The measured probability distribution of the microwave field in an ensemble of strongly scattering samples is far from Gaussian. We show, however, that the field in a subensemble with specified total transmission is a Gaussian random variable. This confirms the central hypothesis of random matrix theory of perfect mode mixing and leads to the conclusion that the field and intensity normalized by their respective average magnitudes in a given configuration and the total transmission are statistically independent. This yields a universal form for the intensity correlation function and explains measurements of steady-state and time-resolved transmission in weakly and strongly scattering samples.