The performance of optical filters with resonant waveguide gratings is investigated numerically in a stochastic context, assuming random fluctuations of various design variables. Specifically, we derive stochastic models based on polynomial chaos expansions, whose involved coefficients are obtained by computing spectral projections via sparse-grid quadrature. The latter exploits purely deterministic results from a rigorous coupled-wave analysis solver and requires less simulation data than standard Monte Carlo (MC) techniques. The statistical moments of the filter's spectral response are calculated reliably, as the comparison against reference results from MC analysis verifies, and the extraction of the Sobol indices reveals the structure's sensitivity with respect to specific design parameters. Moreover, the present analysis clearly points out that neglecting even small geometric variations in the filter design may produce misleading conclusions regarding the corresponding performance, with undesirable consequences in real-life applications.