# Full text of "Mortality laws and statistics"

## See other formats

MATHEMATICAL MONOGRAPHS EDITED BY Mansfield Merriman and Robert S. Woodward. Octavo, Cloth. No. 1. History of Modern Mathematics. By DAVID EUGENE SMITH. |i.oo net. No. 2. Synthetic Protective Geometry. By GEORGE BRUCE HALSTED. $1.00 net. No. 3. Determinants. By LAENAS GIFFORD WELD. $1.00 net. No. 4. Hyperbolic Functions. By JAMES Mc- MAHON. $1.00 net. No. 5. Harmonic Functions. By WILLIAM E. BYERLY. fi.oo net. No. 6. Grassmann's Space Analysis. By EDWARD W. HYDE, f i.oo net. No. 7. Probability and Theory of Errors. By ROBERT S. WOODWARD. |i.oo net. No. 8. Vector Analysis and Quaternions. By ALEXANDER MACFARLANE. Ji.oo net. No. 9. Differential Equations. BY WILLIAM WOOLSEY JOHNSON, Ji.oo net, No. 10. The Solution of Equations. By MANSF:ELD MERRIMAN. |i.oo net. No. 11. Functions of a Complex Variable. By THOMAS S. FISKE. $ i.oo net. No. 12. The Theory of Relativity. By ROBERT D. CARMICHAEL. ji.oo net. No. 13. The Theory of Numbers. By ROBERT D. CARMICHAEL. $1.00 net. No. 14. Algebraic Invariants. By LEONARD E. DICKSON. $1.25 net. No. 15. Mortality Laws and Statistics. By ROBERT HENDERSON. $1.25 net. No. 16. Diophantine Analysis. By ROBERT D. CARMICHAEL. Si. 2 5 net. No. 17. Ten British Mathematicians. By ALEX- ANDER MACFARLANE. $1.25 net. PUBLISHED BY JOHN WILEY & SONS, Inc., NEW YORK. CHAPMAN & HALL, Limited, LONDON MATHEMATICAL MONOGRAPHS EDITED BY MANSFIELD MERRIMAN AND ROBERT S. WOODWARD No. 15 MORTALITY LAWS AND STATISTICS BY ROBERT JIENDERSON, ACTUARY OF THE EQUITABLE LIFE ASSURANCE SOCIETY OF THE UNITED STATES FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITED f> 1915 COPYRIGHT. 1915, BY ROBERT HENDERSON THE SCIENTIFIC PRESS BROOKLYN. N. Y. PREFACE THE present work is designed to set forth in concise form the essential facts and theoretical relations with reference to the duration of human life. A description is first given of those mortality tables which have had the greatest influence on the development of the science of life contingencies or on its appli- cation in this country. A few chapters are then devoted to the mathematical relations between the various functions connected with human mortality, to the analysis of probabilities of death or survival, so as to lead to their simplest form of expression in terms of the mortality table, and to the general mathematical laws which have been proposed to express the facts of human mortality. The connection is then established between the mortality table and mortality statistics and some investigation made of the corrections which must be allowed for in interpreting such statistics. The methods of constructing mortality tables from census and death returns and from insurance experience are then taken up. The methods adopted for the purpose of adjusting the rough data derived from experience are next described and their theoretical basis investigated. Some of these methods of construction and graduation are then illustrated by a new mortality table now first published. In the Appendix ten useful tables are given. The scope of the treatise is confined to life contingencies excluding all monetary applications, so that the combination of the theory of compound interest with that of life contin- gencies is not touched upon. A warning may not, however, be amiss that the present value of a sum of money payable iii IV PREFACE at death cannot properly be calculated by assuming it to be payable at the end of a definite period equal to the expectation of life, nor can the present value of a life annuity be calculated by assuming it to be certainly payable for that period. R. HENDERSON. NEW YORK, March i, 1915 CONTENTS PAGE CHAPTER I. MORTALITY TABLES i II. THE MORTALITY TABLE AND PROBABILITIES INVOLVING ONE LIFE 17 III. FORMULAS FOR THE LAW OF MORTALITY 26 IV. PROBABILITIES INVOLVING MORE THAN ONE LIFE 34 V. STATISTICAL APPLICATIONS 45 VI. CONSTRUCTION OF MORTALITY TABLES 51 VII. GRADUATION OF MORTALITY TABLES 68 VIII. NORTHEASTERN STATES MORTALITY TABLE 95 APPENDIX. DATA FROM VARIOUS MORTALITY TABLES 100 MORTALITY TABLES DR. HALLEY'S BRESLAU TABLE 3 DEATHS AND POPULATION, NORTHEASTERN STATES, 1908-1912 96 THE NORTHAMPTON TABLE 100 THE CARLISLE TABLE 101 ACTUARIES', OR COMBINED EXPERIENCE, TABLE 102 AMERICAN EXPERIENCE TABLE 103 INSTITUTE OF ACTUARIES' HEALTHY MALE (H M ) TABLE 104 BRITISH OFFICES O M[51 TABLE 105 NATIONAL FRATERNAL CONGRESS TABLE 106 NORTHEASTERN STATES MORTALITY TABLE, 1908-1912 107 RATES OF MORTALITY PER THOUSAND ACCORDING TO TWELVE TABLES. . . . 109 DEATH RATES PER THOUSAND ACCORDING TO VARIOUS TABLES no DIAGRAMS 1. COMPARISON OF AGGREGATE AND ANALYZED RATES OF MORTALITY .... 67 2. COMPARISON OF GRADUATED AND UNGRADUATED RATES OF MORTALITY. 93 3. RATES OF MORTALITY BY VARIOUS AMERICAN TABLES 99 v MORTALITY LAWS AND STATISTICS CHAPTER I MORTALITY TABLES 1. THE subject of human mortality is one which, from its nature, is of widespread interest to mankind. It has always been recognized that it is impossible to predict the duration of any individual life and that the only thing that could be taken as certain on the subject was that death would come sometime to each one. In other words it has been recognized that the date of death of any individual is subject to chance. The scientific study of the subject of chance is, however, a comparatively modern development of mathematics and con- sequently the science of life contingencies is also comparatively modern. 2. Like all other events, whether considered as chance events or as certainties, the death of any individual or his survival to any specified date is the necessary result of those forces which have been operating upon him. The cause of our ig- norance regarding the result in the case of an individual is the limitation of our knowledge regarding the forces operating and their effects. We do know, however, that among the important ones affecting the result are climate, sanitary conditions, medical attendance and habits of life. These vary in their tendency and effectiveness as we pass from one locality to another or at different times in the same locality and may even differ in a recognizable way as between different individuals in the same locality and at the same time. The results of the observations made with respect to human mor- 2 MORTALITY LAWS AND STATISTICS tality under any given set of circumstances are frequently set forth concisely in the form of a Mortality Table showing the number surviving to each age out of a given number living at some selected initial age. A brief description is here given of some of the Mortality Tables which have had a relatively im- portant part in the history of the science of life contingencies. THE BRESLAU TABLE 3. This table is of importance because it represents the first attempt to construct a mortality table from which to deduce the probabilities of survival and the values of life annuities. It was formed by the celebrated astronomer, Dr. E. Halley, from returns of the deaths in the City of Breslau, Silesia, during the five years 1687 to 1691 inclusive. Owing to the fact that the births during the five years only slightly exceeded the deaths (the numbers being 6193 and 5869 respectively) he assumed that the population might be considered a sta- tionary one. He therefore appears to have graduated by inspection the average number dying per annum at the various ages and assumed that this gave the decrements of the table. Dr. Halley published his table in two columns, the first headed " Age Current," and the second " Persons," and it appears ' from the explanation given to have been in modern notation equivalent to a table of values of L x -i, the population at age x next birthday. 4. On the basis of this table Dr. Halley solved various prob- lems regarding survival and calculated annuity values, thus laying the foundations of the science of life contingencies and preparing the way for the transaction, on a scientific basis, of the important business of life insurance, although it was /not until nearly seventy years had elapsed after the publi- cation in 1693 of this table that the first company to operate on a scientific basis was established. Mr. E. J. Farren, writing in 1850, said regarding this table: " With respect to its form, as has already been stated, no improvement has as yet been adopted, beyond inserting MORTALITY TABLES the column of differences or deaths, and choosing higher num- bers for exemplification. Of its two principles of construction, viz., as to the number of living being deducible from the num- ber of deaths, by aid of the assumption of a stationary pop- ulation; and as to the number of deaths at contiguous ages after childhood being allied in number; the former principle was generally prevalent in the construction of such tables, until the appearance of Mr. Milne's Carlisle Table in 1815, but is now as generally abandoned; the latter characteristic is still operative and considered as valid in all the best tables." The mortality indicated by this table was considerably higher than that shown by more modern tables. 5. The table as given by Dr. Halley is as follows: Age Cur- rent. Per sons. Age Cur- rent. Per sons. Age Cur- rent. Per sons. Ag, Cur- rent. Per- sons. Age Cur- rent. Per- sons. Age Cur- rent. Per- sons. I 1000 8 680 IS 628 22 586 29 539 36 481 Age. Persons 2 855 9 670 16 622 23 579 30 531 37 472 7 5547 3 798 10 661 17 616 24 573 31 523 38 463 14 4 584 4 760 ii 653 18 610 25 567 32 515 39 454 21 4 270 5 732 12 646 19 604 26 560 33 57 40 445 28 3 964 6 710 13 640 20 598 27 553 34 499 4i 436 35 3 604 7 692 14 634 21 592 28 546 35 490 42 427 42 3 178 49 2 709 56 2 194 Age Cur- rent. Per- sons. Age Cur- rent. Per- sons. Ae Cur- rent. Per- sons. Age Cur- rent. Per- sons. Age Cur- rent. Per- sons. Age Cur- rent. Per- sons. 63 I 694 70 I 204 77 692 43 417 50 346 57 272 64 2O2 71 131 78 58 84 253 44 407 51 335 58 262 65 192 72 120 79 49 100 107 45 397 52 324 59 252 66 182 73 109 80 4^ S., , 46 47 387 377 53 54 313 302 60 61 242 232 67 68 172 l62 74 75 98 88 81 82 34 28 Total 34 ooo 48 367 55 292 62 222 69 152 76 78 83 23 49 357 56 282 63 212 70 142 77 68 84 20 The column on the right hand is evidently a summary of the preceding figures in groups of seven years; with an additional item giving 107 as the population at ages 85 to 100 inclusive. MORTALITY LAWS AND STATISTICS THE NORTHAMPTON TABLE 6. This table was first published in 1783 in the fourth edition of Dr. Price's work on Reversionary Payments. It was constructed by Dr. Price on the basis of a return of the deaths in the parish of All Saints, Northampton, England, during the forty-six years from 1735 to 1780 inclusive. These deaths by ages were as follows: Ages. Number of Deaths. 0-2 2-5 1529 362 5-10 IO-2O 2OI 189 20-30 373 30-40 40-50 50-60 6O-7O 70-80 329 365 384 378 358 8o-QO 199 QO-IOO Total 22 4689 7. Owing to the fact that the total number of baptisms during the same period was only 4220, or almost exactly ten per cent less than the number of deaths, Dr. Price, apparently believing the baptisms to correctly represent the births, as- sumed a stationary population supported by births and immi- gration at age 20. He supposed that the immigration was sufficient to supply thirteen per cent of the total deaths. The actual process followed appears to have been to transfer a sufficient number of deaths from age group 20 to 30 to age group 30 to 40 to equalize the numbers in the two groups. The number of deaths in the various groups was then pro- portionately increased so as to make the total number 10 ooo. Thirteen per cent of this number or 1300 in all were then de- ducted pro rata from the groups above age 20. All the groups both below and above age 20 were then increased pro rata so as to bring the deaths above age 20 up to the same figure MORTALITY TABLES 5 as before. This increased the total deaths to n 650, which was taken as the value of IQ. A subtraction of the deaths in the successive groups gave the values of lz, Is, ho, ho, etc., the intermediate values being afterwards inserted. 8. From the above explanation it will be seen that in this case, as in the case of the Breslau table, only the death returns were available, without any enumeration of the pop- ulation, and the resulting table indicated rates of mortality, especially at the younger ages, which, in the light of subsequent experience, appear unduly high. In fact, instead of being stationary, the population and the births had been regularly increasing and the population and deaths at the young ages were consequently proportionately higher than would have been the case in a stationary population. Thus when it was assumed that the number attaining any given age was equal to the number dying above that age the result was an under- statement of the former number by the amount of the total increase during that period in the population above that age, subject to adjustment for immigration or emigration. This understatement of the denominator of the fraction determining the rate of mortality, of course, overstated that rate. 9. The Northampton Table was adopted by the Equitable Society as a basis for its calculations immediately after its construction. This fact, combined with the success of that Society, caused its adoption for many purposes for which it was not suitable. An outstanding illustration of this is the fact that the British Government based their rates for the sale of annuities upon it and consequently sustained a serious loss because the longevity of the annuitants proved much greater than was indicated by the table. Until within a few years the Northampton Table with five per cent interest was the basis prescribed by the court rules in New York State for the valuation of life interests and dower rights, but the Carlisle Table has now been adopted instead. MORTALITY LAWS AND STATISTICS THE CARLISLE TABLE 10. This table was constructed in 1815 by Dr. Milne and was the first table to take both the deaths and the corre- sponding population into account. It was based on two censuses of the population of the parishes of St. Mary and St. Cuthbert, Carlisle, taken January i, 1780, and December 31, 1787, or an interval of eight years, and on the deaths for the nine years 1779 to 1787. The following schedule shows the data: Age Last Birthday. Population. Deaths 1779 to 1787. Jan. 1780. Dec. 1787. O 39 I 173 2 1029 1164 128 3 70 4^ I 51 5- 9 908 1026 89 10- 14 715 808 34 15- *9 675 763 44 20- 29 1328 1501 96 30- 39 877 991 89 40- 49 858 970 118 5- 59 588 665 103 60- 69 438 494 173 70- 79 191 216 152 80- 89 58 66 98 90- 99 IO ii 28 100-104 2 2 4 Tola 1 7677 8677 1840 ii. The figures for the deaths are derived from a record kept by Dr. J. Heysham, and the population as of January, 1780, is also derived from an enumeration by him, taking account of the ages. The population as of December, 1787, appears to have been merely enumerated in gross and then distributed by ages in the same proportions as had been found in 1780. It will be seen on examination that the proportions on the two dates are the same, which is a condition scarcely likely to be realized in two actual enumerations. It was assumed MORTALITY TABLES 7 that the average population during the period covered by the observations could be represented by the mean of the two censuses. The method followed in deducing the rates of mortality will be described in Chapter VI. This table presented a more accurate statement of the probabilities of death at various ages than any preceding table and was widely used for insurance calculations. It has now been largely superseded for this purpose by tables more recently constructed from the experience of insured lives. It is, however, still in use for special purposes. Owing to the small extent of the data on which it was based and the graphic method adopted in redistributing the pop- ulation and deaths into individual ages, the rates of mortality were somewhat irregular, particularly at the older ages, and various regraduations of the table have been made with the idea of removing the irregularities. THE ENGLISH LIFE TABLES 12. At various times tables of mortality have been con- structed on the basis of the census returns and registration of deaths in England. On account of the fact that an exten- sive series of monetary tables was based on it the most widely known of these tables is the English Life Table No. 3, which was constructed by Dr. Farr on the basis of the censuses of 1841 and 1851 and the deaths of the seventeen years 1838-54. Separate mortality tables were constructed for male and female lives starting with radices of 511 745 and 488 255 respectively or for the combined table, i ooo ooo at age zero. The method followed in constructing these tables will be described in Chapter VI. 13. At about the same time as these tables were constructed other tables, known as the Healthy Districts Life Tables, were also constructed from the census returns for 1851 of the sixty- four English districts having at that time an average death rate below 17 per thousand, and the deaths in the same dis- tricts during the five years 1849-53. These tables were pre- 8 MORTALITY LAWS AND STATISTICS sen ted in threefold form, the radix at age zero for the male table being 51 125 and for the female 48875, the two added together constituting a mixed or combined table with a radix 100 ooo. The Healthy Districts Male Table with certain modifications was used by the Committee of the Actuarial Society of America in charge of the Specialized Mortality Investigation as a basis for the comparison of the mortality in the different classes. 14. The more recent of the series of English Life Tables are designated as Nos. 6, 7, and 8. The English Life Tables No. 6 were based on the census returns of 1891 and 1901 and the deaths of the ten years 1891 to 1900 inclusive and, while not originally prepared by Mr. Geo. King's method described in Chapter VII, have been readjusted by that method. The English Life Tables Nos. 7 and 8 have just been published and were prepared by Mr. Geo. King by his method. The former set are based on the census returns of 1901 and 1911 and the deaths of the ten years 1901 to 1910 inclusive, while the latter are based on the census of 1911 adjusted for increase to the middle of that year and on the death returns of the three years 1910 to 1912 inclusive. The special feature of these tables is that not only do they indicate an improvement in mortality as compared with the earlier tables of the series, but, when compared with one another they indicate that the improvement was still progressing. The No. 7 Tables show a lower mor- tality throughout than the No. 6 and the No. 8 Tables a lower mortality at practically all ages than the No. 7. THE ACTUARIES', OR COMBINED EXPERIENCE, TABLE 15. This table, also known as the Seventeen Offices' Ex- perience Table, was prepared in 1841 by combining the experi- ence, by lives, of the Equitable and Amicable Societies with the experience, by policies, of fifteen other companies as con- tributed in 1838 to a committee of actuaries. It was thus the first example of a mortality table formed by combining the experiences of different insurance companies into one MORTALITY TABLES \J general average. It appears to have covered in all 83 905 policies or lives of which 13 781 were terminated by death, 25 247 were terminated otherwise, and 44 877 were in existence and under observation when the observations closed. The total of the numbers exposed to risk, for one year at each age, was 712 163 indicating an average duration of 8.5 years. 1 6. Probably owing to the mixed nature of the data, which as above stated, was partly by lives and partly by policies, and to the fact that the average duration of the experience contributed by the companies other than the Equitable and the Amicable was only 5.5 years, this table was never widely used in Great Britain for insurance purposes. It was, however, prescribed by the State of Massachusetts as the basis for the valuation of the reserve liabilities of life insurance companies. The example of Massachusetts was later followed by New York and other states with the result that for many years the Actuaries' Table with four per cent interest was the accepted valuation standard in the United States, although the pre- miums actually charged by the companies were as a rule based on a different table. THE HEALTHY MALE (H M ) TABLE 17. This is the most important of the group of tables published in 1869 and representing the results of the Insti- tute of Actuaries' Mortality Experience, 1863. They were based upon data contributed by twenty British life insurance companies regarding their experience up to 1863 on insured lives. The H M table was based on the experience of male lives insured at regular premium rates, and duplicate policies on the same life, whether in the same or in different companies, were carefully eliminated. This table represented a much broader experience than that upon which the Actuaries' Table had been based, confirmed in a general way the results of that experience and obtained immediate acceptance as a fair representation of the average mortality of insured lives. The official H M Table was graduated by Woolhouse's formula 10 MORTALITY LAWS AND STATISTICS but it was subsequently regraduated by King and Hardy according to Makeham's formula, with a modification at the younger ages, and extended down to age zero by means of rates of mortality taken from the Healthy Districts Male Table. This graduation of the table is published in Part II of the Text Book of the Institute of Actuaries. 