In this paper we analyze the mean squared error (MSE) for one-bit compressed sensing schemes based on measurement matrices that correspond to unit norm tight frames. We show that, as in the unquantized case, sensing with unit norm tight frames improves the MSE in the reconstruction of sparse vectors from one-bit measurements using l1 and thresholding algorithms. From our analytical and experimental results we conclude that when implementing one-bit compressed sensing schemes with fixed measurement matrices unit norm tight frames are the measurements of choice