Punched-card machinery is in fairly common use in Life Offices and in the following paper I shall assume familiarity with the more usual machines, i.e. punches and verifiers, sorters and tabulators, together with ancillary machines such as reproducers, collators or interpolaters, and interpreters. For those who are not familiar with these machines several explanations have appeared already in print, a recent example being The development of the punched card method by K. J. Hedley, read before the Actuarial Society of Australasia in the latter part of 1946 and appearing in the Transactions of that body. The purpose of the present paper is to explain with some detailed examples of the method of application how the addition of a multiplying punch to a punched-card installation adds greatly to the power of that installation to perform calculations of certain types important to Life Offices, and how those calculations can be performed with great economy of manpower as compared with existing methods. I do not pretend to make an exhaustive survey of the uses of this remarkable machine.

A point estimate may provide, in logic or in practice, an unsatisfactory answer to some problems, and the additional requirement then arises for estimating limits within which a parameter is thought to lie. An elementary example of such an ‘interval estimate’ occurred in Example 2(c) of § 1, and we may recall the custom of quoting the measurement of a physical constant in such a form as g = 32 ± 2 units—which is an assertion, made according to some convention, that the true value of the quantity g lies between 30 and 34 units. We consider below some of the criteria which have been suggested for interval estimation, some methods of satisfying them, and a few of the results to which they lead.

It is intended in the following paragraphs to give an outline of a very general statistical method which originated in agricultural experimentation and which has since been used in many other fields, including that of mortality investigations. We will start with a special case, that of regression analysis involving only one independent variable. Well-known formulae will be developed with the aid of orthogonal polynomials and it will be shown how this tool can be used for generalizations. A new theorem referring to these polynomials, and its significance for the analysis of variance, will also be mentioned.

In practice it is desired occasionally to calculate the value of an annuity-certain at a rate of interest decreasing steadily from the outset. If the rate of interest for the first year is ix and decreases by h per annum throughout the term n of the annuity, the value required is

say, where

This expression is somewhat laborious to evaluate.