18. In the construction of this table all lives of the same attained age were included together without regard to the period elapsed since medical examination. But an analysis of the experience indicated that the rate of mortality among lives recently insured was much less than among lives of the same attained age who had been insured for a longer period. Accordingly a second table, known as the H M (5) Table, was formed by omitting the experience during the calendar year of issue and the next four calendar years. This table was taken as representing the ultimate rate of mortality after the effects of selection had worn off. The rates of mortality at the young ages are considerably higher in the H M (5> table than in the H M , but the two rates gradually approach one another and coincide at the extreme old ages where there are no recently selected lives. 19. These two tables used together were adopted by many British companies for the valuation of their liabilities, and the H M Table was for many years prescribed for that purpose by the laws of Canada. It will be noticed that the rates of mortality according to the H M Table are lower than those for the same ages in the Actuaries' Table except for ages 46 to 50 and ages 73 to 85 inclusive and ages 95 and over. The difference is not, however, important except at the young ages, where it is considerable. THE BRITISH OFFICES' LIFE TABLES, 1893 20. These tables represent the experience on insured lives of sixty British life insurance companies during the thirty years from the policy anniversaries in 1863 to those in 1893. The data were compiled under the joint supervision of the MORTALITY TABLES 11 Institute of Actuaries and the Faculty of Actuaries in Scot- land and was classified into male and female lives and accord- ing to the plan of insurance issued. The O M Table represents the experience of male lives insured on the Ordinary Life plan with participation in profits. The total number of lives under observation was 551 838, of whom 149 566 were insured prior to 1863. Of these 140 889 died, 148 392 withdrew and 262 557 remained insured in 1893, the total number of years of risk being 7 056 863. The O M ( ' Table represents the same ex- perience, omitting the first five policy years and covers 5 324 862 years of risk and 129 ooi deaths. These tables were grad- uated by Mr. G. F. Hardy. The O M (5) table was first grad- uated by the application of Makeham's formula, the dif- ferences in the values of log p x by the two tables being then graduated by the use of a double-frequency curve. A select or analyzed table was also prepared from the same data and is known as the O [M1 Table. In this table separate rates of mortality are indicated for each age at entry for the first ten policy years, merging into an ultimate table at the end of that time. This select table was also graduated by Makeham's formula, different constants being used for the different policy years. The rates of mortality by the O M table are lower throughout than those in the H M table as graduated by Make- ham's formula and also, with unimportant exceptions, than those in the official H M table. The O M (5) Table is the basis at present prescribed for the valuation of policies in Canada. THE AMERICAN EXPERIENCE TABLE 21. This table was constructed by Mr. Sheppard Homans and was first published in its present form in 1868. No com- plete record has ever been made public of the method adopted in its construction, but it has always been understood that the mortality experience of the Mutual Life Insurance Company of New York was used as a basis. As that experience covered only a few years and therefore did not include any exposures or deaths at extreme old ages it must have been supplemented 12 MORTALITY LAWS AND STATISTICS from other sources. The table is a very smoothly graduated one and evidence has been discovered which seems to indi- cate that the author first constructed a table of values of the reciprocal of the rate of mortality showing the number of lives out of which one death would be expected at each age. From these values the usual columns of the mortality table were then formed. 22. The first publication of the table was in the schedule of an act prescribing it as a basis of valuation in the State of New York and although it was temporarily abandoned in that state for the sake of uniformity it is now the legal standard in practically every state of the Union. The table as originally published was found to conform very nearly to Makeham's law, and was subsequently regraduated in accordance with that law for use in connection with joint life calculations. The American Experience Table has been widely used in America as a basis for insurance premiums even when another table was prescribed as a legal basis of valuation, as it presented a conservative view of the mortality after the effect of selec- tion had worn off. The rates of mortality shown were higher than those in the Actuaries' table for ages 30 and under and for ages 78 and over, but lower between 30 and 78. Com- pared with the H M Table, which was published about the same time, it gave higher rates of mortality for ages under 36 and over 80 and slightly lower for the intervening years. Compared with the O M(6) it shows higher rates of mortality for ages 40 and under and for ages over 70 and lower values for the intermediate ages. It will thus be seen that in general the American Experience Table seems to give relatively low rates of mortality for the central ages and high rates for the young and old ages. THE NATIONAL FRATERNAL CONGRESS TABLE 23. This table was constructed by the Committee on Rates of the National Fraternal Congress, an association of Fraternal Societies in the United States of America, and was presented MORTALITY TABLES 13 in its original form at the annual meeting of that association in 1898. It was based on the experience up to that time of the societies connected with the Congress. It was subse- quently regraduated by Mr. Abb Landis and reported in its amended form the next year. Compared with the American Experience Table the rates of mortality are lower throughout, although the difference is proportionately smaller at the older ages than at the younger. Compared with the M Table the rates of mortality are higher at ages 20 to 27 inclusive and for ages 81 and over and lower at the intervening ages. 24. This table of mortality with interest at four per cent is prescribed as a basis for minimum rates of contribution in fraternal orders by the laws of several States. It is worthy of note that a subsequent investigation was made of the experi- ence during the year 1904 of 43 societies. This experience covered 2 880 166.5 years of exposure and 19 414 deaths, and the rates of mortality in the resulting table were lower than those in the National Fraternal Congress Table for ages up to 52 inclusive and for ages 79 and over, but higher for the inter- vening ages. THE M. A. TABLE OF THE MEDICO-ACTUARIAL MORTALITY INVESTIGATION 25. This table was constructed in 1912 by the joint committee of the Medical Directors' Association and the Actuarial Society of America in charge of the Medico-Actuarial Mortality Investigation into the relative mortality of special classes of risks. It was intended for use as a standard with which to compare the mortality of the special classes. It was there- fore based on the experience of the same companies as con- tributed to the special class experience on policies issued during the same period and observed up to the same date. The data used were based on the experience of the companies on policies issued during the month of January in odd years and July in even years from 1885 to 1908 inclusive, observed to the anni- versaries in 1909. The total number of policies was 500 375, 14 MORTALITY LAWS AND STATISTICS the total years of exposure 2 814 276 and the number of policies terminated by death 20 222. The table is shown in the form of analyzed rates of mortality for the first four policy years with an ultimate table for the fifth and subsequent years. A special feature of this table is that the difference between the rates of mortality in the early policy years and those shown for the same attained ages in the ultimate table is relatively small. This has been explained on the theory that an improve- ment in general mortality conditions was going on during the time of the observations and that, owing to the fact that the observations in the early policy years were on an average made at an earlier date than those for the longer durations, this partly concealed the true effect of selection. This theory was confirmed by investigating separately the experience on policies issued in the years 1885 to 1892 inclusive, those issued in 1893 to 1900 inclusive, and those issued in 1901 to 1908 inclusive. A progressive improvement was shown in passing from one group to the next. In the ultimate part of the M. A. Table the rates of mortality are throughout lower than those for the same age in the American Experience Table, but prac- tically equal at age 69. Compared with the National Fraternal Congress Table, they are lower at ages under 55 and over 80 but higher between those ages. For ages under 70 the rates of mortality are lower than in the ultimate part of the British Offices' O (M1 Table, but after that age they agree exactly with that table. 26. This table was constructed in a special way for the special purpose above indicated and is not recommended by its authors for any other purpose. The experiences, however, of some individual companies which have since been investi- gated appear to confirm substantially the ultimate part of the table as a -fair representation of ultimate mortality of in- sured lives in American and Canadian Companies transacting a normal business. MORTALITY TABLES 15 MCCLINTOCK'S ANNUITANTS' MORTALITY TABLES 27. These tables were constructed in 1899 by Dr. McClin- tock on the basis of experience of fifteen American companies, collected and analyzed by Mr. Weeks. The data comprised the entire experience of the companies on annuities up to the anniversaries of the contracts in 1892. Separate tables were constructed for male and female lives, the number of lives taken into consideration being 4365 males and 4821 females. Although this was an experience of American companies only about one-fourth of the number of annuitants were actually American lives, the remaining three-fourths representing an- nuities granted abroad by the companies. The experience was taken out strictly by lives, all duplicates being carefully eliminated, and in the case of deferred annuities only the ex- perience after the annuity became payable was considered, owing to the uncertainty with regard to the date of death during the deferred period. 28. Each table was graduated by Makeham's formula, (Art. 50), the same value of c being used for the two tables and in consequence of this fact the principle of uniform seniority may be used, although in a modified form, even where the lives are not all of the same sex. The formula adopted was colog />z = log b+c x log h, where Iogc = .o4 and for the male table log b = .003 2 and log log h = 5.5 5; for the female table log b = .0015 and log log ^ = 5.43. The rate of mortality is higher throughout the male table than for the same age in the female table, the difference being proportionately greatest at the young ages. The rate of mortality in the male is higher than in the American Experience Table up to age 62 and lower above that age. In connection with these tables it should be remembered that at the young ages they are purely theoretical, there being only two actual deaths at ages under 40 in the male experience and three in the female. These tables are now prescribed by the law of New York State as the basis for the valuation of annuity contracts issued by life insurance companies. 16 MORTALITY LAWS AND STATISTICS THE BRITISH OFFICES' LIFE ANNUITY TABLES, 1893 29. These tables are derived from the experience of British Offices in respect of life annuitants, male and female, during the period 1863 to 1893, including the British Annuity experi- ence of three American companies. Both select and aggregate unadjusted tables were constructed, duplicates being separately eliminated for each. After the final elimination of duplicates for the aggregate tables the total number of male lives involved was 6728, the number of years of risk 53 599 and the number of deaths 3503. For the female table the number of lives was 1 8 951, the number of years of risk 173 519 and the number of deaths 9107. 30. The graduated tables constructed from these data were shown in the select or analyzed form with separate rates of mortality for each of the first five contract years, merging into an ultimate table at the end of the fifth year. The male table was graduated by Makeham's formula (Art. 50), mod- ified for duration, the value of logioc being .038. The female table could not be graduated as a single series by that law. A second series was therefore introduced and it was assumed that' / w +i = /iS.f+C+i> where /$+, and /$+, each con- formed to Makeham's formula modified for duration. The rates of mortality in the ultimate part of the male table are lower than in McClintock's table for ages under 50 and over 82 and higher for the intervening ages. In the ultimate female table the rates of mortality are higher throughout than hi McClintock's table. The value at 3^ per cent interest of an annuity at date of issue is somewhat higher by the British Offices' Male Table throughout than by McClintock's Table. By the female table the value at date of issue is lower than by McClintock's table for ages under 62 and from age 69 to age 75 inclusive and higher for ages 62 to 68 inclusive and for ages over 75. CHAPTER II THE MORTALITY TABLE AND PROBABILITIES INVOLVING ONE LIFE 31. The mortality table has been defined as " the instru- ment by means of which are measured the probabilities of life and the probabilities of death." It may be considered as primarily a table showing how many on an average survive to each attained age out of a given number living at some selected initial age. The symbol l x is used to denote the number surviving to age x and if a be the initial age selected it is evident that l a represents the given number observed, since they are all living at that time. The mortality table, therefore, asserts that on the average out of /<, persons living at age a, l x will survive to age x, where x is any higher age. But if on the average in each N out of a series of cases in which an event A is in question A happens on pN occasions, the probability of the event A is said to be p. The probability, therefore, that a life aged a will survive to age x is l x /l a . 32. This is a property, however, which is not confined to the initial age a. Consider any third age y greater than x. The probability, then, of a life aged a surviving to age y will be ly/la- But this event may be considered as a compound event, being composed of a life aged a surviving to age x and a life aged x surviving to age y. The probability of the first is lx/ l a ', therefore, by division, the probability of the second is ly/lx- This may also be demonstrated from the considera- tion that the l v survivors at age y are the survivors out of l x living at age x, because they are all included among the l x , and there are none included in l x who, if surviving at age y, would not be included in /. Therefore, again the probability of a life aged x surviving to age y is l v /l x . A single table, 17 18 MORTALITY LAWS AND STATISTICS therefore, of the values of l x gives by a single division the probability of survival for any age and period included in its range. 33. The probability of a life aged x surviving n years is designated by n p x , and since the attained age at the end of the period is x+n we have npx=l x +n/lx ........ (l) The probability of surviving one year is denoted by p x , so that px = lx+i/lx ........ (2) From these equations it is evident that we have the gen- eral relation np X = px-px+l'Px+2 Px+n-l- ... (3) We thus see that from a complete table of the values of p x the probabilities over longer periods can be calculated. Eq. (2) can in fact be stated in the form l s +i=l x px by the appli- cation of which the successive values of l x can be calculated, starting from any given value. 34. Hitherto we have dealt with the probability of sur- vival over a specified period. The complementary probability is that of death within the period. The probability of a life aged x dying within one year is denoted by q x . Since the life must either survive one year or die within the year we have whence we obtain the following (4) It is usual to designate the function (l x l x +i) by d x , so that dx denotes the number dying between the ages x and x+i, or at age x last birthday, out of l x living at age x; or out of l a living at age a. 35. Suppose that w i is the highest age at which any survivors are recorded in the mortality table, so that l w -\=d w -i and /. = o, then MORTALITY TABLE AND PROBABILITIES INVOLVING ONE LIFE 19 or , ' In fact, if a frequency 'curve is supposed to be drawn suchV_^ that the height of the ordinate corresponding to any age x is proportionate to the number dying at that age, then d x will be the area included between the ordinates for ages x and #+i, and l x will be the entire area between the ordinate for age x and the limit of the curve. 36. The probability of a life aged x dying within n years is denoted by \ n q x and we have, since \ n qx-\- n px= i, (6) 37. The probability that a life aged x will die in the nth year from the present time is evidently compounded of the probability that it will survive n i years and that having done so it will then die within one year, and is denoted by n -\\q x . Hence we have n -i\q I = n -ipx'q x +n-i- From this equation or from the consideration that the number, out of l x living at age x, who die in the wth year there- after is d x+n -i we have i _ n-i\Qx- , 'X v'x4-n 1 vi 38. We have heretofore considered only the values of /* for integral values of x, but it is evident that deaths occur at all times throughout each year of age so that l x may be con- sidered as a continuously varying function. Let us investigate its rate of decrease at any particular age. This rate is the limit when n vanishes of the function (l x l x+n )/n which rep- resents the average number dying per annum over a period of n years. This limit may be expressed in the language of the infinitesimal calculus as ^. The ratio of this instan- dx taneous rate of decrease of l x to the corresponding value of l x 20 MORTALITY LAWS AND STATISTICS is called the force of mortality and is denoted by /u*, so that we have Unless, however, it is possible to express l s as an algebraic function of x we cannot determine exactly the value of -, 08 and consequently of n z - Certain approximate expressions can, however, be determined on the assumption that differential coefficients of a high order may be neglected. We have and / =/ ++ + + . etc dx 2 dx 2 6 dx 3 24 dx* I =i -h _i_j+ dx 2 dx 2 6 dx 3 24 dx 4 by Taylor's theorem, whence, x . x . - | -+etc. dx 3 dx 3 d 3 l If we assume that -r-| and higher orders may be neglected, (LOO and put h = i, we have as a first approximation =m -i } dx 2 whence J7 .... (9) d 5 l 39. Next let us assume that - \ and higher orders may be dx- } neglected, and substitute for h successively i and 2. Then we have , , dl x . i d 3 l x dl x . 8 d 3 l x ^--\ y-r. dx dx ' MORTALITY TABLE AND PROBABILITIES INVOLVING ONE LIFE 21 whence and 40. It will be noted that by the principles of the infinitesimal calculus we have d log e l x _dlx/j /I 'V* /T^ U/-k U/* whence we have, _ d \0ge lx / v 41. We have already pointed out that trie distribution of the lives according to duration may be represented by a frequency curve, and in this case as in the case of frequency curves in general, we may seek for some typical value to rep- resent the curve as a whole. There are three quantities some- times used for this purpose. The first is the mode, or that value of the variable for which the probability is the greatest. In the mortality table this corresponds to the age at death for which l x n x is the greatest. The most probable duration of life is therefore the difference between the present age and this definite age provided the present age is less than the age of maximum deaths. 