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Public transit systems are the backbone of urban mobility systems in the era of urbanization. The design of transit schedules is important for the efficient and sustainable operation of public transit. However, previous studies usually assume fixed demand patterns and ignore uncertainties in demand, which may generate transit schedules that are vulnerable to demand variations. To address demand uncertainty issues inherent in public transit systems, this paper adopts both stochastic programming (SP) and robust optimization (RO) techniques to generate robust transit schedules against demand uncertainty. A nominal (non-robust) optimization model for the transit frequency setting problem (TFSP) under a single transit line setting is first proposed. The model is then extended to SP-based and RO-based formulations to incorporate demand uncertainty. The large-scale origin-destination (OD) matrices for real-world transit problems make the optimization problems hard to solve. To efficiently generate robust transit schedules, a Transit Downsizing (TD) approach is proposed to reduce the dimensionality of the problem. We prove that the optimal objective function of the problem after TD is close to that of the original problem (i.e., the difference is bounded from above). The proposed models are tested with real-world transit lines and data from the Chicago Transit Authority (CTA). Compared to the current transit schedule implemented by CTA, the nominal TFSP model without considering demand uncertainty reduces passengers' wait times while increasing in-vehicle travel times. After incorporating demand uncertainty, both stochastic and robust TFSP models reduce passengers' wait times and in-vehicle travel times simultaneously. The robust TFSP model produces transit schedules with better in-vehicle travel times and worse wait times for passengers compared to the stochastic TFSP model.