comparison of two methods of constructing abridged life tables · comparison of two methods of...

INATIONAL CENTER Series 2

For HEALTH STATISTICS Number 4

VITAL axad HEALTH STATISTICSDATA EVALUATION AND METHODS RESEARCH

comparison of two methods of

Constructing

Abridged

life Tablesby reference to a “standard” tableComparison of the revised and the prior methodof constructing the abridged life tables for theUnited States.

Washington, D.C. Revised March 1966

U.S. DEPARTMENT OF

HEALTH, EDUCATION, AND WELFARE Public Health Service

John W. Gardner Wil Iiam H. Stewart

Secretary Surgeon General

Public Health Service Publication No. 1000-Series 2-No. 4

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402- Pric, ! 15 cents

NATIONAL CENTER FOR HEALTH STATISTICS

FORREST E. LINDER, PH. D., ~i?’e7f07

THEODORE D. WOOLSEY, L@@ Director

OSWALD K. SAGEN, PEL D., ~uistmtDirector

WALT R. SIMMONS, M. A,,statistical Advisor

ALICE M, WATERHOUSE, M. D., Medical Advisor

JAMES E. KELLY, D. D.s., Dental 4dviJor

LOUIS R. STOLCIS, M, A., .&tcuth oj’icer

OFFICE OF HEALTH STATISTICS ANALYSIS

IW,IO M. MORIYAMA, PH. D., GViej

DIVISION OF VITAL STATISTICS

ROEIERTD. GROVE, PH. D., Chitf

DIVISION OF HEALTH INTERVIEW STATISTICS

P!mn S. LAWRENCE, SC, D., Cki?f

DIVISION OF HEALTH RECORDS STATISTICS

h40NROE G. SIRKEN, PFL D,, Ckief

DIVISION OF HEALTH EXAMINATION STATISTICS

ARTHUR J. MCDOWELL, Chief

DIVISION OF DATA PROCESSING

SIDNEY BINDER, Chief

Public Health Service Publication No. 1000-Series 2-No. 4

CONTENTS

Page

Inmoduction ---------------------------------------------

Method of Construction -----------------------------------

Constructing the 1959 Abridged Life Tables -----------------

Table 1. Conversion factors based on decenniallifetablesforthe United States, 1949-51-----------------------

Evaluation of the Abridged Life Table Methods --------------

Table 2. Differences between values of expectation oflifeinthe complete life table andinabridged life tables,by color, sex, and age: United States, 1949-51-----

Table3. Differences between values of the probabilityofdyingin the complete life table andinabridgedlife tables,bycolor, sex, andage: United States, 1949-51-----

Appendix: Explanation of the Columns of Table A-----------

Appendix Table A. Computation of abridged life table forthe total population of the United States, 1959----------

1

1

2

3

4

6

7

8

10

SYMBOLS

Data not available ----------------------- ---

Category not applicable ------------------ . . .

Quantity zero --------------------------- -

Quantity more than Obutless than 0.05---- 0.0

Figure doesnot meet standards of

reliability or precision ----------------- *

COMPARISON OF TWO METHODS OF

CONSTRUCTING ABRIDGED LIFE TABLES

INTRODUCTION

The publication of an annuai series of abridgedlife tables for the United States was started in1945. After small biases were detected in thevalues of 1950 U.S. abridged life tables, studieswere undertaken which led to the developmentof a revised method for constructing the U.S.abridged life tables. This report outlines the re-vised method used in constructing the abridgedlife tables since 1954. The construction of thelife table for the total population for 1959 isshown in appendix table A. An earlier report 1out-lined the method used in preparing the abridgedlife tables for the years 1946 to 1953 inclusive,which henceforth will be referred to as theoriginal method.

A test of the accuracy of the revised method ofconstructing the U.S. abridged life tables is pre-sented which involves a comparison of the 1949-51 abridged life tables constructed by the revisedmethod with the complete decennial 1949-51 lifetables which were constructed by elaborate andlaborious methods2. The 1949-51 abridged lifetables constructed by the original method are alsocompared with those derived from the 1949-51 lifetables. Comparing the abridged life tables, con-structed by original and revised methods, with thedecennial life tables provides a test of the relativeaccuracy of these methods of constructing theU.S. abridged life tables.

This report was prepared

of Heaifh Records Statistics.

by Monroe G. Sir&en, of the Division

METHOD OF CONSTRUCTION

The original and the revised methods of con-structing the U.S. abridged life tables have incommon the fact that each involves reference toa standard life table. According to this method ofconstructing abridged life tables, certain relation-ships among the functions of the life table underconstruction are assumed to be the same as thoseof another life table already existing (referred toas the “standard” table). In the calculation of theannual abridged life tables since 1954, the de-cennial U.S. life tables 1949-513 have been used asstandard tables. When the 1959-61 decennial lifetables are constructed, they will become thestandard life tables in constructing the U.S.abridged life tables.

