SIFT descriptors are broadly used in various emerging applications. In recent years, these descriptors were deployed in compressed and binarized forms due to the computationalcomplexity, storage, security and privacy cost incurred by working on real data. At the same time, the theoretical analysis of SIFT feature performance in different applications remains an open issue due to the lack of accurate statistics of binarized SIFT descriptors. We address this problem and statistically analyse projected binarized SIFT descriptors in this paper. The methodology is based on dimensionality reduction using random projections with binarization. Furthermore, we investigate the statistical models of intra- and inter-descriptor dependencies for various distortions. Finally, we demonstrate a simple heuristic to distinguish between descriptors from identical but distorted images and descriptors from non identical images.