42. The second quantity corresponds to the median in the theory of frequency curves and is the duration which the life has an even chance of surviving. It is known as the vie probable sometimes translated into probable lifetime. Its value for any age x is determined by solving for n the equation npx = \ Or l x +n=%lx- 43. The function most commonly used, however, for the purpose of summarizing the probabilities of survival of a given life is the expectation of life which corresponds to the mean value in the theory of frequency curves. The curtate expectation of life is the average or expected number of com- 22 MORTALITY LAWS AND STATISTICS plete years survived by lives of a given age. Its value may be calculated as follows: Out of l t lives at age x, d z die in the first year without completing a year of life after age x, d x+ i die in the second year after completing one year, d x+2 after completing two years and so on. Therefore if we designate the curtate dura- tion of life at age x by e z we have, lie* = d x+ 1 + 24 +2 +3^+3 +etc. . . . (12) Substituting now for d x +i, etc., their values in terms of 1 X we have s -^+4) +etc., or _r + l+r + 2+r + 3"~. _ n~is+ n / \ *~ ~~ '' (14) This same expression for the curtate expectation may also be obtained by considering separately the number com- pleting each year. The number completing the first year is / I+ i, those completing the second year l x+2 , the third year /z+3, and so on. Therefore the total number of years com- pleted by the l x lives is /z+i-Hz+a+^+s+etc., and the average is obtained as above. 44. The calculation of the curtate expectation takes account only of years of life entirely completed and omits the fraction of a year survived in the year in which death occurs. The complete expectation of life, denoted by e x , includes this fraction of a year. In arriving at a first approximation to the value of e x , it is usual to assume that this fraction, which may have any value from zero to one year, averages half a year, so that we have (15) MORTALITY TABLE AND PROBABILITIES INVOLVING ONE LIFE 23 Or substituting for e x its value in terms of l x , we have o _ \lx + /X + 1 +lx + 2 + lz + 3 +etC. _Z, J +L g +i+L r +2+etc. ; > I 10 / *z where generally, .-i(4+4+i}- 45. The exact expression for the complete expectation of life is evidently Integrating by parts in the regular way this becomes l*e* = - [&+!" + f "/,+<& = rl x+t dt, . . . (18) /0 ^/O since tl x+t vanishes at both limits. This integral cannot be evaluated exactly unless an algebraic expression in integrable form can be substituted for l x+t . An approximate expres- sion can, however, be obtained by dividing the integration into yearly intervals. We have /*< + ! /"I / ,//_ I / .Al, ll-^-lUl I I'x-^l-l-Jl lilt J* Js*, I 1 / /// /f2 fJ2J W /73/ IA .J4/ 1/7 I L****+* I n ** l X + t i '* t* t-I + t , '* C* *3; + / = ' [L Yt + h 1 \r r dt 2 dt 2 6 d/ 3 24 t/^ 4 -I 4- etc.dA. _. i dl s+t , i </ 2 /z+ t , i cPl x+t . i d 4 l x+t >-H"T T] "TT -- Tn ' 775 ' TTt -- r"tC. 2 dt 6 dt 2 24 dt 3 120 J/ 4 from Taylor's theorem; also, i, ,, v , ^i dl x+t i d 2 l x+t ^ i <P/ g+< i d 4 l x+t -- l*+t-i ---- -7: I --- T^ I --- ^s~~l o . rt +etc. 2 a/ 4 rf/ 2 12 rf/ 3 48 fi/ 4 /dlg+t+i _ dfx+t\ _ \ dt dt / d 2 l x+t . i d?l x+t dt 2 2 dP 6 APfc-M+i d*l x+t \_d*l x+t \ dt 3 ' dt 3 ) dfi 24 MORTALITY LAWS AND STATISTICS Hence we have / I2\ dt dt i APU-M d?l x+t \ 720\ Putting then / successively equal to o, i, 2, 3, etc., we get XI / i2\dx dx/ +-L 7 rv^-Ku*+^) T ( dii+z dix+i ] ./ i2\dx dx / i d*l x+1 s t+l , ., , o etc. 72o\ dx* dx* etc. Whence, summing and remembering that at the upper limit l x+t and all its differential coefficients may be assumed to vanish, we have Ci rH-*i j-v 00 / i 1 dl x i d 3 l x l x+ tdt = ?lx-rZi l x + t -\ - --- ^-r+etc., Jo 12 dx j2odx- i 17 1 v7 J 7 I d l x . = 2/ z +2 x l z+t -- l x n x -- -+etc. 12 720 dx 6 i d*l Hence, if we neglect the term - f and all higher differ- "j2odx 3 ential coefficients, we have l x e x = 5^ + 21 l x+t - or ex = ?+ex-^Mx ........ (19) This shows that the correction to the first approximation is approximately faux- As the value of this correction is very small except at extreme old age it is usually neglected and the first approximation used for e x . MORTALITY TABLE AND PROBABILITIES INVOLVING ONE LIFE 25 46. From a table of the expectation of life it is possible to derive directly the corresponding rates of mortality, for we have so that we have or *X-{-\ f \ \ jk / * I \ f \ x ~ \ ^ "T~ @x + 1 / /'z \ ^ ~i ^c -f-l/j" \ 2O J whence, <?* #*=- , and Eq. (21) gives the rate of mortality in terms of expectations and Eq. (20) gives a rule for determining e x from p x and e x+ i. By the successive application of this formula, beginning at" - the oldest age, the expectation of life for all ages may be computed from the rates of mortality without constructing the l x column. 47. The value of n x may also be expressed in terms of complete expectations of life, for we have l x e x = I l x dx, from Eq. (18). Differentiating, then, with respect to x we have, after changing sign, 7 _ / dtx _ j * dx or de HxCx = i +-r = I \ (r- ie x +i], approximately (LX whence 4 CHAPTER HI FORMULAS FOR THE LAW OF MORTALITY 48. Before the various labor-saving devices now in use in connection with the calculation of monetary values from the mortality table had been invented, the desirability of reducing, if possible, the mortality table to a mathematical law in order to facilitate such calculations was especially evident. The first attempt of this kind was made by DeMoivre in his " Treatise of Annuities on Lives," for the purpose of passing from the expectation of life to the value of a life annuity. His assumption was the very simple one that the value of d x was the same for all ages, or in other words, that l x decreased uniformly up to the limiting age. The equa- tion for l x in terms of x can therefore be written l x = a(wx), (i) where x is less than w. Whence dl x , = -a=d x , dx and therefore, a i i ^ q x = n* = r- i ( 2 ) a(w x) wx also /**%/ /*"** \ I e I = \ l x dx= ( a(w x)dx= [(wx) 2 ] v =-(ivx) 2 J* J* whence O J/ \ / \ This equation may also be stated in the form w = x-}-2e x , which shows that, if the formula applied, the function x+2c x would be a constant. The calculation of this function for 26 FORMULAS FOR THE LAW OF MORTALITY 27 two or three ages at intervals, or the examination of the d* columns of any mortality table based on actual experience will show that DeMoivre's hypothesis is only a rough ap- proximation to the truth. While it accomplished its -purpose of enabling approximate life annuity values to be calculated from the expectation it cannot be accepted as a statement of the true law of mortah'ty. 49. In the next attempt the problem was approached directly by an investigation of the causes of death. It was made by Benjamin Gompertz, who assumed that the force of mortality increases in geometrical progression with the age. This may be written as follows: M. = 5c*, (4) where B and c are constants for a given mortah'ty table, but may have different values in different tables. From this equation we have d log* l x _ _ E * UL . dx Whence, integrating with respect to x, we have D log /z = loge &- r C*, logeC or /* = */, (5) TO where \og e g=-- , or B = - log e g log e c. We may thus lOge C express \i x in terms of the constants of the mortality table by substituting this value for B, so that we have x =- (loge g' loge C]C X (6) 50. Gompertz's formula constituted a genuine approxima- tion to the law of mortah'ty, but it was found that it did not apply to the period of childhood, and that even at adult ages it would not cover the complete range without a change of constant at an age in the neighborhood of 50 or 60. To remedy this Mr. Makeham proposed to modify Gompertz's formula 28 MORTALITY LAWS AND STATISTICS in a way actually suggested by the reasoning of Gompertz himself, who had stated that "It is possible that death may be the consequence of two generally coexisting causes; the one chance, without previous disposition to death or deteri- oration; the other a deterioration, or increased inability to withstand destruction." The modification consisted in adding a constant to the expression for the force of mortality, which became x = A+Bc? ........... (7) = - k)g e S- (loge ' loge C)C* ..... (8) by putting A = log fl 5 Substituting then - , f r M* and integrating, we get CLOC loge l x =log e k+X loge S+C* log, g, where log e k is the constant of integration, or This formula has been applied to various mortality tables with considerable success, reproducing them very closely from about age 20 to the end, but not covering the period of infancy. Certain other tables, however, cannot be reproduced by this formula. 51. The simplest way of determining the constants in Makeham's formula is from four equidistant values of log/*. We have. log l x = log k +x log 5 +c* log g, log /,+, =log k + (x+t) log s+c* + ' log g, log l x+2t = log k + (* 4- 2/) log s +c* +2t log g, log / z+3t = log k+(x+st) log s+c x+3t log g. Taking now the differences, we have log l x+t -log /, =/ log *+<?(<? -i) log g, log /x+2 log l g+t =t log 5+c* +l (c l - 1) log g, log /*+; - log l x+2t = Mog *+c* +2 V - 1) log g. FORMULAS FOR THE LAW OF MORTALITY 29 Taking differences again, we have log l x+ 2t - 2 log l t+t +log l x = c*(c l - i) 2 log g, log l x+3t ~ 2 log /*+2* + log l x+t = *+'(<? - l) 2 log g. Dividing the second by the first, we have ^ ^log I***-* log k+2i+log lx+t log /,+ - 2 log /x+t + log /, From this equation c is determined and then in succession log g, log s, and log k. 52. For example, according to the Makehamized American Experience Table, we have, log k + 20 log 5 +C 20 log g = log /20 = 4.96668 log &+40 log S + C 40 log g = log /4Q = 4.89286 log + 60 log J + C 60 log g = log /60= 4.76202 log k +80 log 5 + C 80 k)gg = log/80= 4.l6l22 20 log S+C 20 (C 20 i) log g = .07382 201og5+C 40 (c 20 l) logg= .13084 20 log + C 60 (c 20 - l) log g = .60080 c 20 (c' 20 i) 2 log g = .05702 . ^(c 20 ~i) 2 log g--. 46996 Taking logarithms 20 log c+ 2 log (c 20 -i)+loglogg = "2.756o3W 40 log f + 2 log (C 20 l) +log log g = 1.67206^ 20 log ^ = .91603 log c = . 045801 5 . . (a) Hence c 20 = 8.242 log (c 20 -i)=. 85986 20 log c+ 2 log (c 20 -i) = 2.63575 log log g=^.756o3- 2.63575 =4.12028^ log 8= -.00013191 . (b) 30 MORTALITY LAWS AND STATISTICS 20 log C+log (C 20 -l)-flog log g = 2. 75603**-. 85986 c 20 (c 20 - 1) log g=-. 00787 but 20 log S+C 20 (c~ l) log g = .07382 .'. 20 log 5 = - .06595 log 5 =-.003 297 5 . . (c) 20 log c+log log g = 3. 03 631 n c 20 logg = .00109 20 log 5 + C 20 log g= .06704 but log&-f2ologS + C 20 k>gg = 4.96668 /. logk = 5-0337 2 (<0 53. It is interesting to compare the values thus calculated with those from which the table was constructed. The dif- ferences arise from the fact that in the mortality table the value of l x is expressed to the nearest unit and that we have used five-figure logarithms throughout. The comparison is as follows: Exact Value. Calculated Value. logc .04579609 .045802 logs .003296862 .003298 log g .00013205 .00013191 log& 5.03370116 5-337 2 The value of log c in the Makehamized American Experi- ence Table is one of the highest which it has been found nec- essary to use. The range of values lies between .036 and .046 with a general average close to .04. 54. Unfortunately for the widest usefulness of Makeham's formula it is not possible to evaluate the integral j ks'g^dx otherwise than approximately, so that it does not serve the original purpose for which a mathematical law was sought. Any mathematical law, however, gives a very smooth series FORMULAS FOR THE LAW OF MORTALITY 31 which enables formulas of approximate summation or inte- gration to be used which greatly reduce the labor of cal- culating the values of complicated benefits, and Makeham's law in particular offers great advantage in the calculation of probabilities of survival involving more than one life, on account of the form of the expression for log n p x . This point will be discussed further in connection with those probabilities. The advantage thus secured is so great, however, that it is considered that a mortality table which is to be used for monetary calculations should be adjusted so as to conform to Makeham's law if it can be accomplished without departing too seriously from the facts upon which it is based. The general subject of adjustment or graduation will be taken up in a later chapter. Modifications have been proposed to Makeham's formula for the purpose of making it fit certain tables more closely. These modifications consist in adding terms to the expression for // z . One modification assumes n x =A-}-Hx-\-Bc x , adding the term Hx to Makeham's expres- sion, and another takes the form n x = ma x -\-nb x . Both of these modifications sacrifice a considerable por- tion of the advantage which can be secured from the use of Makeham's formula in its original form. 55. Another formula has been proposed by Wittstein, which is intended to cover the entire range of life from infancy to extreme old age. He assumes that the values of q x may be expressed in terms of x as follows : From the form of this expression it is evident that the first term becomes equal to unity when x = M and that where a is greater than unity and n is positive the value of this term increases regularly up to that value. It appears therefore that M+i must equal the limiting age in the table. Also for infant mortality we have q = a~ M ^-\ , showing that the 32 MORTALITY LAWS AND STATISTICS probability of death during the first year after birth is slightly greater than . Also m ax Now it is evident that this vanishes when M = (Mx) or x = w + T and it will be found that for all cases arising in practice this represents a minimum value of q x . In applying this formula to mortality tables it is found that for normal mortality in temperate climates a is approximately 1.42, n is approximately .63, M is between 95 and 100 and m is not less than 6. With these values it is evident that the second part of the expression for q x decreases rapidly as x increases and becomes negligible at about age 25, so that the first term may be taken to rep- resent adult mortality and the second term to represent the additional mortality of infancy. This formula does not possess the practical advantages of Makeham's and consequently has not been much used in practice. 56. Still another method was adopted by Prof. Karl Pearson, who took the numbers dying at the various ages and analyzed the series into the sum of five frequency curves typical respect- ively of old age, middle life, youth, childhood, and infancy. The table selected was that known as the English Life Table No. 4 (males) and the expression which he deduced for was as follows: ,.22l5(z-71.5) if 4.g-[.Q5524U-41.6)] _j_ 2.6e~ I- 09092 **" 22.5)]* +8.50-2)- 3271 <r- 3271(z - 3) FORMULAS FOR THE LAW OF MORTALITY 33 In the first four curves the maximum values are at ages 71.5, 41.5, 22.5, and 3 respectively, while the fifth theoretically extends below age zero, the ordinate becoming infinite at age .75. The method has not, however, been applied to other tables and it is difficult to lay a firm foundation for it, because no analysis of the deaths into natural divisions by causes or otherwise has yet been made such that the totals in the various groups would conform to these frequency curves. CHAPTER IV PROBABILITIES INVOLVING MORE THAN ONE LIFE 57. IN calculating probabilities involving more than one life it is usual to assume that the probabilities of survival or death of the various lives involved are independent of one another, so that the probability of a compound event is found by simply multiplying together the elementary probabilities of which it is composed. For example, the probability that two lives now aged x and y respectively will both be alive at the end of n years is found by multiplying the probability that the life aged x will be alive, n px, by the probability that the life aged y will be alive, />. For the sake of brevity it is usual to write (x) for a life aged x. If then the probability that both (x) and (y) will survive n years be denoted by n p xv , we have 'x+n' / \ (l) Similarly, where more than two lives are involved, we have npxyz ~ nfx' npy' nPz .... ^2^ 58. These probabilities of joint survival are the elementary forms to which other probabilities are usually reduced. It is interesting to investigate the form which they take when Makeham's law applies. We have \ogs+c x+n = log s+<f(c n -i) logg. Similarly, log nPv = n\ogs +c v (c* - 1) log g. 34 PROBABILITIES INVOLVING MORE THAN ONE LIFE 35 Therefore, }(c n -i} logg Let us now take two lives of equal age w. Then we have log npww = 2n log S+ 2C w (c n - l) log g. If, therefore, 2c v> = c x +c v , the value of n pww will be the same as that of n p xv for all values of n. Thus in all questions relating to the joint continuance of two lives aged x and y we may substitute two lives of equal ages w. The relation 2c v> =c t +c v may be expressed in another form by dividing through by c* when we have ic w ~ x = i +c v ~ x , from which it follows that the value of w x depends only on that of y x and is independent of the actual values of x and y. Similarly for any number m of lives (x}, (y), (z), etc., we have log npxvt . . . (m) = 2 log n p x = mn log s+ (c n - 1) Zc* log g, = m\n\ogs+(c n -i) c"\ogg\=m\og n p w , provided mc" ! = 'Lc z , so that m lives of equal ages may be sub- stituted for any m lives. 59. Under Gompertz's law this relation takes a simple form because the term involving log s disappears and we have log npw ...<> = (c n - 1) Zc* log g = (c n - 1) c w log g = log n p w , provided c w = Zc*, so that here a single life may be substituted for any number of lives. In this case, too, the addition of the same number of years to each of the ages x, y, z, etc., will add the same number of years to w. This property of Gompertz's and Makeham's laws is known as the property of uniform seniority. 60. Another way in which the principle can be applied to Makeham's law is by constructing a hypothetical mortality table such that l' x = ks mx g^, so that we have log np'x = mn log S+C*(C B - 1) log g. We have, therefore, ic = nP xvz . . . (m) 36 MORTALITY LAWS AND STATISTICS for all values of n provided c v> = I,c f as in Gompertz's law. It would thus be necessary to construct a special table for each value of m, but once constructed it would apply to all combina- tions of m lives. 61. Hardy's modification of Makeham's law may be written or log l x = log k+x log s+x 2 log r+c* log g, log npx = log l t +m ~ ^g l*=n log S + (2WZ+W 2 ) log T +C*(c" - l) log g, = (w log s+n 2 log r) + 2nx log r -H?(*- 1) log g. log n p xvt ...() = w(w log s+w 2 log r) + 2n 2x log r w log /> = w(w log s+w 2 log r) + 2www log r+mc u (c n - 1) log g. The first term in each of these expressions is the same and the last terms can be made equal by putting, as in Makeham's law, mc w = '2c t . This will leave an outstanding difference in the second term of 2W log r(2x mw). Since an addition of / years to each of the ages will add the same number of years to w and leave this expression unchanged, it follows that its value depends only on the differences of the ages. Since the variable n enters in the same way into this expression as into mn log s, we may consider it as a modification to be applied to the value of s. In fact, if we put / i Zx \ log s =log s +2 log r{ -- w ), \m ) and log n p'w = (n log s' +n 2 log r} + inw log r + (c n - 1) c w log g then we have log npxvz ...(m)=m log n p' w . (3) It will be seen, however, that this modified value of s depends not only on m, but also on the differences of the ages, so that the complications are considerably increased. PROBABILITIES INVOLVING MORE THAN ONE LIFE 37 62. If we assume k-fr^y, or log /, = log k+a* log r+tf log s, we have, log n px = (a n i) a x log r + (b* i) b x log 5. If, therefore, w is determined by the equation tf/V-2 and / is determined so that then we have log np xv * ...()= (<? ~ i) to" log r+ 0" - 1) /&" log $ = / log .. From this it follows that for the m lives (x), (y), (z), etc., we may substitute / lives of equal ages w. The difficulty is that / is not usually integral and it would, in practice, be found necessary to determine any required value by a double inter- polation because w also is usually not integral. 63. We have seen that for a single life (#) we have the relation e x = 2 n p x . Similarly out of a large number N of groups of m lives aged respectively x, y, z, etc., we find Nip xyz . . . (m) complete the first year, Nzpxyz ...(*> complete the second, and so on, so that if we denote by e xyz . . . (m) the average number of years completed during the joint continuance of the m lives, we have Cxyz . . . (m)= ^nPxyz . . . (m) , ..... (4) Consequently where any expression occurs involving a sum- mation with respect to n of the probabilities of joint survival, we may substitute a joint expectation. 64. Heretofore we have dealt with the probability that all of the lives involved shall survive. Similar reasoning will, however, show that the probability that every one of the m lives will be dead at the end of n years is obtained by multiplying together individual probabilities of death. This probability is expressed by \nOxi/z . . . (m) Or I npxvt ...