# Life Tables

The conventional life table

Life tables directly from census data

Life tables directly from death data

Multiple decrement tables

Select tables

Cohort or generation life tables

Further applications

BIBLIOGRAPHY

The life table (also referred to as the mortality table) is a statistical device used to compute chances of survivorship and death and average remaining years of life, for specific years of age. The concept of the life table is applicable not only to humans (Spiegelman 1957) and other species of life (Haldane 1953; Ciba Foundation 1959) but also to items of industrial equipment (Dublin, Lotka, & Spiegelman [1936] 1949) and other defined aggregates subject to a measurable process of attrition. Life tables can also be developed further for computing the chances of other vital events in human life, such as marriage and remarriage, the birth of children, widowhood, illness and disability, and labor force participation and retirement (Spiegelman 1957); and they enter into a wide variety of annuity and life insurance computations (Hooker & Longley-Cook 1953–1957; Jordan 1952).

## The conventional life table

The conventional form of a life table for the general population is illustrated in Table 1. The original data are recorded deaths and the census of population classified according to age (this step is not shown on the table). From these data were computed the rates of mortality, conventionally designated as qx, for each year of age, x. These rates show the proportion of deaths occurring within the year of age among those who attain that age; the rates are usually shown per thousand (1,000 qx). For example, Table 1 shows that of every 1,000 who just attained age 0 (the newly born), 23.55 died before reaching their first birthday; similarly, of every 1,000 who attained age six, 0.53 died within that year of age. Typically, mortality rates for a general population start at a high point in the first year of life, fall rapidly to a minimum at about age ten, and then rise with advance in years. The rise is gradual to about age 40, and then becomes increasingly rapid; since the maximum attainable age for human beings is in the neighborhood of 110 years, life tables seldom go beyond that point.

Once one knows the mortality rates at each age of life, it becomes possible to compute the number of survivors (column lx of the life table) and also the number of deaths (column dx). It is usually most convenient to start the population life table with a base (radix) of 100,000 newborn individuals. In the example presented here, where there is a death rate of 23.55 per 1,000 at age 0, among the 100,000 newly born there must be 2,355 deaths in the first year of life. The number of survivors to attain age 1 is then 100,000 – 2,355 = 97,645. With a mortality rate of 1.89 per 1,000 at age 1, among the 97,645 who attained that age there are

The number of survivors to age 2 is then calculated in the same way:

97,645 – 185 = 97,460.

This procedure is continued to the end of the life table. Obviously, the number in the survivorship column, lx , at any attained age is equal to the sum of the deaths in the dx column for that and all higher ages.

To compute the expectation of life (ėx) , or average future lifetime, for any attained age, it will be assumed that deaths,dx, are uniformly distributed over the year of age,x. Equivalent to this is the

Table 1 – Life table for white females, United States, 1949–1951°
RATE OF MORTALITY PER 1,000
OF 100,000 BORN ALIVE
Year
of
oge
Number dying
belween ages x
and x+1 among
1,000 living of oge x
Number
survining to
exact oge x
Number dying
belween ages x
and x+1
Number of years
lived by the cohort
between ages x
and x+1
Total number of years
lived by the cohort
from age x on, Until all have died
Average numbor
of years lived
after age x par person surviving
to exact age x°
x1 ,000qxlxdxLxTx̊ex
a. Based upon recorded deaths in the United States during the three-year period 1949–1951, recorded births for each year from 1944 through 1951, and the census of population taken April 1, 1950; for details, see U.S. Public Health Service 1959, pp. 149-158.
b. Represents complete expectation of life, or average future lifetime.
Source; U.S. Public Health Service 1954–1955, p. 18.
023.55100,0002,35597,9657,203,17972.03
11.8997,64518597,5527,105,21472.77
21.1297,46010997,4067,007,66271.90
30.8797,3518597,3086,910,25670.98
40.6997,2666797,2336,812,94870.04
50.6097,1995997,1696,715,71569.09
60.5397,1405297,1146,618,54668.13
70.4897,0884697,0656,521,43267.17
80.4497,0424397,0206,424,36766.20
90.4196,9993996,9806,327,34765.23
100388.392941142375661.92
101407.52180731433291.83
102426.0010746841861.74
103443.676127481021.66
104460.76341626541.59

