Optimum procedures for the statistical improvement, or adjustment, of an existing data evaluation are redeveloped from first principles, consistently employing a minimum-variance viewpoint. A set of equations is derived that provides improved values of the data and their covariances, taking into account information from supplementary measurements and allowing for general correlations among all measurements. The minimum-variance adjustment equations thus obtained are found to be equivalent to a method suggested by Linnik and applied by a number of authors to the analysis of fission reactor integral experiments. The minimum-variance solution is also shown to give the same results as the commonly applied normal equations, but with reduced matrix inversion requirements. Examples are provided to indicate some potential areas of application.