ON SOME METHODS OF INTERPOLATION APPLICABLE TO THE GRADUATION OF IRREGULAR SERIES, SUCH AS TABLES OF MORTALITY, &c., &c.

BY ERASTUS L. DE FOREST, M. A.,

Of Watertown, Connecticut.

[The portions of the following methods of interpolation comprising the formulas 2, 3, A, B, C, D, E, F, 11, 12, 13, 17, 19, 20, 21, 24, 25, 26, 27, 28, 30, 43, 44, 45, 46, 48, 49, and 50, were presented to the Smithsonian Institution for publication in the year 1868. The method of constructing tables of mortality from two successive census enumerations was first given in January, 1869, and the formulas 40, 41, 42, 53, 54, 55, 56, and 59, in January, 1870.—J. H.]

We have no analytical formula which expresses the law of mortality with precision, and at the same time with such simplicity as to be prac tically useful. For all the purposes of life insurance and life annuities, it is expressed by numerical series. The law is known to vary in dif ferent localities, and even in the same locality at different epochs. That which prevails in any community, at a given period, can be ascertained by enumerating the persons living at the various ages, and the deaths which annually occur among them. Reduced to one of its usual forms, it is expressed in a statistical table, showing, out of a certain number of persons born, how many survive to complete each successive year of their age. These numbers of the living form a diminishing series of about one hundred terms, whose first differences are the numbers dying during each year of age. We have reason to believe that a true law of mortality is a continuous function of the age, free from sudden irregu larities, so that in a perfect table the second, third, &c., orders of differences of the series ought to go on continually diminishing, and each order by itself ought to show a certain degree of regularity; in other words, the table should be well graduated. But, in point of fact, all purely statistical tables are irregular, especially when the popula tion observed has been small, and every table of mortality now in use has been graduated artificially. It was not strange that the Carlisle table, derived from records of population and deaths in a single town, should show many irregularities. They have been adjusted to some extent, but very imperfectly. The Combined Experience table, also, which was compiled from the records of seventeen British life insurance offices, owes its better graduation to art rather than to nature. Farr's English life-table, No. 3, for males, derived from the census returns of 1841 and 1851, and from the registry of deaths in England and Wales

for the seventeen years from 1838 to 1854, though perhaps the best expression we have for the law of general mortality, is by no means well graduated. In this case the population observed was so large that if the tables had been formed directly from the enumeration of persons living and persons dying in each single year of age, and if these observations could have been relied upon as accurate, any irregularities then existing in the series might possibly have been thought to result from something peculiar in the law of life at certain ages. But it was necessary to combine the single years of age into groups, owing to the impossibility of ascertaining ages with precision. All persons were required to give their exact ages at last birthday, but the reports state that round numbers, such as 50, 60, &c., were disproportionately numerous, showing that the ages were not always correctly given. In forming the life-table No. 3 the years of age were grouped together into decennial periods chiefly, and the whole term of life was then divided into five unequal parts, so as to form a chain of sub-series, each of the fourth order, and not continuous at their points of junction. We must conclude, then, that the great irregularities now found at certain points in the series result from imperfect distribution, and not from any irregularity in the true law of mortality.

A good system of distribution or adjustment, though not positively essential in practice, is nevertheless desirable, first, because a judiciously adjusted table probably comes nearer to the truth than an unadjusted or ill-adjusted one; that is, nearer to what the statistics would show if the population observed could be made indefinitely large, and if the numbers for each year of age could be independently determined. Secondly, if the primary table is well graduated, all the various series of numbers derived from it, forming the usual "commutation tables" and tables of premiums and valuations of assurances and annuities, will be well graduated also, and this will sometimes facilitate the computation of such tables, because a part of the tabular numbers can be accurately found by ordinary interpolation, and errors of computation can be discovered by the method of differences. Many writers on the law of mortality have treated of the subject of adjustment, as may be seen in the pages of the London Journal of the Institute of Actuaries and Assurance Magazine, and elsewhere. The rule of least squares was used to adjust the American table given in the report of the United States census of 1860. (See the Appendix on Average Rate of Mortality, pages 518 and 524.) The series there given, however, is not very thoroughly graduated, as can easily be shown by taking its successive orders of differences. In England, the "law of Gompertz" has been chiefly taken as a basis. But it is not necessary to adopt any exclusive theory respecting the precise nature of that function which expresses the law of mortality. The following system of distribution and graduation is based upon principles which apply to any continuous series of numbers, and is analogous to the ordinary methods of inter

