We consider polarization properties of the unpolarized emission of an ensemble of classical emitters with randomly varying polarization. The light is supposed to be unpolarized in the sense that all three polarization-related components of its Stokes vector are zero. At the same time, the mean-square values of these components should not be necessarily zero, may differ from each other, and, therefore, may provide additional information about properties of individual emitters. Experimentally, this information is revealed as dependence of the polarization noise on the azimuth of the quarter-wave plate placed before the polarization-sensitive detector. This dependence appears to be different for the emitters randomly polarized over the equator of the Poincaré sphere, or preferentially located on its poles, or uniformly covering the whole sphere. We show that full quantitative analysis of the polarization-noise anisotropy allows one, in the framework of the proposed model, to get information about polarization characteristics of individual emitters hidden in the emission of the ensemble. Vitality of the method is illustrated by its application to polarization analysis of the polariton laser emission, which is shown to predominantly arise from linearly polarized emitters.