The impressive success of recent deep neural network (DNN)-based systems is significantly influenced by the high-quality datasets used in training. However, the effects of the datasets, especially how they interact with each other, remain underexplored. We propose a state-vector framework to enable rigorous studies in this direction. This framework uses idealized probing test results as the bases of a vector space. This framework allows us to quantify the effects of both standalone and interacting datasets. We show that the significant effects of some commonly-used language understanding datasets are characteristic and are concentrated on a few linguistic dimensions. Additionally, we observe some “spill-over” effects: the datasets could impact the models along dimensions that may seem unrelated to the intended tasks. Our state-vector framework paves the way for a systematic understanding of the dataset effects, a crucial component in responsible and robust model development.