This paper investigates cooperative adaptive cruise control (CACC) for mixed platoons consisting of both human-driven vehicles (HVs) and automated vehicles (AVs). This research is critical because the penetration rate of AVs in the transportation system will remain unsaturated for a long time. Uncertainties and randomness are prevalent in human driving behaviours and highly affect the platoon safety and stability, which need to be considered in the CACC design. A further challenge is the difficulty to know the exact models of the HVs and the exact powertrain parameters of both AVs and HVs. To address these challenges, this paper proposes a data-driven model predictive control (MPC) that does not need the exact models of HVs or powertrain parameters. The MPC design adopts the technique of data-driven reachability to predict the future trajectory of the mixed platoon within a given horizon based on noisy vehicle measurements. Compared to the classic adaptive cruise control (ACC) and existing data-driven adaptive dynamic programming (ADP), the proposed MPC ensures satisfaction of constraints such as acceleration limit and safe inter-vehicular gap. With this salient feature, the proposed MPC has provably guarantee in establishing a safe and robustly stable mixed platoon despite of the velocity changes of the leading vehicle. The efficacy and advantage of the proposed MPC are verified through comparison with the classic ACC and data-driven ADP methods on both small and large mixed platoons.