() 38 MORTALITY LAWS AND STATISTICS We have therefore, |9**... (>= In?* -Uv |?*... = (l- n />z)(l ~nA,)(l ~npz}- (5) In this symbol the bar over the letters denoting the ages of the lives involved signifies that the last survivor of the lives is in question, the probability designated by n q xyt . , . (w) being that the last survivor of the m lives shall have died before the end of the nth year. 65. The complementary probability is npxvz . . . (m) I ~ ( I ~ npx ) ( I ~ npy) , and is evidently the probability that at least one of the lives will survive n years. By expanding the product and reducing the equation may be written as follows: nPxyz . . . On) = ^npx ~ ^npxy + ^npxyz ~ etc. . . (6) In this equation the summation extends over all probabilities similar to the one under the 2, that is, involving the same number of lives. 66. Let us now investigate the probability that exactly r out of the m lives will be alive at the end of n years. This probability is designated by n p M The probability xyz . . . (m) that r particular lives, (#), fy), etc., are alive and the remain- ing (mr) lives (z), (w), etc., all dead is evidently nPxv . . . (T)-\ n (fzw . . . (m-r) Or n p xv . . . (r) ( I n p z ) ( I n /> w ) . . . and the total probability sought is the sum of these probabilities for all the combinations r at a time of the m lives, or pxt . . . <r)(l -/>)(! ~npw) ... (7) (m) From the form of the expression it is evident that it may be expanded in a series of probabilities of joint survival involving from r up to w lives; also that each probability involving more than r lives will appear more than once in the expression because it will appear once for each combination r at a time of the lives involved in it; also that the sign of any probability in- PROBABILITIES INVOLVING MORE THAN ONE LIFE 39 volving r-\-t lives is positive or negative according as t is even or odd. Thus we have v . . . (r+1) r+2Crnxy . . . (r+2) etc. . . . (r) Bw . . . (r+1) 67. This may be verified by supposing all the ages x, y, z, etc., to be equal, in which case 2 B /> Z1 , 2 . . . (r+0 becomes equal to m c T+ tnpx T+t , because m c r+t is the number of terms included in the summation and each term becomes equal to n p x r+t . The whole expression therefore reduces to m^T npx T+lCr'mCr+1 ' npx \r+2Cr'm.Cr+2nPx etc., = mCr'nPx mfr'm-r^l 'npx imCr'm-rCz'nPx CtC. This is evidently the proper expression for the probability in question, because the probability that any particular r of the m lives aged x are all alive and the remaining (mr) all dead is n px(i n px} m ~ T , and there are m c r different groups of r lives included among the m. 68. A very convenient symbolic notation is sometimes used to condense the form of Eq. (8) by substituting Z l for 2 n pxvz ...> when the equation takes the form m p in =Z r - zyz . . . (m) I ' 2 r + 2 ) (r + 3 ) zr+3 1-2-3 In this connection it is to be remembered that the expres- sion is purely symbolic and that no operations can be performed upon it which in any way disturb the meaning of Z'. 40 MORTALITY LAWS AND STATISTICS 69. Let us now investigate the probability that at least r out of m lives will survive n years. This is denoted by ,/> r _ and it is evident that we have xyt . . . (m) xt/z . . . (m) xyz . . . (m) xyz . . . (m) From this we see that the expression may be written in the form nPxyt . . . (r+2)+etC. xvz . . . (m) where a\, az, etc., remain to be determined. But from Eqs. (8) and (10) we have -\-' +(-!) r+fCf, r+t- id +r+t- 1^2) ~ (r+t- 1 Therefore, we have A *V A i Pxyz . . . (r) ~"^nPxyz . . . (r+1) T *>*& . . . (r+2) . . . (m) etc., -Z* ' (n) where the same meaning is assigned to Z r as before. 70. It is to be noted that although the relation is purely symbolic and the function of Z has no meaning except as expanded in ascending powers and then interpreted, we have the following relation: n p . . . (m) =Z r (i+Z)- (r+1) +Z r+1 (i+Z)- (r+2) +Z r+2 (i-f-Z)- (r + 3) + . . . etc., =/> M +/> fr+ir +/> fr+2] +etc.. xyz . . . (m) xyt ... (m) xyz . . . (m) as in Eq. (10). PROBABILITIES INVOLVING MORE THAN ONE LIFE 41 71. Also if we have m lives aged respectively x, y, z, etc., the expected number of survivors at the end of n years is Srft iri , where the summation extends over all values xyz . . . (m) of r from unity to m. Expressed symbolically this becomes, from Eq. (9), Z(i+Z)- 2 +2Z 2 (i+z)- 3 + 3 Z 3 (i+Z)- 4 +etc., {i + 2 Z(i+Z)- 1 + 3 Z 2 (i+Z)- 2 +etc.}, (12) This may be also verified by reasoning similar to that by which Eq. (n) was deduced. We thus see that the expected number of survivors out of any group of lives is found by adding together the individual probabilities of survival. 72. Another class of probabilities involving more than one life relates to the order in which the deaths occur. The probability that (x) will die in the nth year from the present time is l . ~npx, - lx lx Px- and the probability that (y) will be alive at the end of the wth year is n p v . The probability therefore that (x) will die in the wth year and (y) will be alive at the end of that year is Summing this function for all values of n from unity up, we get the total probability that (y) will be alive at the end of the year in which the death of (x) occurs This sum is Similarly, the probability that (x) will be alive at the end of the year in which the death of (y) occurs is - e x -^\ e^, Pv-i 42 MORTALITY LAWS AND STATISTICS and the probability that both deaths will occur in the same year is the complement of the sum of these probabilities and is there- fore, i i , *< * . y , *t Y M. Px-l py-l It may be assumed that where both deaths occur in the same year the chances are even that the death (x) will occur before that of (y) . The total chance therefore that (x) will die before (y) denoted by Q\ u is 0, f i _ e \ + lL + 2e i c _ i 1 \Px-l X Z j 2\ p x _i X p y _i X " l j = l| I+ _l_ e x . l e x .\ . (13) 21 px-l Py-1 } 73. The same probability may be otherwise expressed in terms of the infinitesimal calculus by indefinitely reducing the intervals considered. The probability that the death of (x) will occur in the interval of time between t and t-\-dt will d I 4. be dt/lg. and the probability that (y) will be alive at I at that time is . The total probability therefore that ly (y) will be alive when the death of (x) occurs is dl x 7 /" // fl f i X l -y U'V^y0 ~T ;" * ~T \*X*V^Xy ) y Ix'v dx i d PROBABILITIES INVOLVING MORE THAN ONE LIFE 43 It> i ii / dx r dx ' K^z-i :v~ 6 Wi :v) approximately. (14) 74. Similarly, where m lives, (x), (y), (2), etc., are involved, the probability that (x) will die first is dlx+t* i 7, (TO) I ( ~Y~J **'v ... 70 7, at . . . (m) : yz . . . (m))- 75. Also from the fact that the probability that (x) will die rth in order out of the group of m lives may also be stated as the probability that, when (x) dies, there will be exactly m r survivors of the m i lives other than (x), we may express this probability by the use of Eq. (9) in terms of probabilities of dying first. In fact, if we denote by Y l the sum of the values of Ql vz ... for (x) along with all groups / at a time of the (m i) lives (y), (z), etc., we have where the summation included in Z covers only the m i lives (y), (z), etc. X=o // z+l ,p x Z l dt = Y 1 for all values of /. Therefore expanding, integrating, and condensing, we have For m = 3 we have Qxvz = ^xxvz-\-^(t-x-l:vz~ e x+i:vz), - - (i?) ^ z = F(i + F)- 2 = F- 2 F 2 , (18) ..... (19) 44 MORTALITY LAWS AND STATISTICS 76. Where Makeham's law is assumed to hold, the prob- ability Ql v takes a special form. We have generally Qxy~ 77" I ly+t ,. dt = I Iv+Jx+tVx+tdt, IxlyjO WvJo c' t p xv dt. Similarly, Therefore, or B I c' t p xy dt = * y (i 2 A e xv }. Jo c -\-c Substituting this in the expression for Ql v , we have ^\\ A O C/ / A I - C -A- -e xv (20) 77. For Gompertz's law we have A=o, and this equation takes the form We thus see that according to this law the probability of survivorship depends only on the difference of ages. CHAPTER V STATISTICAL APPLICATIONS 78. SUPPOSE that in a certain country there is neither emigration nor immigration and that the number of births occurring each year is uniform and equal to IQ. Then it is evident that on the assumption that the law of mortality remains unchanged the number of inhabitants attaining age x in each year is l x , being the survivors out of the IQ who were born x years before. Therefore, at any moment the number in the existing population whose age is between x and x+dx will be l x dx, being the survivors out of the I dx who were born in the interval dx years x years earlier. The total population at age x last birthday or between ages x and x+i is, therefore, If now we assume that the deaths are evenly distributed, so that we have for the population at age x last birthday denoted by L x the following L x = f Q \l x -td x )dt = l x -%d x = $(l x +l x+1 '). . . (i) Also the deaths occurring per annum between the ages x and x+dx will be l x n x dx = -~ dx. Therefore, the total deaths dx per annum at age x last birthday or between ages x and x+i rx+i $ will be -j^dx = l x l x +i=d x . Summing this for all ages J z dx x and over we see that the total deaths for those ages is l x , so that the aggregate number of deaths per annum at all ages 45 46 MORTALITY LAWS AND STATISTICS will be /o, which is also the number of births. The population is therefore constant in total number and also in age com- position. 79. The total population at age x and over will be 2 X L X , which is usually denoted by T x , so that we have The total population at all ages will be T Q = l<>eo, ....... (3) The general average death rate is obtained by dividing the total number of deaths per annum by the total population, its value is therefore equal to lo/T = lo/loe = i/ei or the re- ciprocal of the complete expectation of life at birth. Similarly, the average death rate at ages x and over will be lx/T x = l x /l x e x = i/e x . The total number of deaths per annum between ages x and x-\-n will be l x l x + n , and the total population at the same ages will be T x T x+n , therefore, the average death rate between those ages will be ~ * +n . When n is equal to i , this be- * x 1 x+n comes the central death rate for age x last birthday and is denoted by m x , so that we have l x l x +i _d x _ d x d x / v mx- - i/;,/ \~j_ij' ' ' w JL x -I x+1 J^x 2 WE "T Ix+l) h -2 a x If we divide both numerator and denominator by l x , we have m x in terms of q x , as follows: *, =-V or ---i ..... (s) i-rffc m * Q* So. Assuming uniform distribution of deaths within each year of age, the sum of the ages at death of all those dying in a year at ages x and over will be = (x+e x )l x . STATISTICAL APPLICATIONS 47 Since the total number of deaths at those ages is l x it follows that the average age is x+e s . Putting x equal to zero, we see that the general average age at death is e . The aggregate of the ages at death of those dying between ages x and x-\-n is evidently and the number of deaths is l x l x + n , so that the average age is (T x - T x+n ) +xl x -(x+n)l x+n _ T x -T x+n -nl x+n l 1 / / l x t x +n "X "x+n 81. We have seen that the population at age x last birthday is L x , and the total population is T , so that the proportion of the total population at age x last birthday is L X /T and the proportion between ages x and x-\-n is (T x T x+n )/T . Sup- pose, for example, that all young men are required to serve in the army from age 18 to 21, then, assuming a stationary pop- ulation, the proportion of the total male population so serving will be (Tig 2n)/2V 82. We have hitherto assumed that we are dealing with a stationary population. A consideration, however, of the ques- tion leads to the conclusion that such a condition never exists, but that, owing to various disturbing factors, the percentages of the total population at the various ages will not be exactly the same as in the assumed stationary population derived from the mortality table representing the actual death rates experienced. It is evident that, if for any reason we have in one community more than the normal percentage of the pop- ulation at those ages where the death rate is low and in another community less than the normal percentage at those ages, then, even though the death rate at every individual age might be the same in the two communities, the general average death rate in the first will be less than in the second. It cannot be assumed, therefore, that a higher average death rate nec- essarily means a more unfavorable mortality experience. A correction must first be made for the difference in age dis- tribution. 48 MORTALITY LAWS AND STATISTICS 83. One method of making this correction is to construct the mortality tables representing the observed death rates, analyzed by ages, in the two communities and calculate from such tables the complete expectation of life at birth. From the fact that in a stationary .population the general average death rate is the reciprocal of this expectation it is readily seen that this amounts, in effect, to substituting for each actual population the stationary population corresponding to its actual mortality. It is readily seen that this method may be applied to the death rates for ages above any assigned age, or within given limits, by constructing the corresponding por- tion of the mortality table and calculating the average death rate in the stationary population. For example, it might be desired to compare the mortality in two communities for ages 15 and over or for ages 15 to 64 last birthday inclusive. The corrected death rate for the former would be i/Ji 5 , and _ j * 15 5 for the latter =^&f-. The labor of constructing a il5~-*65 2i 5 L z mortality table is, however, considerable and other methods of correction are usually followed. 84. Although the stationary population is largely of theo- retical interest the notation derived from it is useful with cer- tain modifications in connection with actual population statistics. For this purpose 6 X represents the deaths between age x and age x+i, and X z = ^+0 z+ i+etc., is the total number of deaths at age x and over, but is not equal to the number attaining age x. For the population between ages x and x+i the symbol L x is retained. The symbol T x is also used to denote the total population at ages x and over, so that we have as before. The general average death rate is then \o/T , but is not equal to i/Jo except for a stationary population. Simi- larly for the average death rate at age x and over we have \x/T x but not i/l z , and for ages between x and x+n we have \c~ \c+n/*x~ 1 x+n- STATISTICAL APPLICATIONS 49 85. One method of correcting the death rates of different communities is to analyze each into certain age groups, usually quinquennial up to age 15, then decennial up to age 85, with a final group for ages 85 or more last birthday, the average death rate for each group being used. These death rates are then applied to a standard proportionate distribution of the population into these age groups. One standard which has been used is the age distribution of the population of England and Wales according to the Census of 1801. The general average death rate for the standard population on the basis of the observed group rates for each community is thus calculated and this is considered as the corrected death rate for the com- munity. In this way all communities entering into the com- parison are placed on the same footing with respect to age distribution. The same method may be extended to cover varying proportions of the two sexes by analyzing the statistics for the different communities and also the standard popu- lation in this way. It may, in fact, be extended to cover any factor, such as occupations, considered as having an impor- tant bearing on the mortality to be expected and for which the necessary data can be obtained. 86. Another method of comparison is to use a standard scale of death rates for the different groups into which the actual populations are analyzed. The actual population in each group is then multiplied by the standard death rate and the expected deaths according to the standard are thus cal- culated. The total of the actual deaths in the community is then expressed as a percentage of the expected and these percentages for the different communities are compared. 87. These two methods have been described as applying to a whole community, but it is evident that they apply also to a part, such as those aged x and over, or those whose ages lie between x and x+n, or those who are engaged in a certain occupation. In fact, what may be considered as mortality index numbers for various occupations have been formed from the census and death returns (in England. A standard population is taken, analyzed into the five decennial age groups 50 MORTALITY LAWS AND STATISTICS between 15 and 65, the aggregate population being such that the expected deaths according to the general average death rates for occupied males in the various age groups will total up to 1000. The actual death rates for the various age groups in each occupation are then applied to this standard population and the t resulting total of expected deaths gives a number whose r^tio to 1000 measures the general mortality of the occupation. This is in effect the standard population method above described with the addition that instead of recording the corrected average death rate we record its ratio to an average death rate based on the same standard population combined with standard group death rates. 88. The standard population method is the one most used for the comparison of general population mortality statistics, while the standard death rate method is most used in con- nection with the mortality of insured lives. In connection with such insurance statistics three modifications are made. The first is that the actual experience is usually analyzed into individual years of age and sometimes also into years elapsed since medical examination. The second is that the rate of mortality or probability of dying within one year is usually used instead of the death rate or average force of mortality, and that along with it the exposed to risk of death, which is dis- cussed under the head of construction of mortality tables, must be used instead of the population. The third is that amounts insured or amounts at risk are frequently taken into account instead of lives, so that we compare actual losses with expected losses rather than actual deaths with expected deaths. 89. In this chapter it has been assumed that the period covered by the statistics is one year. Where a period other than one year is dealt with, we must take the average deaths per annum, and in any event whether for a period of exactly one year or otherwise the average population during the period must be taken. The ratio will, of course, be the same if both of these are multiplied by the period, so that we have on the one hand the total deaths and on the other the aggregate number of years of life during the period. CHAPTER VI CONSTRUCTION OF MORTALITY TABLES 90. In the second chapter it was shown that in any mor- tality table we have the relation l x+ il x p x for all values of x and that consequently if we have a complete table of the values of p x we can, by starting at the initial age and working forward progressively, construct a complete mortality table. A little consideration also shows us that there is an insuperable practical difficulty in the way of constructing the l x column of a mortality table by taking a large group of lives of a given age and following them throughout the balance of their lives, observing the number surviving to each age. This difficulty arises not only from the length of time that would necessarily be consumed in waiting for the last one to die, but also from the fact that out of any large number some are certain to pass out of the knowledge of the observers and from the moment that any do so disappear the further observations are nullified by our ignorance of the time of their death. A correction is therefore necessary and this correction can be most con- veniently applied by a method which also obviates the neces- sity of waiting until some particular group of lives selected at a young age have all died. This method is to use the rela- tion already quoted and to determine separately the values of pi for each year of age. By this method the observations do not necessarily extend over a longer period than one year, although a longer period is usually taken in order to eliminate the effect of special conditions. In that event the observations at different times for the same year of age are combined. 