assumption that each of the persons dying lived one-half year after the last birthday. Thus, among the 294 in Table 1 who attained age 100, there were 114 deaths during that year of age, and these individuals lived ½ × 114 years after their last birthday. Similarly, the 73 who died at age 101 lived 1½ years each after attaining age 100, and the 46 who died at age 102 lived 2½ years each after attaining age 100, and so on, to the last death. Altogether, the total number of years of life lived from age 100 on by the 294 who attained that age is (½ × 114) + (l½ × 73) + (2½ × 46) + (3½ × 27) + … = 566. This is the figure for age 100 in the column headed Tx. Since the 294 who attained age 100 lived a total of 566 years from their 100th birthday until the death of the last survivor, the average remaining lifetime was

This is more commonly known as the expectation of life,̊ex ; as an average, it is not applicable to any specific individual.

In Table 1 the life table symbols at the head of each column are defined by the terms above them. Reference has already been made to each, except Lx, which denotes the total number of years lived within the year of age by the number,lx, who attain that age. It has been assumed that each of the persons dying lived only one half year after the last birthday. Accordingly, among the number, lx who attain age x, the years of life lived by those dying during that year of age is ½dx. The years of life lived by the survivors is lx+1, which is equal to lxdx. The sum of ½dx and lxdx is the total number of years lived within that year of age. Thus,

Since Tx is the total number of years lived from age x on by those who attain age x, it follows that

Tx = Lx + Lx+1 + Lx+2 +….

and also that

Tx = Lx + Tx+1.

It should be recognized that except for the mortality rates, which represent an actually observed situation, all other columns of figures in the life table represent a hypothetical situation. Thus, the survivorship column and the column of life table deaths show only the expected number of survivals and deaths for successive ages, on the assumption that the mortality rates observed during the specified calendar period continue without change over time. The same assumption underlies the column of figures for expectation of life.

### Life table formulas

It will be seen from the preceding discussion that the construction of life tables rests upon a small number of elementary assumptions, which can be summarized in the following formulas :

Moreover, it is evident that if pf denotes the probability of surviving one year after attaining age x, then

Similarly, if npf denotes the probability of surviving n years after attaining age x, then

where nqx is the probability of dying within n years after attaining age x. Thus,

Another measure of mortality is the “force of mortality.” This measure takes into account the fact that mortality varies continually with advance in age. In this sense, the rate of mortality in the brief instant after attaining exact age x will be different from that for the brief instant just before leaving age x to attain exact age x + 1. The force of mortality, µx, is the annual rate of loss of lives, corresponding to the loss, at any instant of time, per head surviving at that time. In terms of the calculus,

where d/dx denotes the derivative of the specified function with respect to x.

The force of mortality at age x may be approximated by

or, more closely, by

The relevant approximation formulas have been discussed by Jordan (1952, pp. 19-21).

### Life table computation

The first task to be carried out in computing a life table for any specific population is to convert the central deathrate, mx—that is, the average annual death rate for persons of a given age—into a mortality rate, qx, such as has already been described. A means of doing this is illustrated as follows. In any specified community, let Dx denote the number of deaths recorded within a calendar year of individuals at age x on last birthday (or average annual deaths for a calendar period). Also, let Px denote the number of people at age x on last birthday on the mid-date of the calendar year or period; this is an approximation to the average number living and, therefore, to the number of years of life lived within the year of age. Then the central death rate at age x for the community is

The problem is to convert the central death rate, mx, into a mortality rate,qx.

