Stochastic computing (SC) is a promising technique with advantages such as low-cost, low-power, and error-resilience. However so far SC-based CNN (convolutional neural network) accelerators have been kept to relatively small CNNs only, primarily due to the inherent precision disadvantage of SC. At the same time, previous SC architectures do not exploit the dynamic precision capability, which can be crucial in providing efficiency as well as flexibility in SC-CNN implementations. In this paper we present a DPS (dynamic precision scaling) SC-CNN that is able to exploit dynamic precision with very low overhead, along with the design methodology for it. Our experimental results demonstrate that our DPS SC-CNN is highly efficient and accurate up to ImageNet-targeting CNNs, and show efficiency improvements over conventional digital designs ranging in 50~100% in operations-per-area depending on the DNN and the application scenario, while losing less than 1% in recognition accuracy.