5. Statistical Applications of the Mortality Table

GEORGE KING (1902)

From Institute of Actuaries' Textbook, Part II, Second Edition, pages 56-58, 63. London: Charles and Edward Layton.

We omit several numerical examples given by King of the uses of the life table, which concern populations at various ages, annuity payments, and the effects of immigration on observed death rates. The article introduces the life table in modern notation.

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D. P. Smith et al., Mathematical Demography© Springer-Verlag Berlin · Heidelberg 1977

4. The fundamental column of the Life Table is the column of l3:. The first value in that column, 10 , called the radix, is the number of annual births in the imaginary population; and the succeeding numbers show how many persons, out of 10 born alive, complete each year of age.

In the table the number of annual births is 127,283; and we observe that 100,000 live to complete the tenth year of their age; 89,685 live to complete the thirtieth; and so on. We also see that only 4 live to complete the century; and that, although 1 survives 101 years, all die before reaching the age 102. Age 102 is therefore the limiting age of the table, being the year of age on which some lives enter, but which none complete. To the limiting age, the Greek letter (() is assigned for symbol; and therefore 1ro=O. Also, the difference between the limiting age, and the present age, is called the complement of life; so that, at age x, the complement of life is (()-x; in the case of our table, 102-x.

5. The column of it3: contains the differences between the numbers in the column of l3:; and shows how many, out of 10 persons born alive, die in each year of their age. Thus, by the table, out of 127,283 persons born alive, 14,358 die before completing their first year; 691 survive to age 30, but die before reaching 31; and so on. The number, then, in column it opposite any age, x, is the number who complete that year of age, but die before completing the next; that is, the number in column it opposite age x, is the number who die in the (x+1)th year of age. As all born must die, it follows that the sum of all the numbers in column it is equal to lo: also, the sum of the numbers in column it, from age x to the oldest age, is equal to 13:.

6. To find how many die aged between x and x+n, we may take the sum of the numbers in column it, from it3: to it3:+n-l, inclusive; but unless n be very small, it will be easier to obtain the result by means of the 1 column; because l3:-l3:+n= it3: + it3:+l + &c. + it3:+n-l . Thus, by the table, the number of persons, out of 127,283 born alive, who die between ages 20 and 30, is ll1A)-l30=6376.

7. Passing now to the column L, we have the population living in a stationary community. Such a community, sustained by lo annual births, will, on the supposition of uniform distribution of births and deaths, always contain t(lo+ll)=Lo children in the first year of their

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age; i(ll + 12) =L1 in the second year of their age; and so on. Thus, by the table, a population sustained by 127,283 annual births, will always contain 95,787 young persons aged between 20 and 21. The total population at all ages will be the sum of all the numbers in column L ; and that is given in column T. By the table, the total population that would be supported by 127,283 annual births, is 6,082,OJn=To. The column T bears exactly the same relation to the column L, that the column 1 bears to the column it: that is, Tz is the sum of the numbers in column L, from age x to the oldest age: therefore Tz is the total popu­lation, aged x and upwards, in the community. In the community of To inhabitants, there must be 10 deaths annually; because there are 10 births; and, the population being stationary, the deaths must be equal in number to the births. Similarly, there must be 1z deaths annually of persons aged x and upwards; and 1:tJ-1z+n deaths annually, of persons aged between x and x+n. Also, the number of inhabitants aged between x and x+n, is Tz-Tz+n ; and, therefore, the proportion of

deaths to population, between ages x and x + n, is ;z-~~+n ; and, for the z- ;c+n

whole community, the proportion of deaths to population is ~: . When n

• . 1z-1:lHn itz h 1 d th t t B IS umty, T T = L- = m;c, t e centra ea ra e a age x. y

;c- z+n z

the table, the proportion of deaths to the population for the whole

community is 127,283 = ·020928 or about 21 per thousand. The 6,082,031

. 127,283-72,795 proportion for the population aged less than 50, IS 6,082,031-1,475,603

_ 54,488 = .011829, or not quite 12 per thousand, while the pro-4,606,428

. . 72,795 049332 portion for the populatIon aged 50 and upwards, IS 1,475,603· ,

or over 49 per thousand. These figures illustr;ate the remarks made in Chapter iii, Art. 13. They show that if from any cause there is an unusual proportion of young persons in a community, the ratio of deaths to population will be diminished; but, as previously remarked, it does not follow that therefore the members of that community enjoy unusual longevity. . ..

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We have seen that, of l:z; persons who attain the precise age te, 1:Z;+l will complete a year of life in the first year; and a:z; will live on the average half a year each; therefore the quantity of existence in the first year due to the l:z; persons will be 1:Z;+1 +ta:z;=L:z;. Similarly for future years; there­fore lL:z;=T:z; will be the total future existence due to the l:z; persons;

giving ~: =e:z; years to each; and the average age at death of the 1",

persons will be te+e.v years. The existence within the next n years due

t th 1 . T T .. T.v- T:z;+n I 0 l' h o e.v persons, IS :z;- .v+n; gIvmg 1.11 = nC:e years lor eae .

Of these years, 17, X l.v+n are due to those who complete age te+n; leaving T:e-T"'+n-nl.v+n for those who die between age te and age te+n. But l.v-1.v+n persons die between these ages; therefore the average amount of existence between ages te and te + 17" belonging to those who die

• T:z;-T",+n-nl:z;+n . between these ages, 18 1 1 ' and theIr average age at death

:»- :z;+n • T",-T:z;+n-nl:z;+n F if IS te + 1 1 • or example, te=20 and 17,=10, we find

:»- :»+n o T20-Tao

itOC20= ~ =9'680, Also, the existence within the 10 years, due to

all those who reach age 20, is T20-Tao=929,902; the existence in the period due to those who survive it is 10130 = 896,850; leaving 33,052 years to the 12G-1ao=6,376 persons who die in the 10 years;

or 5'184 to each. The average age at death of those who die between 20 and 30, is therefore 25'184, Similarly the average age at death of

, . To-T20- 20120 those who dIe below 20 IS lo-~ = 3'734.

13. We have seen, Art. 7, that the ratio of deaths to population

in a stationary community is ~:: also (Chap. iii, Art. 16), that the

complete expectation of life is ~:. It therefore follows that the ratio

of deaths to population is equal to the reciprocal of the complete

expectation of life.

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