91. The observations are not, in fact, made directly on the value of p x , but rather on that of m x determined by the relation (i) 51 52 MORTALITY LAWS AND STATISTICS where 6 X represents the deaths observed at age x, last birthday, and L x is the corresponding population. But we have in terms of the mortality table m x _ d z _l x l x +i i Px from which we have _2m x 2+m x and *"*>WI q* = T--px = 92. In connection with population statistics it has been usual to calculate m x from the data and then to pass to q x and p x . In connection with observations on insured lives, on the other hand, the practice has been to determine the value of L X +%6 X denoted by E x for each age and so to pro- ceed directly to q x by the equation q x = 6 x /E x . The problem therefore reduces *to the determination of the values of B x for each value of x, and of the corresponding values of L x or E x . The methods followed vary, of course, with the form in which the facts are presented, and the conditions in connection with general population statistics differ so much from those in connection with insured lives that it is well to take up the two cases separately. 93. In the case of general population statistics the in- formation regarding the deaths is usually derived from the registration returns and it is a necessary condition, for their use in the determination of death rates, that the registration should include all the deaths coming within the scope of the investigation. It is evident that, to the extent that the returns are incomplete, the numerator of the fraction determining the death rate is understated and consequently the death rate itself is also understated. For this reason the statistics can be used of only those countries, states or municipalities in which the laws and their enforcement are such as to secure CONSTRUCTION OF MORTALITY TABLES 53 substantial accuracy in the death returns. In the United States those states and parts of states which, in the opinion of the Federal Census Bureau, comply with this requirement con- stitute the registration district. The area included in this district is extended from time to time as the registration becomes more complete. Particulars of the deaths in the various com- ponent parts of the registration area are published annually by the Census Bureau. 94. For information regarding the population corresponding to the deaths reported we must depend upon the census results. As a census is made only periodically, some means must be devised of passing from these figures at periodical intervals to the average population by age groups during the interval covered by the observed deaths. The census returns and the death returns are also frequently given only for groups of ages, and we have therefore an additional problem to solve, namely, that of passing from age groups to individual years of age. 95. Let us take first the problem of finding the average population in a certain age group during a specified period. For the sake of simplicity we will first suppose that the total population analyzed by age groups is known for the beginning and end of the period and that the whole period may be con- sidered for this purpose as a unit of time. Let the total pop- ulation at the beginning be PO and at the end PI, also let the population in any particular age group at the beginning be aP and at the end (a +b) PI. Then, evidently, the sum of the values of a for all age groups must be equal to unity and this is also true for the values of (a +6), so that the sum of the values of b must be zero. Also suppose the ratio of increase of the total population during the period is r, so that we have Pi=rPo. Then it is assumed that at any time t during the interval the population in the age group is P (a-{-bt)r t , the sum of the values of which for all age groups is evidently PQ^. In other words the total population is supposed to vary in geometrical progression, while the percentage of that total in the particular age group is supposed to vary in arithmetical 54 MORTALITY LAWS AND STATISTICS progression. On these assumptions the average population during the period is r i P = -M log e r [log e r (log e r) 2 ]' r ] 'loge 96. It is evident that if the period covered by the obser- vations were the interval between two censuses, the census returns would give directly the values of P . P\, a and b. But the dates upon which the census is taken do not usually coincide with the limits of the period of observations. Suppose, therefore, that we have the results of two censuses taken at the times t\ and fa counting from the beginning of the period and that the corresponding total populations are PS and P 4 , also that the populations in the age group are A PS and BP 4 . Then, according to the assumptions already made, we have or P 4 = r"P or (h - /i) log r = log P 4 - log P 3 , logr = (logP4-logP 8 )/(/2-/i), .... (5) log P = log Ps-h Iogr = (t 2 log Ps-h log P 4 )/(/ 2 -/i). (6) a+bfa =A, a+bt 2 = B, (7) (8) 97. The expression for the average population in the age group, when we substitute in Eq. (4) these values of a and b, takes the form CONSTRUCTION OF MORTALITY TABLES 55 B-AI r i T> ' I f* "1 i *<N \. : r- 7-1 7 log* r I /2 *i fe <i\f I loge ry _ r-ii t 2 A A t r i \ = -^oi ii r~] rl 1 ) log e r[/2 h t 2 ti\ri log e r) B / r i tz ti t 2 ti\ri loge r ri\t 2 i / r i = A P og,r p r-i i / r i \ _ ti ,v lo ge rl/ 2 -/iV-i log e r/ h-til' Since A For 11 and J5P ^ 2 are the numbers shown in the two censuses for the age group in question, it follows that we obtain the average population for any age group by multi- plying the numbers shown in the two censuses by /I fe ^l\r I loge P. and ..r i Iog e r[t 2 -ti\r-i \og e r respectively, and adding together the products. These factors are the same for all age groups and may be calculated once for all. 98. The average deaths per annum may be obtained by dividing the total deaths during the period by the number of years included, or the same object can be accomplished by multiplying up the average population by the number of years to get the aggregate population or years of life corresponding to the total number of deaths. The latter is the course usually followed. 99. Having, then, the total deaths and the corresponding population by groups of ages the remaining problem is to as- certain the death rates for individual ages. An approximation which was formerly used was to divide the total deaths by the total population, and assume that this represented the force of mortality at the middle of the interval, or, in terms of the 50 MORTALITY LAWS AND STATISTICS notation explained in Chapter V, n x+ ? = ~ ^f^_ Where 2 T x T x+n n is odd this gives directly the value of m x +j(-i), the two functions being approximately equal and each equal to d t+i(n _ !)//*+, Thus &+}(,_!> is obtained by Eq. (3) of Chapter VI. Where n is even, however, x-\-\n is an integer. The value of q x+ m is then determined on the assumption that during the year n x increases in geometrical progression at the ratio r determined from the values of /JL X for the neighboring groups. We have then since n x + t = - , x+t at From these values of q x the intermediate values are found by a formula of interpolation. 100. It was always recognized that the quinquennial age group from 10 to 15 required special treatment, and it has recently been shown by Mr. Geo. King that the method under- states the death rate at the older ages. This can be seen by taking any mortality table and, assuming a stationary pop- ulation, comparing the values of ^ ~* with those of m x+2 . T- x~ 1 x+5 In view of this fact some more accurate method is desirable. Greater accuracy has been attained by distributing the total deaths and population of each age group into individual years of age. 101. In the construction of the Carlisle table this distribu- tion was effected by a graphic method. On a base line dis- tances were laid off consecutively representing the number of years included in the successive age groups. On these bases rectangles were constructed whose area represented the total number (of deaths or of population as the case may be) in the age group. The heights therefore represented the average number per year of age in each group. A continuous CONSTRUCTION OF MORTALITY TABLES 57 line with continuous curvature was then drawn through the tops of these rectangles such that the area included between it and the base was the same in each interval as that of the corresponding rectangle. The base was then subdivided to represent individual years and ordinates erected to the curve so drawn. The area between the base and the curve in the interval representing each year of age then represents the number assigned to that year. And the total should agree with the total for the group. 102. Under this method, however, it was found difficult to read off the diagram with sufficient accuracy and an analyt- ical method of redistribution has been devised. If we take the population for the various age groups and sum from the oldest group downwards we obtain a series of numbers rep- resenting the total population older than the respective ages which are the points of division between the groups. In other words the values of T x are given for a series of values of x. If then the values of T x can be interpolated for unit intervals we can calculate the values of L x because we have L X = T X T X+ I. Also, if the deaths are similarly treated, we have a series of values of \ X = I,6 X , from which, by interpolation and differencing, the successive values of 6 X may be deter- mined. The successive values of m x , q x , or p x may then be determined from the relations already given. 103. The intervals used in tabulating death returns and population statistics in the publications of the United States Census Bureau are individual years from birth to age 4 last birthday, inclusive, and five-year intervals thereafter. The returns of some countries, however, give only ten-year intervals, beginning with age 15. This grouping is adopted in order to avoid a transfer of lives from one group to another arising from a tendency to state ages at a multiple of ten years. Where the interval is ten years it is readily subdivided into five-year intervals by the finite difference formula: . (n) This may be easily demonstrated by expanding by Taylor's 68 MORTALITY LAWS AND STATISTICS theorem each of the functions on the right and assuming that fourth and higher differential cofficients vanish. This formula does not apply to the last interval, where we use instead the equation 4 (f( x )+f(x-2t)\=6f(x-t)+f(x-$t)+f( X +t), (") which may be similarly demonstrated. 104. For the sub-division of the five-year intervals a special interpolation formula is used which ensures a continuous series. The first and second differential coefficients are determined at each point of junction by the formulas I2fif"(x) = (14) These formulas may be obtained by expansion as above, except that the fourth differential coefficient is not neglected. For each interval a function is then found such that the values of the function itself and of its first and second dif- ferential coefficient at the beginning and end of the interval will be equal to those so determined for those points. As there are six conditions to be satisfied it follows that if a rational algebraic function is to be used it must be of the fifth degree. It may readily be demonstrated that the following function satisfies the conditions for the interval from x to x+t: t /"(*) V(t-hY , ~ fi f(r I fi ,'(* + *) ~~1*~~ ~T~ ~p ' ' ' This may be seen by differentiating with respect to h and then putting h equal to o and / successively. CONSTRUCTION OF MORTALITY TABLES 59 105. This equation takes a simpler form when expressed in terms of central differences as follows: Let 5 denote an operation such that 8*f(x)=f(x+2f)-4f(x+t)+6f(x)-4f(x-t)+f(x-2t), etc. Then, from Eq. (13) we have or (x)}. (16) Similarly, (f (*+<) = tA*+/) H-WCs+O -iWGH-0} - 1/0) -5 2 /(*)Ki7) And from (14) we have similarly, (*), .... (18) Substituting, then, these values in Eq. (15) and collecting like terms, we have, after reduction, -4' 106. This method cannot be applied in the above form below age 15 because f(x 2t) enters into the formula and the mortality differs so much at infantile ages from that at other ages that it is not safe to assume that /(o) can be determined from the same rational algebraic function as the values of f(x) above age 5. Having determined, however, /(i6) the first 60 MORTALITY LAWS AND STATISTICS and second differential coefficients at age 10 may be determined by the equations . . . (21) These values enable us to interpolate the values of f(x) for ages n to 14 inclusive by the use of Eq. (15). The values for ages 6 to 9 inclusive may then be filled in by determining values for/'(5) and/"(5) such that if 5 is put for x in Eq. (15) the values of 7(3) and 7(4) will be determined by putting h successively equal to 2 and i . 107. Sometimes it is found preferable to interpolate by the above methods values for log T x and log X^ instead of those of T x and X z , but the principle is the same. In fact, any single valued reversible function of T x or \ x can be used if it is found to furnish a series more appropriate for interpolation. One of the most valuable suggestions in this line is probably the use of the ratios of the values of T x and X^ to their values in a stationary population derived from some standard mortality table from which all minor irregularities have been removed. The further treatment of mortality statistics of the general population will be considered under the heading of graduation, in the next chapter. 1 08. When dealing with the mortality experience of a life insurance company or group of such companies the prob- lem is a different one, because in this case information is usually available as to the exact date when each life first came under observation, so that the death if it had occurred would have been included, as well as the exact date when it passed out from observation. The problem in this case is to determine the most convenient way in which the data can be analyzed and how labor can be saved without sacrificing accuracy. It is not proposed to describe how the data can best be collected as that will depend upon various circumstances, particularly as to the mechanical sorting and tabulating devices which may be available and as to the nature of the records from which the information is to be extracted. Attention CONSTRUCTION OF MORTALITY TABLES 61 will be confined to the principles to be adopted in classification. Aggregate mortality tables will first be dealt with, because, even where tables are constructed that are analyzed both by age and by policy duration, the differences in duration are usually neglected when the duration is in excess of some assigned limit and an aggregate table used for all longer durations. 109. The first point to which attention is directed is the analysis of the deaths, three essentially different methods having been used. The first method is known as the age year method and might otherwise be described as the exact method. Under this method the date of birth is noted and the deaths are classified precisely according to age last birthday at the time of death. In this case it is necessary to determine the number observed within each year of age and as the same life is usually observed through a series of ages the calculation can usually be most conveniently made by a continuous process, the value of E x+i being determined from that of E x by the proper modification. The particular form which the modi- fication will take will depend on the treatment of the new entrants and withdrawals. The deaths at age x last birthday are always treated as included in E x for the full year or as included in L x for an average of half a year. In the case of new entrants and withdrawals four different methods are now available. Under the first method the exact age at entry or withdrawal may be noted and the new entrant treated as exposed for the fraction of a year of age after entry, the with- drawals being similarly treated as exposed for the fraction of a year elapsed since birthday at the time of withdrawal. Let us denote by n x the number of new entrants at age x last birthday and by f x the aggregate of the fractions of a year since last birthday at time of entry. Also, let w x denote the number of withdrawals and g x the aggregate of the fractions at withdrawal. Then we have, evidently, E x+ i = E x +(n x+l -e x -w x+1 )-(f x+l -f x ) + (g x+1 -g I ). (23) 62 MORTALITY LAWS AND STATISTICS no. Under the second method of treating new entrants and withdrawals the exact fraction in each case is not cal- culated, but a general relation is assumed such a.sf x =fn x or gx = gw x , based on an examination of part of the data taken at random. In this case we would have E,+ 1 = E x + (n x+ i-e x -w x+ i) -f(n x+ i - n x ) +g(w x+l - w x ) = E x -e x +{fn x +(i-f)n x+1 }-\gw x +(i-g)w x+l \. . (24) Sometimes it is assumed that/ and g are each equal to \. In this and the preceding section those under observation at the commencement of the observations are treated as entrants at that time and those under observation at the close as withdrawals. in. Under the third method the new entrants and with- drawals are classified according to mean age at entry or with- drawal, the mean age being calculated by deducting the calendar year of birth from the calendar year of entry or withdrawal. On the assumption that birthdays and dates of entry and withdrawal are evenly distributed over each calendar year this will give approximately correct results, the cases in which the age is overstated balancing those in which it is under- stated. The maximum difference between the exact age and the mean age is one year. Where the observations are closed at the end of a calendar year with a number still under obser- vation, or started in the same way with a number already under observation, these cases must be specially treated, as the assumption of distribution of entry or exit over the cal- endar year does not apply. It may, however, be assumed that on an average half a year has elapsed since the last birthday. If then n x be the new entrants at mean age x, w z the withdrawals at the same age, a x those under observation at age x last birth- day when the observations began, and e x the corresponding number at the close of the observations, we have \. (25) CONSTRUCTION OF MORTALITY TABLES 63 112. Under the fourth method the age nearest birthday at entry or exit is taken instead of the mean age, the average being again correct on the assumption of uniform distribution. In this case those under observation at the opening and closing of the observations do not require special treatment, but may be grouped at age nearest birthday. If then <r x and e x be the numbers of such cases at age x nearest birthday, we have 113. The second method of analyzing the deaths is by the policy year method under which the completed age at death is determined by adding the curtate duration (or integral part of the duration) to the age at entry determined according to whatever rule may be adopted for that purpose. Under this method the age at entry is usually taken as the age nearest birthday, but may be taken as the mean age. The age at withdrawal and of the existing at beginning or end of obser- vations is determined by adding the duration to the age at entry. The duration may be calculated exactly in each case for the withdrawals, but it is usual to open and close the obser- vations on policy anniversaries in chosen calendar years when the policy year method is adopted, so that there will be no fractional exposures in the case of the existing. We have, therefore, e x + l . . (27) where w x is the number of withdrawals at completed age x determined as for the deaths, and g x is the aggregate of the fractional durations. Approximate methods of calculating the values of g x are sometimes used. The duration of the withdrawals may also be taken as an exact number of years in each case, the mean duration or the nearest duration being used. When this is done fractional durations disappear and we see that in this case Eq. (26) holds. 64 MORTALITY LAWS AND STATISTICS 114. The third method of analyzing the deaths is by cal- endar years, the completed age at death being the age nearest birthday at the beginning of the calendar year of death. Under this method the ages at entry and withdrawal may be deter. mined by any of the methods outlined for the policy year method and the equations of relation will be the same as for that method. In many cases where the calendar year method is applied, the age at the beginning of the calendar year is not taken as the age nearest birthday, but is calculated by adding the curtate duration plus half a year to the age nearest birth- day at entry or by adding the curtate duration to the age next birthday at entry. In the former case the ages for which the rates of mortality are determined will not be integral, but will be of the form x+%, where x is integral. 115. To summarize, let x be the exact age at entry, x\ the age last birthday, so that the age next birthday is x\ + i, (x) the age nearest birthday, \x the mean age, and [x] the the assumed age at entry. Also let / be the exact duration, t\ the curtate duration, (/) the nearest integral duration, and {/| the mean duration. Then, under the age year methods the completed age at death is taken as x+t\, under the policy year methods it is taken as [#]+/|, and under the calendar year methods it is taken as either [#] + |/| , or |ff-H| | or \x !