A critical overview of the current doubtful practice on presentation of correlated data in the physics literature and in the scientific and technological databases is presented. The simple rules to calculate the rounding thresholds to preserve the positive definiteness of the covariance and correlation matrices as well as the rounding thresholds for the components of the mean vector to keep them inside the "non-rounded" scatter region are formulated. Evidence that in the multivariate case there are severe limitations on the applicability of the linear differential law of uncertainty propagation is presented. The explicit relation of the number of input random variables I, the number of output variables D, and the order T of Taylor polynomials sufficient to preserve the self-consistent numerical presentation of the mean value of the vector function and its covariance matrix under nonlinear differential propagation procedure is obtained. It is stressed that the rounding thresholds for the safe rounding of correlated data impose the severe requirements on the storage and exchange formats of the correlated data that could not be met in the traditional publications on the paper but could be realized in the electronic media.