Statistics forms a not unimportant part of the actuary's studies, and is understood to be attracting greater attention than was the case, say, fifty years ago. To some extent this is undoubtedly due to an increasing recognition of its importance in this field, but we must also recognize that it is a subject in which big advances have been made on the theoretical side during the present century. Now there must always be of necessity a great gulf fixed between the papers of an Aitken and those of the Institute's examinations. It does not follow, however, that the elements of the subject must continue to be taught in the same old way. The enigma facing the student when extensions to knowledge are being continually made can be resolved, and very often is resolved, by a complete rewriting of the elementary text-books. To give an example, we have all been taught at one time or another how to test the difference between two sample means by using the standard error, on the large sample theory.