In the life table the number of years of life lived during the year of age x is Lx and deaths during age x number dx, so that the central death rate mx is

Since dx = lx · qx,

Solving for qx yields

In terms of the recorded (observed) deaths and population,

In practice, however, the mortality rates at the very early ages are usually computed on the basis of a population estimated from recorded births and deaths, since census data for this stage of life are usually unreliable. The risk of mortality in infancy is highest in the first month following birth, and decreases rapidly thereafter; accordingly, the assumption of a uniform distribution of deaths is not valid for the first year of age. For the terminal ages of life, the basic data are usually meager and un-reliable; various artifacts are therefore used to compute these mortality rates. The mortality rates for the broad range of intervening ages are generally subjected to mathematical procedures of interpolation and graduation in order to produce a smooth progression of figures (Spiegelman 1955, p. 72). A complete life table shows the figures in each column for every age of life. An abridged life table shows figures for only selected ages, such as every fifth or tenth year of age.

## Life tables directly from census data

Where death data are grossly inadequate or lacking, a life table may be approximated from the age distributions of population in two consecutive censuses, as in the following simplified example.

Assume two censuses, five years apart, with correct reporting of ages and with no migration. Then, clearly, the population at age x + 5 in the second census,P"x+5, consists of survivors of the population five years younger at the time of the first census, P'x. The ratio of P"x+5 to P'x accordingly is a five-year survivorship rate for a population at age x last birthday. Assuming a uniform distribution of population over the year of age, this population is approximately at an average attained age x + ½. Thus,

Having arrived at a series of values of 5px+½ according to age, it is possible to work back to a series of mortality rates, qx. In using this method, allowance may be made for migration (Mortara 1949).

There is also a method of life table estimation that can be used when a population age distribution is available from only one census (Stolnitz 1956). If there is good reason to believe that the size of the population of a community has been virtually stationary over time and that mortality according to age has remained essentially unchanged over time, then its age distribution is clearly ery much like that of the life table column Lx. In other words, the number living, Px, at age x last birthday is proportionate to Lx. Thus,

and qx may be estimated, as in the case with two consecutive censuses.

Consider now a population that may be regarded as stable, in the sense that it is growing at a constant annual rate, r, and that mortality at each age is also constant over time. This growth results solely from an excess of births over deaths each year; there is no migration. Then, for an interval of five years,

Likewise, P"x+5 consists of survivors of P'x as before. It follows that

so that use is made of the population at the second census only. Stolnitz generalized this approach by tracing the populations and from their respective births,x + 5 and x years previously, namely Bx+5 and Bx. For this, he introduced survival factors to the same attained age x last birthday, namely and and made use of the five-year survivorship ratio, 5px+½. Thus,

Stolnitz shows how the birth ratio and the ratio of survival factors may be estimated from other experiences. With such estimates it becomes possible to compute 5px+½ from the age distribution of a single census.

### Model life tables for developing areas

In the developing areas the problem is to estimate a life table for a population with scanty mortality data or from data gathered in a special survey. Since the mortality rate in infancy or the first few years of life is frequently indicative of the general level of mortality, such a rate may be used as the basis for estimation of life table values. Such an observed mortality rate, with suitable adjustment to enhance its validity, is used as a key to select one of a series of life table mortality rates (q0,4q1, and 5qx for x at five-year intervals) from 40 theoretical model series (United Nations 1955a). These models were derived from a study of the patterns of mortality rates in existing life tables. For refinement, the series of life table mortality rates may be selected by interpolating among the models on the basis of the key rate. Further refinement is possible by computing from the equations used to derive the models. Although these model life tables of the United Nations have been subject to technical criticisms, they are widely used (Gabriel & Ronen 1958; Kurup 1966). A more extensive set of model life tables, prepared at Princeton University, takes into account variations in the patterns of mortality between four broad geographic regions, defined as East, West, North, and South, in addition to variations in the level of mortality within each region (Coale & Demeny 1966).

