Equating functions are supposed to be population invariant, meaning that the choice of subpopulation used to compute the equating function should not matter. The extent to which equating functions are population invariant is typically assessed in terms of practical difference criteria that do not account for equating functions' sampling variability. This article shows how to extend the framework of kernel equating so that the standard errors of the root mean square difference (RMSD) and of the difference between two subpopulations' equated scores can be estimated. An investigation of population invariance for the equivalent groups design is discussed. The accuracies of the derived standard errors are evaluated with respect to empirical standard errors. This evaluation shows that the accuracy of the standard error estimates for the equated score differences is better than for the RMSD and that accuracy for both standard error estimates is best when sample sizes are large. (Contains 3 tables and 10 figures.)