We investigate the observational signatures of many-field inflation and present analytic expressions for the spectral index as a function of the prior. For a given prior we employ the central limit theorem and the horizon crossing approximation to derive universal predictions. These include a specific dependence on the prior choice for initial conditions that has not been seen in previous studies. Our main focus is on quadratic inflation, for which the initial conditions statistics decouple from those of the mass distribution, while other monomials are also briefly discussed. We verify the validity of our calculations by comparing to full numerical simulations with 100 fields using the transport method.