The title of this lecture was chosen over a year ago before I had formed a clear picture of its scope. It is, however, adequately descriptive of my purpose which is to develop consistent general principles, as opposed to detailed methods, which can serve as a guide to the individual computer—primarily one working alone with an ordinary calculating machine. I shall endeavour to illustrate the principles by actual examples, but it will be outside my present compass to concentrate upon those methods which rely upon brilliance of technique rather than on soundness of principle.

It is clearly desirable to say precisely what ‘computation’ embraces, and I therefore put forward the following definitions of ‘calculation’ and ‘computation’—in each case it being understood that the word is prefixed by ‘numerical’:

Calculation is the application of the four rules of arithmetic to the numerical evaluation of an algebraic expression.

A Few years ago two estate duty valuations were made in somewhat unusual circumstances.

A bequeathed an annuity of £75 equally amongst his five daughters, expressed to be payable for their joint lives, the whole of the annuity being subject to remainder to the survivors or survivor for lives or life. The annuity was charged on a leasehold property which was bequeathed absolutely to B, the son of A, subject to the joint-life annuity.

One of the daughters died, and for the purposes of estate duty it was necessary to know the present value of the proportion of annuity which passed on her death, i.e. the value of the annual sum of £15 payable until the last survivor of four lives aged respectively 81, 78, 77 and 71.

These notes were written after a meeting of the Students' Society at which Mr A. W. Joseph submitted some problems in reversions concerning the valuation of an annuity. It has been my experience that where a case is submitted to an actuary ‘for valuation of an annuity’ it is imperative to obtain precise instructions on the circumstances and purpose of the valuation and the exact nature of the interests to be valued. It frequently turns out that the valuation to be made does not entail the valuation of an annuity. The following paragraphs may help to indicate what may be required in a number of cases.

The present note is contributed, not because the numerical results obtained will be of use to actuaries, but in the belief that the methods adopted may be of interest to students as illustrating the application of the concepts of theoretical statistics to a problem which arose in practice.

The theory and technique of statistical Quality Control have been described by H. C. H. Carpenter in J.S.S. Vol. v, 1946, p. 207, and it is shown there how the control lines inserted on the charts are based on the assumption of a normal distribution.

The variance of a continuous frequency distribution is often calculated by concentrating the frequencies in intervals of width h at the mid-points (vide Kendall, Advanced Theory of Statistics, Vol. 1, §3.16). Under certain conditions this approximation can be corrected by subtracting from the calculated variance. This note gives an interpretation in elementary terms of the ‘reason’ for such a correction.

We consider three distributions, all having the same frequency in any one of the intervals of width h (Fig. 1):

C the original continuous distribution;

H where the frequency density is uniform within any one interval;

P where the frequency in an interval is concentrated at the mid-point.

It has long been the desire of administrators, particularly those concerned with personnel selection problems, to find an objective method of dividing people into types. So far, the accepted method of obtaining data to achieve this object has been to submit the individuals to some form of testing procedure. Although the objective type of test, scored quite independently of the opinion of the marker, has done much to dispense with criticisms against the examination system, the interpretation of the results is obviously of little value unless it is known precisely what the tests are measuring. It is, for instance, obvious that the score in a test, ostensibly of mathematical ability, in general depends to some extent on the ability of the candidate to read, and, in fact, all tests probably measure several aspects of attainment. For this reason it is evidently desirable that some method should be found of separating test scores into components which can be ascribed to specific attainments, and the process of doing this is called Factor Analysis.