Both rupture and wave propagation affect strong ground motions on the surface. The complexity of finite earthquake source process plays a significant role in determining near-source ground motion characteristics, especially for large events. Spontaneous dynamic rupture modelling has great potential to study ground motions dominated by earthquake source and has been efficiently adopted for physics-based source and ground motion modelling for the last several decades, but the input of the state of initial stress, friction laws and their parametrization are still not well constrained in general. We demonstrate that at least two statistical measures, that is 1-point and 2-point statistics, are needed to quantify the heterogeneity of spatial data and that 1-point statistics such as mean, standard deviation and shape of probability density function of dynamic input parameters, is a separate quantity that we need to consider in addition to 2-point statistics in earthquake source modelling. We show that 1-point statistics of input dynamic parameters such as stress drop can significantly affect resulting kinematic source and near-source ground motions. The standard deviations of input stress drop affect both 1-point and 2-point statistics of kinematic source parameters derived from spontaneous rupture modelling significantly and even systematically. They also strongly control near-source ground motions, especially in the rupture directivity region. Quantifying the characteristics of both earthquake source and ground motions in the same format of 1-point and 2-point statistics may help us to construct a consistent framework for studying the effect of finite earthquake source process on near-source ground motions. © The Authors 2013. Published by Oxford University Press on behalf of The Royal Astronomical Society.