If by pa be indicated the probability that a person of the precise age z will survive one full year, and if = number of persons living at the beginning of the year x, of whom +1 survive to the end of the year, then survivors at end of year Px= = number living at its beginning. It can be shown 1 also that By this formula a very simple relation is found to exist between the probabilities of life and the rate of mortality. From the census returns and the death-registers, m, for all ages, i.e. for all values of x, can easily be obtained; then by the above formula pa for each age can be calculated, and this enables the life-table to be constructed. No attempt is made here to describe the construction of a life-table in detail. For this, reference must be made elsewhere.2 It is necessary, however, to understand the exact meaning of each column in a life-table, and this is best grasped by knowing how the column has been built up. First of all, as indicated above, from the population figures and the deaths for each individual year of life the death-rate and the probability of living a single year are calculated as described above. Then it is usual to start with 100,000 children at birth, males and females in the proportion shown at the last census. Thus, supposing they were 51,195 and 48,805 respectively, as in the Brighton Life-table. These numbers are entered at the head of the le columns for males and females respectively. Then starting with 51,195 male infants at birth, the number living at the end of one year is obtained by multiplying this number by the probability of surviving to the end of the first year. Thus 51,195 x 84608 = 43,315 surviving at the end of the first year of life, 43,315 × 93392=40,452 surviving at the end of the second year of life, and so on. Thus the 43,315 males surviving to the end of the first year of life out of 51,195 born will have each lived a complete year in the first year, or among them 43,315 years. Similarly, 40,452 males will live another complete year each in the second year, or among them a further 40,452 complete years; similarly 39,456 complete years of life will be lived in the third year; 38,723 in the fourth year, and so on, until the males started with become extinct. It is evident, therefore, that the total number of complete years lived by the 51,195 males started with at birth will be the sum of the products of the multiplication of the number starting at each year of life into the probability of surviving to the end of each year. Thus in the above example it is 43,315 + 40,452 + 39,456 + 38,723 + 1=2,206,174 years. As this number of years is lived by 51,195 males, the number of complete years lived on an average by each male .... This result is known as the curtate expectation of life. The above represents the complete years of life, no account being taken of that portion of lifetime lived by each person in the year of his death. 1 See Elements of Vital Statistics, A. Newsholme, p. 259. 2 Op. cit. pp. 255 et seq. In some instances this may only be a few days, in others nearly an entire year; but it may be assumed with a fair degree of accuracy, taking one person with another, that the duration of life in the year of death will be half a year. If we add this half-year to the curtate expectation of life, the complete expectation of life is obtained. Thus the complete expectation for males at birth in Brighton on the basis of the experience of 1881-9043·09 + 0.5= 43.59 years; at the age of ten years=48.62 + 0·5=49·12 years. This method is accurate for most ages, but for the first year 0-5 is too much. The following tables are extracted from the English life-table and from the Institute of Actuaries life-table. They illustrate the two methods of nomenclature previously mentioned. Note that Only the last years of life in each table have been taken for the sake of brevity, but the method of construction of the life-table can be studied in them exactly the same as if the life-table had commenced at birth. LIFE-TABLE FOR ENGLAND AND WALES, BASED ON THE MORTALITY Duration of Life.-By means of a life-table the average duration of life can be accurately determined. For, although nothing is more uncertain than the duration of life, there are, as Babbage has said, "few things less subject to fluctuation than the duration of life in a multitude of observations." This principle enables annuities and life assurance to be made the subject of exact and definite calculations, which among a large class of persons will be found to be approximately accurate. INSTITUTE OF ACTUARIES H.M. (HEALTHY MALES) LIFE-TABLE The duration of life as determined by a life-table is known as the expectation of life or mean after-lifetime. It should be observed that the expectation of life is the average number of years which persons of a given age, taken one with another, live, assuming that they die according to a given table of the probabilities of life. The term "expectation of life" does not imply that an individual may reasonably expect to live a given number of years. The excess of those who die late is distributed among those who die early, "those who live longer enjoying as much more in proportion to their number as those who fall short enjoy less of life." Thus the expectation of life has no relation whatever to the most probable lifetime of any given individual. Many attempts have been made in the absence of a life-table to ascertain the expectation of life, but none of them give exact results. The formula of Willich gives approximate results for ages between 25 and 75. It is as follows: If x = expectation of life, and a = present age, then x = {(80-a). Thus at the age of 45 years the expectation of life according to this formula (80-45)=23-3 years. According to the English life-table it = varies from 22 for men to 24 for women. Farr gave the following formula for obtaining an approximation to the expectation of life at birth when the birth-rate and death-rate are known. If b = birth-rate and d-death-rate per unit of population, then the 1 (× 1 b expectation of life -(x)+( × }) = Thus in the decennium 1881-90 the birth-rate in England and Wales was '03234, and the death-rate 01908 per unit. The expectation of life at birth X = (3 × 01908) + (3 × 03234) · 234)=45-6 years, while the expectation of life shown by the English life-table for males and females combined was 45.4 years. These formulæ might be used without misleading to any considerable extent when applied to a given community for successive years. They would be misleading if employed for comparing communities having widely different birth-rates. Females. The expectation of life at different ages, and according to different lifetables, is shown in the following table:: EXPECTATION OF LIFE (MEAN AFTER-LIFETIME) ACCORDING TO The comparability of the expectation of life according to different tables, of which examples are given in the preceding table, depends on their being constructed by similar methods. From this standpoint it is unfortunate that, as has been shown by Dr. T. E. Hayward,1 a different and improved method has been employed in the construction of the national life-table for 1891-90 from that employed by Drs. Farr and Ogle. The chief differences are at the two extremes of life, the values of Po to P4 and from P65 onwards being considerably in excess of what they would be had Drs. Farr and Ogle used the same method as was employed in the most recent national life-table. For details as to the methods employed see Dr. Hayward's papers.2 The differences in the p column are as follows:: Dr. Hayward has recalculated Ex values for five decennia on a uniform basis. The following is his statement of the method employed and the results obtained : (1) The foundation numerical facts have been taken from the Census Reports and from the Annual Reports and Decennial Supplements of the Registrar-General. (2) The method of calculation used has been the modified short method which was described in the Journal of the Royal Statistical Society for September and December 1899, vol. Ixii. parts 3 and 4, and which has been shown to give results practically identical with those to be obtained by the extended method described in the same paper. (3) The mean population numbers for each decennium have been worked out by the new method described by Mr. A. C. Waters, in the Journal of the Royal Statistical Society for June 1901, vol. xliv. part 2. (4) In order to economise space, only the final E, values of the lifetables are given in tabular form, and they are left without comment. TABLE SHOWING EX VALUES (i.e. MEAN AFTER-LIFETIME OR EXPECTATION OF LIFE AT AGE x), BASED ON THE YEARS OF LIFE AND ON THE NUMBER OF DEATHS IN ENGLAND AND WALES FOR EACH DECENNIUM. |