## Life tables directly from death data

As pointed out before, in a population that is virtually stationary, with mortality rates essentially unchanged over time, the age distribution corresponds closely to that in a life table. Only in such a situation is it feasible to cumulate the distribution of deaths according to age, starting with the highest age and noting the total for each age, running back to birth, in order to approximate the survivorship column of the life table. This approach is not applicable in any other situation, since the age distribution of deaths will be influenced by the age distribution of the population. Thus, a population with a large proportion of aged persons will have a large proportion of its deaths at the older ages, irrespective of the level of its mortality rates.

## Multiple decrement tables

In a multiple decrement table, the survivorship column of the life table is split, in passing from one age to the next, into two or more component parts, on the basis of changes in status or of newly acquired characteristics (Jordan 1952, pp. 237, 251; Bailey & Haycocks 1946). One example is the case where the survivorship column is split, on the basis of marriage rates according to age, to distinguish those who marry from those who remain single. In another example, shown in Table 2, the

Table 2 — Example of a double decrement table, with decrements by death and by disability
Of 1 00,000 born alive
RATE OF MORTALITY
PER 1,000
DISABILITY
RATE PER 1,000°
NUMBER SURVIVING TO
EXACT AGE x
ACTICE
LIVES
DISABLED
NUMBER DYING BETWEEN
AGES x AND x+1
Year of age, xAmong active livesAmong disabled livesAmong active livesTotalAs active livesAs disbled livesBetween ages x and x+1TotalAmong active liesAmong disabled lives
a. The radix in the source (100,000 at age 10) was changed to 100,000 at birth.
b. Per 1,000 active lives at exact age x.
c. Assuming no lives were disabled before age 15.
Source: Adapted from Hunter et al. 1932, p. 92.
157.552670.58766,94966,9490405115056
167.472540.58466,43866,404343950949613
177.402410.58165,92965,869603850748720
187.402290.57865,42265,344783850648422
197.402170.57564,91664,822943850448024
Table 3 — Example of select and ultimate table, showing probabilities of remarriage during widowhood
012345 or more
Age of Widowhood      Attained age
Source: Adapted from Myers 1949, p. 73.
350.02010.04900.03860.03760.02300.016340
360.01840.04490.03540.03450.02110.014941
370.01690.04120.03240.03160.01930.013742
380.01550.03770.02970.02900.01770.012643
390.01420.03450.02720.02660.01620.011544

SELECT TABLEULTIMATE TABLE

survivorship column of the life table is split to show those who become permanently disabled lives apart from those who remain as active lives. This table shows, in addition to the numbers surviving to successive ages as active lives and as permanently disabled lives, the rates of mortality for each of these categories and the rates at which active lives become permanently disabled. The column of life table deaths is also split to show the number of deaths among the permanently disabled separately from that among the active. It is assumed that the number of newly disabled lives in any year of age is uniformly distributed over that year; consequently, they are exposed to the mortality rate of the disabled for an average of one half of a year.

## Select tables

The life table has been described in terms of rates of mortality dependent only upon attained age; in describing multiple decrement tables, reference was made to rates of disability and of marriage according to attained age. In select tables, rates of mortality (or other rates) are shown on the basis of both the age at acquisition of a new characteristic and the duration since that acquisition (Jordan 1952, p. 26). This two-way classification constitutes a select table, since some selective process is present at the time of acquisition. For example, mortality rates for permanently disabled lives may be shown not only for the age at which disablement occurred but also separately for each subsequent year of disability. Another example of a select table is the two-way classification of rates of remarriage for widows, in relation to both age at widowhood and years since that event. Such a two-way classification of rates is shown in Table 3. In that table, remarriage rates after the fifth year of widowhood are shown only on the basis of attained age, since duration in this case is only of minor influence upon the rates. The table as a whole is known as a select and ultimate table. That portion showing rates according to duration since widowhood is the select table; the ultimate table is that portion showing rates only according to attained age, since duration is no longer of any importance. Select and ultimate tables are used in life insurance mortality investigations. The choice of the number of durations to be shown for the select period is a matter of study in each experience.