| + |^|. I n the last expression x \ \ is used to designate the mean age six months before entry. In connection with the age at entry we may have [x] taken as x, x\+f, (x) or x\, and in connection with the withdrawals the age at withdrawal may be taken as x+t, (x-\-t], \x-\-t\, x+t\+g, [x]+t, 116. The census methods may also be conveniently applied to a life insurance company's experience, a classification of the lives insured being made at the close of each calendar year, giving the population in each year of age at that time. The completed age may be taken as x+t\ or [#]-H|, usually the latter. If, then, the deaths of a given calendar year be analyzed by completed age at death, the average population during that year for any age year can be determined by taking CONSTRUCTION OF MORTALITY TABLES 65 the mean of the populations at the beginning and end of the year. The exposed to risk will then be determined by adding to this mean population one-half of the deaths. Each success- ive calendar year is treated in the same way and the experi- ence gradually accumulated, the experience of as many years as desired being combined. By a modification of this method the deaths are observed from the middle of one calendar year to the middle of the next, or from the policy anniversaries in one calendar year to those in the next, and the population at the end of the first year is taken to represent the corresponding average pop- ulatfon. 117. Where an analyzed mortality table is to be constructed the policy year method is usually adopted and the experience during each year of duration is treated separately so far as is considered necessary. This amounts to the same thing as treating each age at entry separately for as many years of duration as is decided upon. This latter method is the more convenient for descriptive purposes, because new entrants are thereby eliminated after the start and we need consider only deaths, withdrawals, existing at the beginning of obser- vations and existing at end. Assuming that experience before the policy anniversary in the first calendar year or after it in the last calendar year of the observations is neglected and that nearest durations are taken for withdrawals, we have (20) (29) 118. Having obtained the values of qw+t, those of l [x]+t , and d lx]+t are obtained as follows: The ultimate mortality table representing the mortality after the effect of selection is assumed to have disappeared is first constructed in the way indicated in Chapter II by taking an initial value of l x and multiplying successively by the successive values of p x . The values of l lx]+t are then constructed in reverse order, 66 MORTALITY LAWS AND STATISTICS beginning with the value of t at which the analyzed mortality merges into the ultimate. The working formula is (3) The values of d lx]+l are then derived by differencing from the relation 119. Diagram No. i is shown to illustrate the relation between the rates of mortality in an aggregate table and in a select or analyzed table based on the same data. In this diagram, in order to avoid the necessity of a change of scale, the ordinates are made proportionate to log (1 + 100^), instead of to q x . A scale is given on the margin, however, showing the values of q x corresponding to different lengths of ordinates. The tables selected to illustrate this point are the O M and O IM1 tables. It will be noted that at the young ages the rates of mortality in the aggregate table approach those shown in the analyzed table for the first year, whereas at the older ages they approach and become indistinguishable from those in the ultimate part of the analyzed table. CONSTRUCTION OF MORTALITY TABLES 67 \ x 3 a O O I I 3 I 2 S -9 -C & 8 1 + 1 -2 . 'S tn g S3 "2 2 6 o, 2 9 H h o S to S ^ M <U CHAPTER VII GRADUATION OF MORTALITY TABLES 1 20. WHEN the probability of dying within a year is q x and n lives are exposed to the risk of death, the expected number of deaths is nq x . This means that in a large number of such instances, where n lives are observed in each instance, the average number of deaths occurring will be nq x . It does not mean, however, that in every instance exactly nq x deaths will occur. In fact, unless the values of n and q x happen to be so related that nq x is an integer, the exact relation cannot hold. Any number of deaths from none up to n is theoretically possible. The probability of exactly r deaths is shown by In the principles of the theory of probability to be , <f x p x n ~ rn r and in that case the deviation of the actual number of deaths from the expected will be rnq x . The mean value of the square of this deviation may be shown to be np x q x . Where n is large this deviation is equally likely to be positive or neg- ative and there is approximately an even chance that it will exceed \^npxq x in absolute magnitude. If, then, a value 7* q' x = be determined by dividing the observed number of n deaths by the number exposed, there is an even chance that this value will differ from the true probability q x by at least as much as -*/, and the deviation is equally likely to 3\ n be positive or negative. We have no means, however, of deter- mining from the observations themselves at age x whether q' x is greater or less than q x and in the absence of further 68 GRADUATION OF MORTALITY TABLES 69 information the hypothesis is adopted that q x = q' x , because \n that is the value of q x which makes the probability = p x n ~ r q x r \r\n-r of the observed facts a maximum. 121. If, now, we have made similar observations at a series of consecutive ages for which the true probabilities of death are q x , q x+l , etc., the principle of continuity would lead us to expect that the successive values of q x for the different ages would form a smooth series. Where the values of q' x are calculated each will differ from the corresponding value of q x by a quantity which may be large or small, positive or negative, a positive deviation being as likely to be followed by a negative one as by a positive. The theory, however, indicates that the probable values of these deviations in the value of q x decrease as n increases and that positive and neg- ative values tend to counterbalance one another. It follows from the above that we must expect the values of q' x to form a somewhat irregular series, the amount of the irregularity depending on the numbers under observation. The problem of graduation is to remove the irregularities from this series and to approximate as closely as possible to the true values of q x . 122. The methods which have been adopted for this purpose come under four general classes. Under the first method a diagram is made to represent graphically the observed facts and a continuous curve is then drawn as a basis for the graduated series. Under the second method the graduated series is formed by interpolation on the basis of values determined for fixed intervals, these values being so determined as to give an interpolated series fitting as closely as possible to the ob- served facts. Under the third method the individual terms of the graduated series are each determined by a summation of adjacent terms of the original series, a correction being introduced to allow for the curvature. Under the fourth method a mathematical formula containing arbitrary constants is used to express the series and the constants are determined so as to agree in certain respects with the observed facts. 70 MORTALITY LAWS AND STATISTICS 123. After a graduated series has been constructed it is usually tested with respect to the two points of smoothness and closeness to the observed facts. With respect to smooth- ness the fact that a series is determined by a mathematical formula is usually taken as sufficient, but when it is not so determined the criterion usually adopted is the smallness of the third differences in the graduated series. This smallness is sometimes tested by inspection of the differences after they have been taken out, but in comparing two different graduations of the same series, if it is desired to have a numerical measure of their departure from absolute smoothness, the sum of the squares of the third differences or the sum of the absolute values irrespective of sign of such differences may be taken as such measure. 124. With respect to closeness to the observed facts the requirements usually made are (i) that the total number of expected deaths and their first and second moments about any assigned age shall be the same as for the actual deaths, and (2) that the departures in individual groups shall not, on the average, materially exceed in magnitude those expected in accordance with the theory of probability. This comparison is usually made by recording the difference between the actual and expected deaths at each age. A continuous summation of these deviations is then made with due regard to sign. The smallness of the numbers in this column of accumulated devia- tions, the frequency of changes of sign and the extent to which positive and negative terms balance one another indicate the extent to which the first requirement is complied with. The sum of the deviations without regard to sign tests directly the second requirement if the average deviation which is approximately equal to %^/np x q x be calculated for each age for the purpose of comparison. The comparison may also be based on the sum of the squares of the departures or on the sum of those squares each divided by its mean value np x q x . The test in this last form is supported logically by the fact that if the number observed is large the quantity so arrived at is proportional to the logarithm of the ratio between the GRADUATION OF MORTALITY TABLES 71 probability of the observed facts and that of the expected according to the graduated table. 125. The graphic method of graduating mortality tables arises naturally from the graphic method of representing them. Under this method the values of q x or of m x are rep- resented by points in a diagram. For convenience in plotting the diagram accurately ruled section paper is ordinarily used, the years of age being represented by equal intervals along the base line and the rate being represented on a suitable scale by the distance of the point from the base. When the points corresponding to the successive ages are plotted and joined by straight lines it is found in an ungraduated table that the result is a zigzag line full of minor irregularities, but showing indications, the strength of which depends on the volume of the observations, of an underlying regular law. The graduation of the table is effected by drawing among these points, but not necessarily through any of them, a regular curve to rep- resent this law. Preliminary groupings not covering equal intervals, but so arranged as to produce the greatest attainable regularity are made in order to bring out this law. After the curve is drawn, the values of the ordinates are read off and the results corrected to remove any irregularities due to errors in reading. A comparison is then made between the expected and actual deaths on the lines indicated above and, if a rel- atively large and persistent deviation in either direction is accumulated in any section of the table, the curve is amended to reduce or eliminate it. 126. In applying this method the difficulty is found that, if the scale of* the diagram is sufficiently large to permit of accurate reading in one part of the curve, it will be too large in another. This difficulty is met by plotting not the actual value of q x but some more slowly varying quantity from which it can be determined. One method is to take as a basis some mortality table already constructed from a mathematical formula such as Makeham's and to plot either the ratio of the observed q x to the rate at the same age in the table, or the difference of the rates. Another method is to use log (i + 100^) 72 MORTALITY LAWS AND STATISTICS or log(i + io<fc) instead of q t . Specially ruled section paper has been prepared with the vertical spacing so arranged that if the values of q z are plotted according to the ruling the actual distances from the base will be log (1 + 1009*), so that, if this paper is used, the whole diagram is reduced to practicable dimensions without altering the scale where q x is small, and after the curve is drawn the values of q x may be read off directly. 127. In the interpolation method the problem may be divided into three parts. The first is to determine which func- tion of the mortality to interpolate, the second is to deter- mine the values for the selected ages of that function and the third part is to interpolate the intermediate values. Various func- tions have been used as a basis for interpolation and the method of determining the values at the selected ages will depend on the function selected. Where l x or log l x is used we must have an ungraduated mortality table as a basis and we obtain our points for interpolation by simply taking the values at the selected intervals from this ungraduated table. Where, how- ever, q x or some function of q x , such as log q x , m x , or log m x , is selected for interpolation we must group the data so as to determine with as much precision as possible the value of the function for the selected ages. For this purpose the exposed to risk, or the population, and the deaths are usually com- bined into age groups. In the case of the earlier English Life Tables the simple but somewhat inaccurate assumption was made that the total deaths in an age group of five or ten years divided by the total population corresponding to them would give the force of mortality at the central age of the group as explained in Chapter VI. 128. Where more accurate values are desired the redis- tributed values of the deaths and exposures or population for some one year of age in each group may be calculated by methods similar to those described for census statistics in the preceding chapter. For this purpose quinquennial age groups are usually used, and in the case of general population statistics these groupings are given to us ready made or if not can only be formed by subdividing decennial groups. In the case, how- GRADUATION OF MORTALITY TABLES 73 ever, of the experience of insurance companies the groups are formed by combining the figures for individual ages and we have freedom of choice as to the limits of the groups. 129. If the central year of each quinquennial group is taken, it is usual to assume that the population and the deaths for the successive years may be expressed by a rational algebraic function of the third degree, so that we have, for each function, an equation of the form Summing this from h= 2 to h=-\-2 inclusive we get for the total number included in the group of which x is the central age, 5/00 +5/" GO- This may be expressed by the equation Wr = 5/00+5/"00> if we use w x to denote the total of the group. Similarly, summing from h= r j to h=+ f j inclusive we have Eliminating/" 00 between these two equations, we have or 5f(x}=iv x -^8 2 w x . ....... (i) It thus appears that the adjustment to be made to the total of the groups to obtain the number for the central year is the same in form for the deaths, the population, or the exposed to risk. 130. Where the first year of each group is taken, it is necessary to use four age groups in order to determine the value of We have ' '" = S f(x) - 40/(*) + i6 5 r 00 - 74 MORTALITY LAWS AND STATISTICS TOO. ~6~ J - 2 .*. 6257(0;) = 25( f ( ^_2w a; _3 g+2 -3 +a ( . J()= 25 625 131. On account of the lack of symmetry of the above formula, it may be considered desirable to select a year of age, one-half of which will be in one age group and the other half in the next group. In this case and in cases where the force of mortality is to be determined it is more convenient to work with a function <f>(x) representing the number per unit of age at exact age x, so that and T 5 Jo or /*+2i Wx _ =J +(x+h)dk. GRADUATION OF MORTALITY TABLES 75 Expanding <j>(x+h) and integrating, we have J-io 2 6 24 24 != ( <t>(* Jo Wx+2' 2 4 24 j 2 C -- " But /(*-*)= r + w-f-*)^= s ^(*)+- L ./-i 24 and 12 2 ). (3) This covers the case where half of the year of age is in each group. 132. Where the force of mortality is required at the age corresponding to the point of division we have or IO<j>(x) = (w t -3-%8 2 'W I -3) + (Wx+2-%& 2 Wx+2), . (4) which is the formula to be used for the deaths and population. 76 MORTALITY LAWS AND STATISTCIS 133. For the age corresponding to the center of the group we have X + 2J *t* 2J 24 24 whence or T <Z<b(x) =Wr- I 5 2 W-r ( $) J-r\ / X J I ) \J/ 2 4 134. These adjustments enable us to calculate the values of q x , m X) or n x for values of x separated by quinquennial ages, and the remainder of the problem consists in interpolating intermediate values. For this purpose a formula of osculatory interpolation will be found most satisfactory. Three such formulas have been proposed. The first, which is the basis of the other two, is the one described in Chapter VI in con- nection with the redistribution of population and deaths, the simplest form for practical application being that given in Eq. (20) of that chapter. The other two formulas are simplifications of this by omitting some of the conditions, the first differential coeffi- cient only being determined for the points of junction. 135. In Karup's form, which is the simpler, this first dif- ferential coefficient is determined on the assumption of a curve of the second degree for/(#), the value so derived being 2 lj'( x ')j(x-\-f}f(x t}. A curve of the third degree is then GRADUATION OF MORTALITY TABLES 77 determined so as to have the required values and first dif- ferential coefficients at the points of junction. The equation of this curve is found to be (7) 136. Greater accuracy without material increase of work can be obtained by determining the first differential coefficient on the assumption of a fourth difference curve by Eq. (13) of Chapter VI, namely, This condition is then found to be satisfied by the equation H^yV/C*) -***/(*))] } 137. When making this interpolation it is usually necessary to assume the age at which q x becomes equal to unity and the age chosen should be consistent with the data at the old ages. When it has been chosen it will be advisable to arrange the division into groups, if possible, in such a way that the ages for which the function is determined will form with the lim- iting age a regular series of differences. For example, if it is assumed that (7102 = 1 or w 10 2 = 2, then we should so arrange the groupings as to determine the rates for ages ending in 2 and 7, starting the groups at those ages if it is intended to use the first year of each group and starting the groups at ages ending in o or 5 if the central age of each group is to be used. 138. If we have interpolated a series of values of M* for each age, then we can pass to values of p x by the approximate formula 78 MORTALITY LAWS AND STATISTICS +l fx colog e />* = I Jx 2 as 6 ax 2 24 24 / v 139. The summation methods of graduation can be applied only when we have constructed a complete table of the ungrad- uated values of the function to be graduated. In investigating the effect of a graduation formula we may consider the un- graduated value of a function as consisting of two parts, one the true value V x of the function which would result from an indefinitely extended experience and the other the error or deviation E x of the observed value from the true value. The fundamental assumption which is the basis of all gradua- tion is that the values of V x form a regular or smooth series and that the values of E x form an irregular series, the fluctua- tions in the value of any one term being independent of those of the neighboring terms. It is also assumed that the mean value of each error, E x , is zero when the sign of the error is taken into account. 140. Let us first assume that over a short range of values of x, the values of V x may be considered as forming an arith- metic series. Then it is evident that if we take the arithmetic mean of an odd number of terms of the series of values of V x it will be exactly equal to the middle term. Consequently, if we take a similar average of the ungraduated values of the function it will differ from the true value V x corresponding to the middle term by the average of the values of E x , and it follows from principles of averages that the mean absolute value of this resultant deviation will be much smaller than that of the individual ungraduated values. In this case, therefore, a simple average of a number of terms will constitute a grad- uation of the series. It is evident that the same process may be repeated without disturbing the value of V x , but the effect on the values of E x diminishes with the successive summation GRADUATION OF MORTALITY TABLES 79 because the neighboring values are no longer independent, each original deviation now affecting, although to a smaller extent, a number of successive values. For example, suppose we take the average of five successive terms, then the error in the resulting average will be %(E x -2+E x -i+E x +E x+ i+E x+2 ). If we assume that the mean value of the square of each of these primary errors is jU2, then it is evident that the mean value of the square of the error of the averages will be |/*2. If we repeat the process a second time, the expression for the error will be the mean value of the square of which is 85 17 17 i ,V2 = - M2 = --- M2. 