the representation of the movement path. This paper shows that discretizing correlated random movement paths creates a biased path by creating correlations between successive turning angles. The discretization also biases statistical tests for correlated random walks (CRW) and causes an overestimate in distances travelled; a correction is given for these biases. This effect suggests that there is a natural scale to CRWs, but that distance-discretized CRWs are in a sense, scale invariant. Perhaps a new null model for continuous movement paths is needed. Authors need to be aware of the biases caused by discretizing correlated random walks, and deal with them appropriately.