Event-by-event fluctuations in the elliptic-flow coefficient are studied in PbPb collisions at  TeV using the CMS detector at the CERN LHC. Elliptic-flow probability distributions for charged particles with transverse momentum and pseudorapidity are determined for different collision centrality classes. The moments of the distributions are used to calculate the coefficients based on cumulant orders 2, 4, 6, and 8. A rank ordering of the higher-order cumulant results and nonzero standardized skewness values obtained for the distributions indicate non-Gaussian initial-state fluctuations. Bessel–Gaussian and elliptic power fits to the flow distributions are studied to characterize the initial-state spatial anisotropy.