A challenge in accelerator magnet design is the strong nonlinear behavior due to magnetic saturation. In practice, the underlying nonlinear saturation curve is modeled according to measurement data that typically contain uncertainties. The electromagnetic fields and in particular the multipole coefficients that heavily affect the particle beam dynamics inherit this uncertainty. In this paper, we propose a stochastic model to describe the uncertainties and we demonstrate the use of generalized polynomial chaos for the uncertainty quantification of the multipole coefficients. In contrast to previous works we propose to start the stochastic analysis from uncertain measurement data instead of uncertain material properties and we propose to determine the sensitivities by a Sobol decomposition.