Empirical versions of appropriate centering and scale constants for random variables which can fail to have second or even first moments are obtainable in various ways via suitable modifications of the summands in the partial sum. This paper discusses a particular modification, called censoring (which is a kind of winsorization), where the (random) number of summands altered tends to infinity but the proportion altered tends to zero as the number of summands increases. Some analytic advantages inherent in this approach allow a fairly complete probabilistic and empirical theory to be developed, the latter involving the study of studentized or self-normalized sums. In particular, the joint asymptotic distributions of the empirically censored quantities of center and scale are determined as well as precise criteria for convergence to each of the allowable limit laws. Applications to the Feller class and domains of attraction are also considered.