25- 125 25 5 If it is repeated a third time the error is the mean value of the square of which is I75 1 I0 3 J 7 i - J - i ViU2 = - ---- M2- 1252 125 25 5 141. Let us, however, investigate the effect on the third differences of the series. On the assumption made, that the values of V x form an arithmetic series, the third differences will arise entirely from the errors, the expression for the third differences in the ungraduated series being the mean value of the square of which is 20/12- When the first average is taken the third difference becomes the mean value of the square of which is 12 ^ M2= -- 20/*2. 25 125 80 MORTALITY LAWS AND STATISTICS For the second average the third difference is the mean value of the square of which is 12 13 M2 = --- -20M2- 625 25 125 For the third average the third difference is the mean value of the square of which is 20 iii 125 15 25 125 It will be noted that for successive summations the effect on the smoothness of the series of errors does not diminish as rapidly as the effect on the absolute values. 142. The summations do not necessarily all extend over the same number of terms nor is the number of terms in each summation necessarily odd because, while an average over an even number of terms of a series does not give a term of the series but a term midway between two of them, a second average over the same or a different even number of terms will bring us back to the original series. An even number of summations over an even number of terms each may there- fore be introduced and the resulting averages will correspond to terms of the original series. For example, suppose, instead of taking the average in fives three times we take the average in fours, fives and sixes. Then the expression for the resulting error will be the mean value of the square of which is -^-=-M2i r a little I20 2 less than for the third average in fives. The expression for the third difference is -\-E x +E x - 1 +E x - 2 E x -6, GRADUATION OF MORTALITY TABLES 81 Q the mean value of the square of which is -^2- We thus I20 2 see that by making the periods unequal a slight increase in weight is obtained on the individual term and a great increase in weight in the third difference. In other words, with unequal summations a much smoother series is obtained and it is at the same time a little more accurate. 143. Thus far we have assumed that the series of true values is an arithmetic series, the general term of which may be expressed by a linear function. It is necessary, however, to take into account the second and higher differential coeffi- cients. The summation graduation formulas ordinarily used contain a correction so that a series the general term of which is of the third degree will be reproduced. Where the function is of the third degree, we have, generally, If, then, we add together n terms of this series, we have - lt/ (s) + ^y^)V t( *- l)( ^^^ -L ~i But n(n 2 24 If, then, we denote V / \ / n 82 MORTALITY LAWS AND STATISTICS by (n]/(rc), we have, generally, or n 24 144. The operation of taking the average of n successive n 2 i terms has therefore introduced an error / (*) into the value 24 of f(x). If, then, we repeat the process with the same or a different number of terms, an additional error of the same form is introduced. We also see that, since according to our assump- tions /"(x) vanishes for all values of x and consequently J"(x] is linear in form, the error introduced by the first average is carried forward unchanged. If, then, three successive averages cover p, q, and r terms, we have, pqr 24 24 24 24 Where the number of terms in each average is the same and each equal to n this may be expressed as follows: 145. A correction must accordingly be introduced to com- pensate for this error. If, then, we use y n f(x) as a short expres- sion iorf(x-\-n)+f(x n) we have or we have, therefore, or ( i + 2a + 2b + 2c)f(x) - (ayi + by GRADUATION OF MORTALITY TABLES 83 If, therefore, we substitute, (i + 2a + 2b + 2c)f(x)-(ayi+by 2 +cy3)f(x) for/(X), before averaging, we have pqr If then a, b, and c are so determined that /> 2 +g 2 +r 2 -3 = ^ -^ 24 a series of the third degree will be exactly reproduced. We have here three unknowns and only one condition, so that the equation is an indeterminate one and the various formulas result from the use of different values of p, q, and r and of a, b, and c, the condition in some cases not being exactly satisfied. 146. The first summation formula correct to third differences was that devised by Woolhouse. It was not originally con- structed as a summation formula, but it was afterwards found to take that form. In this formula p = q = r = 5, so that = 24 Also a = 3, and b = c = o, so that the formula takes the form M 3 -{7 37ii/0*0- Woolhouse's formula when expanded as a function of the terms of the series becomes li7( 2 5 + 2 47l + 2172 + 773+374 -276-377)/(*). 147. J. A. Higham, who first showed that Woolhouse's formula might be applied by the summation method, suggested an alternative compensating adjustment which was equiv- 84 MORTALITY LAWS AND STATISTICS alent to putting a=-i, b = i and c-o, thus still keeping = 3. His formula therefore took the form which expands into 7^(25 +2471 + 1872 +1073 +374-270- 277 -ys)f(x). 148. Karup's graduation formula uses summations in fives with the compensating adjustment a= f, 6 = 0, c = f, so that the formula becomes, 149. G. F. Hardy used the same compensating adjustment as Higham, but used successive summations in fours, fives, and _____ sixes, so that for his formula ^3iV> an ^ ft is 24 not fully compensated, the remaining second difference error being T V /"(#). This is, however, small and is approximately counterbalanced for some functions such as q x and m x by the fourth difference error. A lack of compensation to this extent is therefore considered admissible. This formula becomes 150. The most powerful summation formula which has been put to practical use is probably that of Spencer, which includes when expanded 21 terms of the original series. In this formula two summations in fives and one in sevens are used, so that p -rq -rr 3 = 4 ^y so a= , 6=0, and c = \, so that the 24 formula reduces to 175 35 151. A still more powerful formula would be given by __ putting p = 5, q = 7, and r=n, so that - ^ = 8, and 24 GRADUATION OF MORTALITY TABLES 85 in the compensating adjustment we have a= i, 6=0, and c = i , so that the formula becomes, 152. The weight of a graduation formula is found by expand- ing the formula, adding together the squares of the coefficients and taking the reciprocal, that being the average proportion in which the mean square of the errors is reduced by the gradua- tion. The smoothing coefficient is found by similarly expanding the third difference of the graduated terms, adding together the squares of the coefficients, dividing by twenty and extracting the square root. It measures the effect of the graduation on the mean absolute value of the third differences. 153. The following table shows a number of summation formulas and their weights and smoothing coefficients: Author. No. of Terms. Formula. Weight. Smoothing Coefficient Error. Finlaison Woolhouse. . . J. Spencer . . Higham G. F. Hardy. J. Spencer . . . Karup 13 IS 15 17 17 19 19 21 21 27 [sP 125 [slMV 8.92 5-50 5-4 5-87 6.07 6-73 6. 14 6.98 6.70 9.11 I 125 I 15 I 60 I 56 I 95 i 85 i 105 i !2.6/ ly (x) 320 !4l3M 3*1 WtsH^l 1 20 tfizL i M "- t /J. Spencer (a) J. Spencer (b) Henderson. . . UH5H6], 1 60 i 141 i 326 600 l3 " ISluHUjir I > 12 g 1131 Taj 154. One difficulty in connection with the application of summation formulas to the graduation of tables is that in 86 MORTALITY LAWS AND STATISTICS order to determine any graduated value of the function it is necessary to know a number of ungraduated values above and below it and, unless the function is such that it disappears and the ungraduated values beyond a certain limit may be assumed to be zero, there will be a portion at either end of the table for which values will not be obtained, and some sup- plementary means must be adopted of completing the table if it is necessary to have it complete. In the case also of insurance companies' experiences, the values of the functions near the limits are usually derived from very limited data and are consequently very irregular. Both of these difficulties may be overcome by taking the three last graduated values of the function that are considered as determined with sufficient accuracy as a basis and determining a function of x of the third degree which will reproduce these three values and will also make die total expected deaths for ages beyond equal to the {expected)? The general expression for a function of the third degreein x such that f(a) = u a , /(&) = u b , and f(c) = u c is _i_ i __ . / T\/ \~*a i /-L \/L \*& ' / \ / lA** (a b)(a c) (b a)(b-c) (c-a)(c b) where k is an arbitrary constant. 155. Taking up next the application of a general law with arbitrary constants to the graduation of tables the law which is most frequently used is that of Makeham, according to which n x = A -i-Bc 1 , or l x = ks x g cx . The constant k is merely in the nature of a radix and does not affect the rates of mortality. This formula may be applied in two general ways: first, to construct a graduated mortality table from the original data of the exposed to risk (or population) and deaths without the explicit deter- mination of the ungraduated rates of mortality; and second, to graduate a rough table without direct reference to the original data. 156. Probably the simplest method of determining the Makeham constants from the original data is to group the exposures (or population) and the deaths into quinquennial GRADUATION OF MORTALITY TABLES 87 age groups and then, by the process already described in con- nection with interpolation methods, determine the values of q x , m x or n x at quinquennial intervals. Where q x or m x is determined, we can proceed immediately to colog p x from the known relations. Now we have colog p x = log l x - log 1 I+ 1 , = (log k+x log s+c x log g) It is therefore of the same form as n x . If, then, we neglect the values at young ages and at extreme old ages as derived from insufficient data and start with some age y in the neigh- borhood of age 30, we have colog />j,+3 colog A+5 + 5 colog /Wio+5 col g / +3 colog = i8a+/3c"(i Similarly colog A/+15+3 colog A+20 + 5 col S ^+25 + 5 colog p v+30 +3 colog />+36+colog ^+40 = S 2 = i8a+/3c" +15 (i +C 5 )(i +c 5 +c 10 ) 2 , and colog &+ 30 +3 col S ^+35 + 5 colog p v+4 o + 5 colog ^ + 3 COlog p y + 50 + COlog />, + 56 = From these equations we see that 88 MORTALITY LAWS AND STATISTICS l8 = TT " = _ 1 3 . Thus, c, a, and are determined. Similarly, the values of c, A, and B may be determined from the values of Hx and from either a and /3 or A and 5 we may proceed to the values of s and g. 157. If a further refinement is required, we may assume that the values determined as above are only approximate and that colog p x = (a+Mh) + (p+Mk)c x (i \ where h, k, and / are small quantities and M is the modulus of common logarithms, or logio e. Then approximately q x = q' x +hp' x +kc x p f x +lxc x p f x , where q' x and p' x are derived from the constants a, /3, and c. We have, then, three unknowns, h, k, and /, to determine and three equations are required. These equations may be ob- tained by making the total number of expected deaths and the first and second moments of the expected deaths equal to those of the actual deaths. The equations so obtained are: ' ' x +k2xc*E x p' ' x +l?x 2 c x E I p ' x = Zx(6 x -E x q' x ), ' x +k2x 2 c x E x p' x +l2x?c I E x p' x = '2x These equations are seen to be equivalent to = Z(x-a)(6 x -E x q' x ), x - d) 2 E x p' x + (k +al)Z(x - a) 2 c z E x p' x +l2(x - a GRADUATION OF MORTALITY TABLES 89 where a is any suitable quantity used for the purpose of re- ducing the numbers involved. From these three equations the values of h, k, and / are determined, and the values of a, 0, and c are corrected accordingly. 158. A shorter process is to assume that the value of c is accurate, and consequently l = o, and to determine h and k from the first two equations or to determine A and B directly from two equations depending on the relation The two equations are 159. Where a mortality table has been constructed and it is desired to graduate it by Makeham's formula, the simplest method is to determine the constants from the values of log l x at four equidistant ages by the method described in Chapter III. Unless, however, the ungraduated table is already com- paratively smooth the constants so determined will depend to too great an extent on the particular ages selected. To minimize the irregularity we may take, instead of individual values of log l x , the sums of a number of consecutive values. Then we have: C I log l x = n log * + W (*+/)+-- log s -i). - log g, C 1 "~ 1 _ | _ | + -'logg, C ~ ~ A 90 MORTALITY LAWS AND STATISTICS I _ L , -S 2 = nt log 5-f - - log g, c i . , 4-03 = / log sH -- - - log g, C ~"~ X I ci = c t . 160. This method does not, however, entirely eliminate the objection *hat special importance is given to special points of division. To obviate this it has been suggested to use all the values of log/ z except those at extreme old ages and at young ages and to so determine the constants that the sum of the values and the first, second, and third moments will be reproduced. This can best be expressed in terms of sum- mations as follows: Suppose a is the youngest age to be included and let n be the total number of ages to be included. Then = (log k -{-a log s) +x log 5 +cV log g, = log k'+x log s+c* log g'. Where log k' = log k -{-a log s and log g' = c log g. Also (2) C 1 I i5 2 = 2S~ 1 log a+z = x\ogk'-i logs-| -log/, 2 C I ,.(2) ^.(3) ( r x t v 1 , lg 24 ( C* I X X (2) ~M 7 vi~7I rv5~T7I T\ I S S i GRADUATION OF MORTALITY TABLES 91 ' log k'-\ --- log s 24 120 Where # (f>) =#(# i)(# 2) . . . (# r+i). From these four equations, after putting x equal to w in each, log k', log 5, and log g' may be eliminated, leaving an equation in c for solution. This equation will be of the (n i)th degree and the numerical solution may be obtained to any required degree of accuracy. After the value of c is obtained those of log k', log s, and log g' follow readily. 161. In the preceding discussion it has been assumed that the mortality table is an aggregate one, or in other words that it is analyzed only according to attained . age, and where select or analyzed tables are required some. , modification of the method is necessary. This usually consists in making a and ft, in the equation functions of the duration, so that we have as the general expression The constants for each year of duration may be deter- mined by any of the methods already described, the data for each year of duration being treated as representing a mortality table complete in itself but the same value of c being used for all. 162. The values of f\(f) and/2(/) so derived will, however, be somewhat irregular, so that they themselves require further graduation. It is usually assumed that they become con- stant after some definite duration, such as five years or ten years, the constant values being determined from the aggre- gate experience for all longer durations. In selecting formulas for graduating the values of these functions during the period of selection, the following conditions should be satisfied: (i) a smooth junction between the curves representing the select 92 MORTALITY LAWS AND STATISTICS and ultimate tables; (2) an agreement between the graduated and ungraduated values in the first year, as special importance Js attached to the rate of mortality in that year; (3) an agree- ment between the aggregate graduated and ungraduated values of these functions during the period between the date of entry and the ultimate table. Considerable experimenting will usually be necessary to determine the function complying with these conditions. 163. In considering the method of graduation to be adopted in any particular case it is evident that a graduation by Make- ham's formula possesses an advantage over the others on the scale of smoothness and since three arbitrary constants are available the sum of the deviations and of the accumulated deviations can be made to vanish. In view of these advan- tages and of its other advantages in connection with the cal- culation of joint contingencies, a graduation by that formula will be the best, provided the absolute deviations in groups do not materially exceed the expected and provided there is no characteristic feature of the experience which will not be reproduced by the formula. 164. Where a mathematical law cannot be applied it will usually be found that where the data is very scanty the graphic method will produce the best results as irregularities will occur of wide range, such as neither the interpolation nor the summation method is competent to remove. Where, how- ever, the data is more extensive, so as to give a satisfactory degree of regularity under the operation of the interpolation or of the summation method, those methods will be the more satisfactory as the values derived do not depend on the judg- ment of the operator except as exercised in the selection of the particular graduation formula to be used, and they can be obtained to a greater degree of accuracy than is possible under the graphic method. 165. Diagram No. 2 illustrates the relation between an ungraduated series of rates of mortality and a graduated series. The irregular line represents the rate of mortality shown in the ungraduated experience, during the ten-year period from s I 111 1 |1 slill 111 94 MORTALITY LAWS AND STATISTICS the policy anniversaries in 1899 to those in 1909, of a large American life insurance company on policies issued in the Northern States and in force more than five years at the time of observation. The ordinates are proportionate to log (i + ioo^ z ) . The regular lines represent the same rates graduated by Spencer's formula with a preliminary adjustment at the extreme ages and the rates in the M(5> table which was graduated by Make- ham's formula. The rates by the M. A. table are not shown, partly-because they . jKOiild .be practically indistinguishable in the diagram from those by the graduated experience of the company from age 35 to age 65 and from those by the M(5) table from age 70 to age 100. The comparatively wide fluc- tuations in the rates by the ungraduated table at the extreme ages should be noted. The irregularity in the M(5) table at the extreme old ages is due to the fact that the values of l x and d x were tabulated to integers only and the values of q x recalculated from them instead of being calculated directly from the constants in the formula. CHAPTER VIII NORTHEASTERN STATES MORTALITY TABLE 1 66. Some of the methods described in the preceding chap- ters will be illustrated by the construction of a mortality table for the Northeastern States. The data used will be the death returns for the five calendar years 1908 to 1912 inclusive and the census returns as of June i, 1900, and April 15, 1910, for the New England States and the three Middle Atlantic States, New York, New Jersey, and Pennsylvania. Table I shows the total deaths in these states for the five-year interval arranged by age groups and the total population at the two dates arranged in the same way. The average population by age groups for the five-year interval is also shown. 167. In this table the average population in each group is determined by means of Eqs. (5) and (9) of Chapter VI, only the population for which the ages are stated being taken into account. The cases where the age is returned as unknown are an extremely small percentage of the total and the effect of their omission is negligible. June i, 1900, is 7 T 7 T years before the beginning of the observations, which cover five years, there- fore, /i = |V April 15, 1910, is 2-sir years after the beginning therefore, /2 = ii- Also ^3 = 21004724 and P = 25 836 088, so that we have log r = . 0455239 and the two factors entering into the determination of the average population are .03154363 and 1.0304817 for June i, 1900, and April 15, 1910, respectively. The total years of life are then obtained by multiplying the average population by five. 168. We then apply Eq. (3) of Chapter VII and obtain the values of Lg k , Z, 141 , L m , etc., and of 14J , 6 m , 24i , etc. The value of 6 9 j is determined by leaving out of account the deaths 95 96 MORTALITY LAWS AND STATISTICS TABLE I DEATHS AND POPULATION NORTHEASTERN STATES. 