## Cohort or generation life tables

In the foregoing account of the conventional life table and the related multiple decrement and select tables, the rates of mortality and other rates of attrition were based upon observations during some specified year or other period. The hypothetical nature of the conventional life table with respect to the time period of observation has already been indicated. A realistic picture of the mortality and survivorship experience of a cohort traced from birth is obtained by observing these events each year in a generation born at the latest 100 years ago. In that way, a record would be obtained of the number surviving to successive ages in successive years and also of the corresponding number of deaths at each age; the mortality rates according to age in successive years may then be computed. After the last death, it would be possible to compute the average length of life of the generation and the average years of life remaining after each age. Such a table is called a generation life table, since it reflects the actual mortality rates of a cohort as it ages in successive years. The table derived from mortality rates for a calendar year or period is called a current life table (Dublin, Lotka, & Spiegelman [1936] 1949, p. 174; Jacobson 1964). Thus, the expectation of life computed from mortality rates observed in 1850 will understate the average length of life of the generation born that year, because of the reductions in mortality since then. In general, with the trend toward lower mortality, the expectation of life at birth computed from a current life table understates the average length of life of a newly born generation.

## Further applications

In addition to the applications of the life table that are mentioned in the opening paragraph of this article, increasing use is being made of it as an analytic tool in social and economic problems. Several interesting and important examples may be cited in the field of demography (for references to examples, see Spiegelman 1955; 1957). John Du-rand (1960) made use of the United Nations model life tables, cited previously, as an adjunct in arriving at estimates of expectation of life at birth for the western Roman Empire. The life table is funda-mental in the stable-population theory developed by A. J. Lotka (Dublin & Lotka 1925) and also in Lotka's work on the structure of a growing population (1931). In the field of education E. G. Stock-well and C. B. Nam (1963) prepared school life tables to show the joint effects of death and school dropouts on school attendance patterns. B. C. Churchill (1955) studied the mortality and survival of manufacturing, wholesale trade, and retail trade firms in the United States; in similar fashion A. J. Jaffe (1961) has used data from the censuses of manufactures in Puerto Rico to prepare survival curves according to the age of the establishment.

Mortimer Spiegelman

## BIBLIOGRAPHY

A very elementary account of the essentials of the life table is given in Dublin, Lotka, & Spiegelman [1936] 1949. A wholly nontechnical account of the life table, including double decrement and select tables, with brief descriptions of applications, will be found in Spiegelman 1957. The beginner in graduate study who has a nonmathematical background but a sense of arithmetic will find the chapter on the life table in Barclay 1958 a good introduction to the subject. A corresponding account of the life table, with some further development, is contained in Pressat 1961. U.S. Bureau of the Census 1951 provides step-by-step directions for elementary life table construction, as well as exercises for the beginning student. More technical is the exposition of the life table in Benjamin 1959. The student ivith a background in the calculus seeking a more comprehensive understanding of the life table, double decrement tables, and select tables may start with Hooker & Longley-Cook 1953–1957; Jordan 1952. The theoretical aspects of double and higher-order decrement tables are discussed in Bailey & Haycocks 1946.A firm understanding of the techniques of life table construction requires a good background in the means for estimating the exposed-to risk, as given in Gershenson 1961, and also for a grasp of the elements of graduation and interpolation, as given in Miller 1946. The principal techniques used in the construction of life tables are described in Spiegelman 1955,which also treats, in detail, the special situations at the early ages, where the assumption of a uniform distribution of deaths over the year of age is not applicable, and at extreme old age, where artifacts are used to complete the column of mortality rates.

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Spiegelman, Mortimer 1955 Introduction to Demography. Chicago: The Society of Actuaries. → See the references in Chapter 5 for materials on life tables.

Spiegelman, Mortimer 1957 The Versatility of the Life Table. American Journal of Public Health 47:297–304. → Contains a list of references.

Stockwell, Edward G.; and NAM, CHARLES B. 1963 Illustrative Tables of School Life. Journal of the American Statistical Association 58:1113–1124.

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