Testing connected cyber-physical systems (CPS) is a complex task. Connected CPS feature complex stochastic dynamic behaviour in interaction with the physical and human environment as well as communication over networks. Devising an oracle for testing connected CPS is a challenge; the oracle should be able to quantitatively reason about the stochastic nature of the interactions between the CPS and its environment. The quantitative reasoning should be sensitive to significant deviations in the dynamics and neglect minor deviations, e.g., due to measurement errors. To address this challenge, we provide the mathematical framework for conformance testing of connected CPS. We define a quantitative measure of closeness for two distributions of trajectories (i.e., output distributions from two distinct stochastic systems that are provided with the same input stimuli) that allows for capturing significant temporal and spatial deviations and neglecting subtle ones. This measure forms the basis for our notion of stochastic conformance, which determines when two stochastic systems conform to each other. We implement our proposed notion of stochastic conformance and compare our notion against a state-of-the-art baseline by applying both approaches to a case study involving a platoon of connected vehicles. Our notion detects a variety of different types of faults whilst allowing subtle deviations resulting from naturally occurring perturbations inherent to CPS.