Population invariance is an important requirement of test equating. An equating function is said to be population invariant when the choice of (sub)population used to compute the equating function does not matter. In recent studies, the extent to which equating functions are population invariant is typically addressed in terms of practical significance and not in terms of the equating functions' sampling variability. This paper shows how to extend the framework of kernel equating to evaluate population invariance in terms of statistical significance. Derivations based on the kernel method's standard error formulas are given for computing the standard errors of the root mean square difference (RMSD) and of the simple difference between two subpopulations' equated scores. An investigation of population invariance for the equivalent groups design is discussed. The accuracy of the derived standard errors is 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.