polation. It is not without interest when regarded from a purely mathematical point of view. The general question as to how an irregular series can be made regular is answered by means of the obvious principle that, although single terms in a series may deviate considerably from the normal standard, yet the arithmetical means of successive groups of terms will be less fluctuating, because the errors of the single terms which compose each group tend to compensate each other, and also because the means of two groups which are partly composed of the same terms must necessarily approximate toward each other as the number of terms common to both is increased. In ordinary interpolation, we proceed from some known single terms in a series to find the values of other terms; in the present case, however, all single terms are unreliable, and the problem is to determine the single terms in a series when only the arithmetical means of some groups of terms are given. To find expressions for the sum, and consequently the mean, of the terms in any group, we shall make use of the known principle that, in a continuous series whose law is given or assumed, the sum of a limited number of terms can be regarded as a definite integral, which is the aggregate of a succession of similar integrals corresponding to the terms considered.*

FIRST METHOD OF ADJUSTMENT.

We know that when equidistant ordinates are drawn to the parabolay=A+Bx+Cx2

they form a series of the second order; that is, their second differences are constant. Let c represent the distance from one ordinate to another; the area of the curve included between two such ordinates will be

where x is the abscissa corresponding to the middle ordinate of the area. Since this area is a function of the second degree in x', it follows that when values in arithmetical progression, such as 1, 2, 3, &c., are assigned to a', the resulting areas will form a series of the second order. This being premised, let us assume any three areas, S1, S2, S3, so situated that the middle ordinates of S, and S, shall fall respectively to the left and right of the middle ordinate of S2, which is taken as the axis of Y. Let n1, n2, n3, be the portions of the axis of X which form the bases of these areas, and let a1 and a3 be the portions of the same axis intercepted between the axis of Y and the middle ordinates of S, and S, respectively. Then we have

See a note by M. Prouhet, appended to Vol. II of Sturm's Cours d'Analyse de l'Ecole Polytechnique.

Let S be a fourth area whose base is n, and let x' be the abscissa corresponding to its middle ordinate; then

Eliminating A, B, C, from the above four equations employing P, Q,E as auxiliary letters, and dropping the accent from x', we have—

P=a3[x2+11⁄2(n2—n22)]—x[a32+¿1⁄2(nz2—n22)]

2

-

P+Q
R

(2)

S3

= n [ ( 1 − P + o ) ( §; ) + ( {{Q+QQ

This enables us to find the magnitude S of an area whose position only is given, when the three other areas S1, S2, S3, are given both in magnitude and position.

Now let each of the four areas be divided by equidistant ordinates into as many subdivisions as there are units in the bases n1, N2, N3 and n respectively, these bases being supposed to represent whole numbers and let a1, a3, and x be each a whole number or a whole number and a half, according as n1+n2, n2+n3, and n2+n are respectively even or odd: then all the subdivisions of the areas will be so situated that the ab scissas corresponding to their middle ordinates will be terms in an arithmetical progression, and, consequently, the subdivisions themselves will be terms in a series of the second order. We may regard these subdivisions as representing not areas merely, but magnitudes of ary kind, and the areas S1, S2, S3, and S being the sums of groups of subdivisions, we see that formula (2) enables us to find the sum S of any group of consecutive terms in a series of the second order when the sums S1, S2, S3, of the terms in any other three groups in the series are given. From the sums of the terms in each group their arithmetica means are known, and vice versa, for n1, n2, n3, and n are given, and these are the numbers of terms which the several groups contain. The groups may be entirely distinct, or they may overlap each other so that some terms belong to two or more of them at once. The intervals be tween the middle point of the group S2, and the middle points of the groups S1, S3, and S are a1, a3, and a respectively; the interval between the middle points of any two consecutive terms being unity. We must regard a1 and a3 as always positive, while x may be either positive or negative. When n is made equal to unity, the formula gives the value of a single term S by means of the sums S1, S2, S3, of the three given groups of terms. The results are exact when the series taken is of the second order, but if it follows some other law, or is irregular, approxi mate or adjusted values for S will be obtained, and if the same groups