Sign random projections (SRP) is a technique which allows the user to quickly estimate the angular similarity and inner products between data. We propose using additional information to improve these estimates which is easy to implement and cost efficient. We prove that the variance of our estimator is lower than the variance of SRP. Our proposed method can also be used together with other modifications of SRP, such as Super-Bit LSH (SBLSH). We demonstrate the effectiveness of our method on the MNIST test dataset and the Gisette dataset. We discuss how our proposed method can be extended to random projections or even other hashing algorithms.