1908-1912 Age Last Birthday. Deaths, 1908-1912. Population. Total Years of Life. April 15. 1910 June i, 1900. Average 1908-1912. 397 985 574480 476 810 576951 2 884 755 I 84939 55 632 426 773 507 583 2537915 2 357S7 553 698 448816 556 419 2 782 095 3 22 I ID 541 178 449855 543 484 2 717 420 4 15694 515976 442067 517 760 2588800 0-4 556 491 2 690 964 2 244321 2 702 197 13510985 5-9 42475 2 400 180 2 IIO 213 2 406 778 12033 890 10-14 25885 2 285 642 I 908 183 2 295 121 11475605 iS-iQ 42677 2 385 256 i 888 668 2 398 388 II 991 940 20-24 64 604 2 554 686 2 O24 318 2 568 703 12 843 515 25-29 71 700 2 405 723 i 977 342 2 416 681 I 2 083 405 30-34 75273 2 121 420 i 738 577 2 131 243 10656 215 35-39 86752 I 984 723 i 562 115 i 995 946 9 979 730 40-44 86342 I 671 571 i 305 952 i 681 329 8 406 645 45-49 90895 i 399 363 i 053 884 i 408 775 7 043 875 5>-54 99493 i 174 250 899 808 i 181 660 5 908 300 55-59 103 030 840 368 697 132 843 994 4 219970 60-64 118 590 686 755 575 880 689 523 3 447 615 65-69 128 504 5H970 418332 5I747I 2 587 355 70-74 128 074 355 427 292 946 357 020 i 785 loo 75-79 "3 343 2IO 122 177814 2IO 918 i 054 590 80-84 82 412 IO2 741 88019 103097 5I548S 85-89 45 174 39617 3i 504 39831 I99I55 90-94 15993 10 198 7923 10259 Si 295 95-99 3497 I 851 i 523 1859 9 295 zoo and over 678 26l 270 260 i 300 All known I 981 882 25 836 088 21 004 724 25 961 053 129 805 265 Unknown 1038 32485 4I97I All I 982 920 25 868 573 21 046 695 at ages o to 4 and assuming that d 2 wr is equal to 8 2 w\2 in the formula. The exposed to risk at each of these ages is then determined by the formula E x L x -\-\d x . The same formula also applies at ages i to 4 inclusive, but for age zero it is found that the average age at death of those dying within one year of birth is only three-tenths of a year, so that we use instead E Q = L Q -}-^QQ. From the values of log 6 X and NORTHEASTERN STATES MORTALITY TABLE 97 log E x the values of log q x are then determined. These figures are shown in Table II. TABLE II CALCULATION OF VALUES OF log q x FOR INFANTILE AND QUINQUENNIAL AGES X ioL x IO8 X ioR x log g z O 28 847 550 3 979 850 31 633 445 1.09972 I 25 379 ISO 849 390 25 803 845 2.51742 2 27 820950 357 570 27 999 735 . 10621 3 27 174 200 221 l6o 27 284 780 3.90879 4 25 888 ooo 156 940 25 966 470 -78i34 9i 23 180 579 57344 23 209 251 39283 145 23 234 918 62 207 23 266 02 1 .42712 195 25 046 068 109 046 25 zoo 591 63793 24* 25 302 916 138 167 25372000 73605 295 22 725 822 145 085 22 798 364 .80372 345 20 660 018 162 848 20 741 442 .89494 395 18 499 612 174 237 18 586 731 97194 445 I537833I 175 75i 15 466 206 2-0555I 495 13 005 892 19 556 13 101 170 .16271 545 10 068 338 201 374 10 169 025 . 29672 595 7 53 953 220 568 7 641 237 .46038 645 6 039 903 249 733 6 164 770 .60756 695 4 35i 046 260 645 4 481 368 76464 74i 2 796 270 246 450 2 919 495 .92642 795 i SGI 735 199 302 i 601 386 i. 09501 84* 650 085 127 131 713651 25077 895 205 286 57085 233 828 38763 945 37512 15 140 45082 .52612 99 3879 2 225 4992 . 64906 169. The value of log <7 4i to serve as an initial term in a systematic interpolation is calculated from those of log <7s, log 5-4, and log<7 9J , on the assumption that for the interval in question log q x may be considered as a rational algebraic function of the second degree in x, the resulting value being 3.72417. An additional term is supplied at the end by assuming <7io4i = i or log <7io4i = .00000. Eq. (8) of Chapter VII is then applied to the interpolation. In this interpolation t is given the value 5 and h takes successively the values, \, f , -f , -J, and f . From these values of log<7 2 , the values of l x and d x are then derived. The following table for the first five ages shows the working process: 2 3 8i 3 + .2 a H > *S 8 S| hie -IM U2 Ti c WOO | K G CO c3 6 <3 Z oj X! NORTHEASTERN STATES NORTALITY TABLE 99 (I) (2) (3) =log (2) (4) (5) =(3) +(4) (6) =antilog (5) Age. log*,. log9 z \ogd x *i IOOOOO 5.00000 i .09972 4.09972 12 581 i 87419 4.94161 2.51742 3-45903 2878 2 84541 .92707 . 10621 .03328 I O8O 3 83461 .92418 3.90879 2.83027 6 77 4 82 784 91795 78134 .69929 500 5 82 284 170. In accordance with the usual custom the values of q x shown in the table have been adjusted to agree exactly with the values of l x and d x and do not agree with the values of log q x used in constructing the table. In the accompanying Diagram, No. 3, the values of log (i + ioo^) are plotted in comparison with the similar functions according to three tables representing mortality among American insured lives. APPENDIX FUNDAMENTAL COLUMNS AND OTHER DATA FROM VARIOUS MORTALITY TABLES x = age ; l s = number living at age x ; d x = number dying at age x last birthday. NORTHAMPTON TABLE X /x dx x /* d z x /* d x o ii 650 3000 35 4 oio 75 70 i 232 80 I 8650 i 367 36 3935 75 7i i 152 80 2 7283 502 37 3860 75 72 i 072 80 3 6781 335 38 3785 75 73 992 80 4 6 446 197 39 3 7io 75 74 912 80 5 6249 184 40 3635 76 75 832 80 6 6 065 140 4i 3 559 77 76 752 77 7 5925 no 42 3482 78 77 675 73 8 S8iS 80 43 3404 78 78 602 68 9 5735 60 44 3326 78 79 534 65 10 5675 52 45 3248 78 80 469 63 ii 5623 So 46 3 J 7 78 81 406 60 12 5573 50 47 3092 78 82 346 57 13 5523 5 48 3014 78 83 289 55 14 5473 5 49 2936 79 84 234 48 IS 5423 50 5 2857 81 85 1 86 4i 16 5373 53 5i 2 776 82 86 145 34 17 5320 58 52 2694 82 87 in 28 18 5 262 63 53 2 6l2 82 88 83 21 iQ 5 199 67 54 2530 82 89 62 16 20 5 132 72 55 2448 82 90 46 12 21 5 060 75 56 2 366 82 9i 34 IO 22 4985 75 57 2 284 82 92 24 8 23 4910 75 58 2 2O2 82 93 16 7. 24 4835 75 59 2 I 2O 82 94 9 5 25 4 760 75 60 2038 82 95 4 3 26 4685 75 61 1956 82 96 i i 27 4 610 75 62 874 81 28 4535 75 63 793 81 29 4 460 75 64 712 80 30 4385 75 65 632 80 31 431 75 66 552 80 32 4 235 75 67 472 80 33 4 160 75 68 392 80 34 4085 75 69 3" 80 100 APPENDIX 101 CARLISLE TABLE X lz d x X /. d x X h d x 1OOOO 1539 35 5362 55 70 2 4OI 124 I 8461 682 36 537 56 71 2 277 134 2 7779 505 37 5251 57 72 2 143 146 3 7274 276 38 5194 58 73 i 997 156 4 6998 2OI 39 5136 61 74 I 841 166 5 6797 121 40 575 66 75 I 675 160 6 6676 82 4i 5009 69 76 1 515 156 7 6594 58 42 4940 7i 77 i 359 146 8 6536 43 43 4869 7i 78 i 213 132 9 6493 33 44 4798 7i 79 i 081 128 10 6 460 29 45 4727 70 80 953 116 ii 6431 3i 46 4657 69 81 837 112 12 6 400 32 47 4588 67 82 725 IO2 13 6368 33 48 452i 63 83 623 94 14 6335 35 49 4458 61 84 529 84 IS 6300 39 5 4397 59 85 445 78 16 6 261 42 Si 4338 62 86 367 71 17 6 219 43 52 4276 65 87 296 64 18 6 176 43 53 4 211 68 88 232 51 19 6i33 43 54 4143 70 89 181 39 20 6090 43 55 4073 73 90 142 37 21 6047 42 56 4000 76 9i i5 3 22 6005 42 57 3924 82 92 75 21 23 5963 42 58 3842 93 93 54 14 24 5921 42 59 3749 106 94 40 10 25 5879 43 60 3643 122 95 3 7 26 5836 43 61 35 2 i 126 96 23 5 27 5793 45 62 3395 127 97 18 4 28 5748 5 63 3268 125 98 14 3 20 5698 56 64 3 143 125 99 ii 2 3 5642 57 65 3018 124 IOO 9 2 31 5585 57 66 2894 123 IOI 7 2 32 5528 56 67 2771 123 102 5 2 33 5472 55 68 2648 123 I3 3 2 34 5417 55 69 2525 124 IO4 i I 102 MORTALITY LAWS AND STATISTICS ACTUARIES', OR COMBINED EXPERIENCE, TABLE X I* d t X l d x X lx d x IO 100 000 676 40 78653 815 7 35837 2327 II 99324 674 4i 77838 826 71 33510 2 351 12 98650 672 42 77 012 839 72 3i 159 2 362 13 97978 671 43 76i73 857 73 28797 2358 14 97307 671 44 753i6 881 74 26439 2339 15 96636 671 45 74435 909 75 24 100 2303 16 95965 672 46 73526 944 76 21 797 2 249 17 95 293 673 47 72582 981 77 19548 2 179 18 94 620 675 48 71 601 I O2I 78 17369 2 092 19 93945 677 49 70580 i 063 79 15277 1987 20 93 268 680 5 69517 i 108 80 13 290 i 866 21 92 588 683 5i 68 409 i 156 81 11424 i 730 22 9i 95 686 52 67253 i 207 82 9694 i 582 23 91 219 690 53 66 046 i 261 83 8 112 i 427 24 90529 694 54 64785 i 316 84 6685 i 268 25 89835 698 55 63469 i 375 85 5417 i in 26 89137 703 56 62 094 i 436 86 4306 958 27 88434 708 57 60658 i 497 87 3348 811 28 87726 7H 58 59 161 i 561 88 2537 673 29 87 012 720 59 57 600 i 627 89 i 864 545 30 86 292 727 60 55973 i 698 90 1319 427 31 85565 734 61 54275 i 770 9i 892 322 32 84831 742 62 52505 i 844 92 570 231 33 84089 75 63 50 66 1 i 917 93 339 155 34 83339 758 64 48744 i 990 94 184 95 35 82581 767 65 46754 2 061 95 89 52 36 81 814 776 66 44693 2 128 96 37 24 37 8 1 038 785 67 42565 2 191 97 13 9 38 80253 795 68 40374 2 246 98 4 3 39 79458 805 69 38128 2 291 99 i i APPENDIX 103 AMERICAN EXPERIENCE TABLE X lx d x X l x d x X lx d x IO IOOOOO 749 40 78106 765 70 38569 2391 II 99 251 746 4i 77341 774 71 36178 2 448 12 98 505 743 42 76567 785 72 33730 2487 13 97 762 740 43 75 782 797 73 3i 243 2 505 14 97 022 737 44 74985 812 74 28738 2 501 15 96 285 735 45 74173 828 75 26 237 2 476 16 95 550 732 46 73345 848 76 23 761 2 431 17 94818 729 47 72497 870 77 21330 2369 18 94 089 727 48 71 627 896 78 18961 2 291 19 93362 725 49 70731 927 79* 16 670 2 196 20 92637 723 5 69 804 962 80 14474 2 091 21 91 914 722 5i 68842 I OOI 81 12383 I 964 22 91 192 721 52 67841 i 044 82 10419 I 8l6 23 90471 720 53 66797 i 091 83 8603 I 648 24 89 75i 719 54 65 706 i 143 84 6955 I 470 25 89032 718 55 64563 i 199 85 5485 I 292 26 88314 718 56 63364 i 260 86 4193 I 114 27 87596 7i8 57 62 104 i 325 87 3079 933 28 86878 7i8 58 60779 i 394 88 2 146 744 29 86 1 60 719 59 59385 i 468 89 I 4O2 555 3 85441 720 60 57917 i 546 90 847 385 3i 84721 721 61 S637I i 628 9i 462 246 32 84 ooo 723 62 54743 i 713 92 216 C37 33 83 277 726 63 53 030 i 800 93 79 58 34 82551 729 64 5i 230 i 889 94 21 18 35 81 822 732 65 49341 i 980 95 3 3 36 81 090 737 66 4736i 2 070 37 80353 742 67 45 291 2 I S 8 38 79611 749 68 43 133 2 243 39 78862 756 69 40 890 2 321 104 MORTALITY LAWS AND STATISTICS INSTITUTE OF ACTUARIES HEALTHY MALE (H M ) TABLE X lx d x X lx dx X /* in 10 IOOOOO 490 40 82 284 848 70 38 124 2371 II 99510 397 4i 81 436 854 71 35753 2433 12 99 "3 329 42 80582 865 72 33320 2497 13 98784 288 43 79717 887 73 30823 2554 14 98 496 272 44 78830 911 74 28 269 2578 15 98 224 282 45 77919 950 75 25691 2527 16 97942 3i8 46 76969 996 76 23 164 2464 17 97624 379 47 75973 i 041 77 20 700 2374 18 97 245 466 48 74932 1082 78 18326 2 258 19 96779 556 49 73850 i 124 79 16068 2138 20 96 223 609 50 72 726 i 160 80 13930 2015 21 95614 643 Si 71 566 i 193 81 "915 1883 22 94971 650 52 70373 i 235 82 10032 I 719 23 94 3 2 i 638 53 69138 i 286 83 8313 i 545 24 93683 622 54 67852 1339 84 6768 1346 25 93061 617 55 66513 1399 85 5422 i 138 26 92444 618 56 65 "4 i 462 86 4 284 941 27 91 826 634 57 63 652 1527 87 3343 773 28 91 192 654 58 62 125 i 592 88 2570 6i5 29 90538 673 59 60533 1667 89 1955 495 3 89865 694 60 58866 i 747 90 i 460 408 31 89171 706 61 S7"9 1830 9i i 052 329 32 88465 717 62 55 289 1915 92 723 254 33 87748 727 63 53374 2001 93 469 I9S 34 87 021 740 64 5 1 373 2076 94 274 139 35 86281 757 65 49 297 2 141 95 135 86 36 85524 779 66 47 156 2 196 96 49 40 37 84745 802 67 44 960 2243 97 9 9 38 83943 821 68 42 717 2274 39 83122 838 69 40443 2319 APPENDIX 105 BRITISH OFFICES' O M < 5 > TABLE X (. dx | X lx d x X lx d x IO 107 324 658 45 82 oio 984 80 15531 2 151 II 106 666 658 46 81 026 1018 81 13380 2 OO7 12 1 06 008 656 47 80008 i 056 82 "373 1847 13 i5 352 655 48 78952 i 096 83 9526 1674 14 104 697 654 49 77856 H39 84 7852 1493 IS 104 043 654 50 76 717 1185 85 6359 1308 16 103 389 654 Si 75532 i 234 86 5051 I 122 17 102 735 655 52 74298 1286 87 3929 943 18 102 080 655 53 73012 1343 88 2986 773 19 101 425 655 54 71 669 i 402 89 2 213 617 20 ioo 770 657 55 70 267 1464 90 i 596 480 21 100113 660 56 68803 1529 9i i 116 360 22 99453 661 57 67 274 1598 92 756 263 23 98792 664 58 65676 i 669 93 493 183 24 98128 667 59 64 007 i 742 94 310 124 25 97461 672 60 62 265 1819 95 1 86 79 26 96789 676 61 60 446 1897 96 107 49 27 96113 68 1 62 58549 1975 97 58 28 28 95432 688 63 56574 2055 98 30 IS 29 94744 694 64 54519 2i33 99 15 8 30 94050 703 65 52386 2 211 IOO 7 4 31 93347 711 66 50175 2285 IOI 3 2 32 92 636 720 67 47890 2355 IO2 i I 33 91 916 732 68 45535 2 421 34 91 184 744 69 43 "4 2478 35 90440 757 70 40 636 2527 36 89683 771 7i 38 109 2565 37 88912 788 72 35544 2 591 38 88 124 806 73 32953 2 6O2 39 87318 825 74 30351 2 596 40 86493 846 75 27755 2 572 4i 85647 869 76 25183 2 529 42 84778 895 77 22654 2 466 43 83883 922 78 20 188 238l 44 82961 95i 79 17807 2 276 106 MORTALITY LAWS AND STATISTICS NATIONAL FRATERNAL CONGRESS TABLE X In d t X / d x X /* d x 20 100 000 500 So 81 702 935 80 20 270 2799 21 99 5o SGI Si 80767 981 81 I747I 2659 22 98999 502 52 79786 i 029 82 14 812 2485 23 98497 503 53 78757 1083 83 12327 2 280 24 97994 505 54 77674 i 140 84 10047 2050 25 97489 507 55 76534 i 202 85 7997 I800 26 96 982 5io 56 75332 i 270 86 6 197 1539 2? 96472 5i3 57 74 062 1342 87 4658 I 277 28 95 959 5i7 58 72 720 1418 88 338i 1023 29 95442 522 59 71302 i 501 89 2358 788 30 94920 527 60 69801 1588 90 i 570 579 3* 94393 533 61 68213 1681 9i 991 404 32 9386o 540 62 66532 1778 92 587 264 33 93 320 548 63 64754 i 880 93 323 161 34 92 772 557 64 62874 1985 94 162 89 35 92 215 567 65 60889 2094 95 73 44 36 91 648 578 66 58795 2 2O6 96 29 19 37 91 070 59i 67 56589 2318 97 10 7 38 90479 606 68 54 271 2430 98 3 3 39 89873 622 69 51841 2539 40 89 251 640 70 49 302 2645 4i 88611 660 7i 46657 2744 42 87951 683 72 43913 2832 43 87 268 708 73 41 081 2909 44 86560 734 74 38172 2969 45 85826 761 75 35203 3009 46 85065 790 76 32 194 3026 47 84275 822 77 29 1 68 30l6 48 83453 857 78 26 152 2977 49 82 596 894 79 23175 2905 APPENDIX 107 NORTHEASTERN STATES MORTALITY TABLE (1908-12) Age. X Number Living. /* Number Dying. d x Rate of Mortality per Thousand. looo q x Expectation of Life. e t o IOOOOO 12 581 125.81 50.41 I . 87 419 2878 32.92 56.59 2 84541 I 080 12.77 57-50 3 83461 677 8. II 57-24 4 82 784 500 6.04 56.70 S 82 284 386 4.69 56.04 6 8 1 898 3" 3-8o 55-31 7 81587 261 3-20 54-Si 8 81326 228 2.80 53-69 9 81098 207 2-55 52.84 10 80891 195 2.41 Si-97 ii 80696 190 2-35 51.10 12 80 506 190 2.36 50.21 13 80316 196 2.44 49-33 14 80 1 20 207 2-58 48.45 IS 79913 222 2.78 47-58 16 79691 244 3-o6 46.71 17 79447 27O 3-40 45-85 18 79 177 299 3.78 45-oo 19 78878 329 4.17 44.17 20 78549 354 4-Si 43.36 21 78i95 374 4.78 42-55 22 77821 390 S- 01 41-75 23 77431 402 5-i9 40.96 24 77029 4i3 5-36 40.17 25 76616 424 5-53 39-38 26 76 192 435 5-71 38.60 27 75757 445 5-87 37.82 28 7S3I2 456 6.05 37 05 29 74856 468 6.25 36.26 30 74388 482 6.48 35-49 31 73906 499 K75 34.72 32 73407 5-i8 7.06 33-95 33 72889 537 7-37 33-19 34 7235 2 557 7.70 32.43 35 7i 795 575 8.01 3 J -68 36 71 220 59i 8.30 3 -93 37 70 629 607 8-59 3-i8 38 70022 623 8.90 29.44 39 69399 639 9.21 28.70 40 68760 656 9-54 27.96 4i 68 104 674 9.90 27.23 42 67430 693 10.28 26.49 43 66737 7i3 10.68 25-76 44 66024 734 II . 12 25.04 45 65 290 758 ii. 61 24-31 46. 64 532 785 12. 16 23-59 47 63747 813 12.75 22.88 48 62934 845 13-43 22. 17 49 62 089 879 14.16 21.46 5 61 210 915 14-95 20.76 Si 60 295 955 15-84 2O.O7 108 MORTALITY LAWS AND STATISTICS NORTHEASTERN STATES MORTALITY TABLE (1908-12) Continued Age. X Number Living. lx Number Dying. d z Rate of Mortality per Thousand. looog* Expectation of Life. e x 52 59340 998 16.82 19.38 53 58342 1045 17.91 18.71 54 57 297 I 096 19 13 18.04 55 56 201 I 153 20.52 17.38 56 55 48 I 217 22.11 16.73 57 53831 1285 23.87 16.10 58 52546 1355 25-79 I5-48 59 5i 191 1424 27.82 14-88 60 49767 1489 29.92 14.29 61 48 278 1547 32.04 13.72 62 46731 i 601 34.26 13 IS 63 45 130 1652 36.61 12.60 64 43478 i 702 39 IS 12.06 65 4i 776 1752 41.94 n-54 66 40024 1802 45-02 ii .02 67 38 222 1850 48.40 10.51 68 36372 1894 52.07 10.02 69 34478 1933 56.06 9-55 70 32 545 1964 60.35 9.08 ?i 30581 2986 64-94 8.64 72 28595 2 OOO 69.94 8.20 73 26595 2004 75-35 7.78 74 24591 1998 81.25 7-37 75 22593 1982 87 73 6.98 76 20 611 1955 94-85 6.60 77 18656 1914 102.59 6. 24 78 1 6 742 1857 110.92 5-90 79 14885 1783 119.78 5-57 80 13 102 1693 129.22 5.26 81 ii 409 1588 I39-I9 4-97 82 9821 1470 149.68 4-69 83 835i 1342 160.70 4-43 84 7009 I 2O7 172.21 4.18 85 5802 1068 184.07 3 95 86 4734 929 196.24 3-73 87 3805 795 208 . 94 3-52 88 3010 669 222. 26 3-32 89 2 .341 554 236.65 3-12 90 1787 451 252.38 2-93 9i 1336 360 269 . 46 2-75 92 976 282 288.93 2-59 93 694 214 308.35 2-44 94 480 157 327.08 2.29 95 323 in 343-65 2.16 96 212 77 363 20 2.04 97 135 Si 377-78 1.91 98 8 4 34 404.76 1.77 99 50 21 42O.OO i .64 100 2Q 13 448.27 1-47 101 16 8 5OO.OO 1-25 102 8 5 625.00 1. 00 103 3 2 666.67 83 104 i I IOOO.OO So APPENDIX 109 RATES OF MORTALITY PER THOUSAND ACCORDING TO VARIOUS TABLES Age. Northamp- Carlisle. English Life No. 3. Healthy Districts, English Life No. 6, North- eastern ton. Mixed. Mixed. Mixed. States. o 257.5I I53-90 149.49 102.95 156.53 125.81 5 29.44 17.80 13-43 10.27 7-43 4.69 10 9.16 4-49 5-73 4-36 2.50 2.41 IS 9.22 6.19 5-36 4.87 3.11 2.78 20 I4-03 7.06 8.42 7-30 4-36 4-51 25 I5-76 7-31 9.38 8.08 5-36 5-53 30 17.10 IO.IO 10.32 8-57 6.48 6.48 35 18.70 10.26 ii .42 9-03 8.36 8.01 40 20.91 13.00 12.87 9.69 10.84 9-54 45 24.02 14.81 14.84 10.82 13.17 ii .61 5 28.35 13-42 17-53 12.62 17.12 14-95 55 33-50 17.92 22.76 15-35 22.77 20.52 60 40.24 33-49 30.66 22.99 32.32 29.92 65 49.02 41.09 43.46 35-35 45-44 41.94 70 64.94 51.64 63.80 53-24 67.20 60.35 75 96.15 95-52 93-94 80.54 95-59 87.73 80 134-33 121.72 I35-5I 120.43 146.00 129. 22 85 220.43 175.28 189 . 29 174.95 215.41 184.07 90 260.87 260 . 56 254.84 245-03 290.56 252.38 95 750.00 233-33 33I-I7 325-11 396.91 343-65 IOO 222. 22 412 .56 413 .04 600.00 448 27 T~T V / Age. Actuaries'. American Experience. Healthy Male. British Offices, OM(5). National Fraternal Congress. Medico- Actuarial. o 5 10 6.76 7-49 4.90 6.13 15 6-94 7-63 2.8 7 6.29 20 7.29 7.80 6-33 6.52 S-oo 25 7-77 8.06 6.63 6.89 5-20 4-7 30 8.42 8-43 7.72 7-47 5-55 49 35 9-29 8-95 8-77 8-37 6.15 5.1 40 10.36 9-79 10.31 9.78 7.17 5-7 45 12.21 ii .16 12.19 12. OO 8.87 7-5 50 15-94 13-78 15-95 15-45 11.44 10.6 55 21 .66 i8.57 21.03 20.83 I5-7I 15-8 60 30.34 26.69 29.68 29.21 22.75 24.0 65 44.08 40.13 43-43 42.21 34-39 39-0 70 64-93 61.99 62.19 62.19 53-65 61.7 75 95-56 94-37 98-36 92.67 85.48 91.9 80 140.41 144-47 144-65 138.50 138.09 137-2 85 205 . 10 235-55 209.89 205 . 69 225.08 203.7 90 323-73 454-55 279-45 300.75 368.79 297-8 95 IOO 584.27 IOOO.OO 637-04 424.73 602 . 74 423-3 576 -4 110 MORTALITY LAWS AND STATISTICS DEATH RATES PER THOUSAND ACCORDING TO VARIOUS TABLES Table. Infancy. Ages 0-2 Child- hood. Ages 3-IO. Youth. ARCS 10-25. Maturity Ages 3S-ds. Old Age. ARCS over 65. MIXED POPULATION TABLES: Northampton 249-3I I30-33 118.19 73-48 116.71 87.25 128.24 80.28 128.07 107-95 66.51 105.30 32.49 24. 21 16.07 10.93 8-57 5.56 16.10 10.85 8.60 16.04 ii .00 8.54 II .60 6.26 6.79 5.89 3-65 3.60 6.62 5-43 3-74 6.97 6-37 3-56 7.14 4-77 5-92 3-84 6.42 7 99 24.23 15-80 16.47 12.66 14.96 13-74 17.03 12.76 16.24 15-90 12.57 13-74 14.86 14.44 15-70 13-36 14.14 13-65 10.92 91.87 89.89 89.45 81.37 92.35 86.69 92.35 83-31 96.79 86.86 79-47 88.88 91 .12 90.8l 91-57 89.16 89.25 90.12 84.50 Carlisle English Life No. 3 Healthy Districts English Life No. 6 Northeastern States MALE POPULATION TABLES: English Life No. 3 Healthy Districts English Life No. 6 FEMALE POPULATION TABLES: English Life No. 3 Healthy Districts English Life No. 6. INSURANCE EXPERIENCE: Actuaries' . . Healthy Male H M Healthy Male H M(5) British Offices' O M British Offices' O M(5) American Experience National Fraternal Congress INDEX Actuaries' Table, 8 Age Year Method, 61 American Experience Table, n Analyzed Mortality Table, Construc- tion of, 65 Breslau Table, 2 British Offices' Life Annuity Tables, 1893, 16 British Offices' Life Tables, 1893, 10 Calendar Year Method, 64 Carlisle Table, 6 Carlisle Table, Method of Construct- ing, 56 Census Returns, Mortality Tables from, S3 Central Death Rate, 46 Combined Experience Table, 8 Construction of Mortality Tables, 51 Death Rate, Central, 46 Death Rate for Communities, Cor- rected, 47 DeMoivre's Formula, 26 English Life Tables, 7 Expectation of Life, 21 Force of Mortality, 19 Gompertz's Formula, 27 Graduation of Mortality Tables, 68 Graphic Method of Graduation, 71 Hardy's Graduation Formula, 84 Healthy Male Table, 9 Higham's Graduation Formula, 83 Insurance Experience, Mortality Tables from, 60 Interpolation Formulas, 76 Interpolation Method of Graduation, 72 Joint Survival, Probabilities of, 34 Karup's Graduation Formula, 84 Karup's Interpolation Formula, 76 Makeham's Formula, 27 Makeham's Formula, Graduation by ; 86 M. A. Table, 13 McClintock's Annuitants' Mortality Tables, 15 Medico-Actuarial Mortality Investi- gation, 13 Mortality, Force of, 19 Mortality Tables, i Mortality Tables, Construction of, 51 Mortality Tables, Meaning of, 17 National Fraternal Congress Table, 12 Northampton Table, 4 Northeastern States Mortality Table. 95 Pearson's Analysis of Mortality Table 32 Policy Year Methods, 63 Population Statistics, 52 Seventeen Offices Table, 8 Spencer's Graduation Formula, 84 Stationary Population, 45 Statistical Applications, 45 Summation Methods of Graduation, 73 Survivorship, Probabilities of, 41 Tests of a Good Graduation, 70 Uniform Seniority under Makeham's Law, 34 Wittstein's Formula, 31 Woolhouse's Graduation Formula, 83 111 21 PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY Jftgffi HM