The method presented here is based on anobserved relationship between the probability ofdeath t. ~, )and the age-specific death rate f,p,).

The function “q ~ is the proportion n; where IXis

the number of survivors to exact ;ge x in thehypothetical life table cohort and ,dX is the number

of the group who die before reaching exact agex + n. The function , P, is the quotient of the num-

ber of deaths between exact ages x and x +ZI duringthe year and the size of the living population be-tween these exact ages. The age-specific deathrate may be defined either in terms of observedpopulation data fnilc?,) or in terms of the stationary

population of the life table ~nmX).The former

(nMX) is the quotient of the number of deaths in a

given calendar year between exact ages x and

1

x+n and the midyear population Delween thoseexact ages. The latter cflrn,) is the number of

deaths (ndXj in the life table divided by the number

of persons f” LX) in the stationary population of

the life table between ages x to x+~.

According to the revised method of con-structing the abridged life table, the relationship

between “q, and .PX is given by the formula

n .Px(1) “q, = 1

1 + f~n/JxJ“Px

(2) :.,=:-+..X. x

It will be observed that formula

2 sets of conversion constantswhether ~PX is defined as nMx

age-specific mortality rate, or as

(2) generates

according tothe observed

.rn X, the age-

specific mortality rate of the stationary population

of the life table. The constants:Mx

are usefd

as adjustment factors to convert the observedpopulation age-specific mortality rates into thevalues on “q ~ of the abridged life table. The

constants ~mx are used to calculate the values

of “L ~ from the values of IXand ~dx in the

abridged life table.Thus ,

nix- L(3) fro,= ~n “

nxor

(4) nLx = ‘]. - ~mx ndx”

Greville4 has also suggested the use of formula(4) to calculate the L-function in the construction

of the abridged life table by reference to a stand-ard table.

The assumption underlying the abbreviatedmethod of life table construction used here is thatin each age interval x to x + n, the constants

:#x <P= M,m] may be regarded as having the

same value in the life table under construction as

in the standard table. The constants a .n!!x ~d

anmx that have been used in the construction of

the abridged life tables since 1954 are presentedin table 1. They were derived by fo~mula (2)according to relationships observed bepween “qx

and “Px in the complete U.S. life table for the

decennial period 1949-51. Until more current

standard tables (U,S. life tables for the decennialperiod 1959-61 ) are constructe~, these constantswill be used each year to Construct the U.S.abridged life tables.

CONSTRUCTING THE 1959ABRIDGED LIFE TABLES

Basic sources of data used in the preparationof the U.S. life tables for 1959 were the annualmortality tabulations of the National Vital Sta-tistics Division and estimates of the populationon July 1, 1959, by age, color, and sex pre-pared by the U.S. Bureau of the Census.

Values of .Mx, the observed population age-

specific mortality rates were obtained from the

basic mortality and population data. The valuesof “q ~ were calculated by formula (1) using the

set of constants:Mx

presented in table 1. (The

method of calculating the values of the proba-bility of death during the first year of life and ofthe final age group 85 years and over is de-scribed below. ) After the values of nqx had been

obtained, the lx and .dx functions were computed

in the conventional manner, according to theformula

.dx= (Ix) (nqx); Ix+n = lx d .-nx

Thereafter, the values of “Lx were calculated by

formula (4) using the set of constants, :m ,x

presented in table 1. The values of ~ were ob-

tained by summing the “Lx column, starting with

the oldest age group. In other words,

Tx=Tx+n+ ~Lx.

2

Table 1. Conversion factors based on decennial life tables for the United States,

Ageinterval(years)

. -- -- -- -. -- -;-:o ----------1o-15---------15-20---------

20-25---------25-30---------30-35---------35-40---------

40-45---------45-50---------50-55---------55-60---------

60-65---------65-70---------70-75---------75-80---------80-85---------

- - ---- -- -- --L;o----------1o-15---------15-20---------

20-25---------25-30---------30-35---------35-40---------

40-45---------45-50---------50-55---------55-60---------

60-65---------65-70---------70-75---------75-80---------80-85---------

1949-51

TotalMale Female

popu-lation

Tots 1 White Nonwhite Total White Nonwhite

;Mx= =+9nxnx

18.7253 19.075517.1188 16.1574

.27.7680 -28.311910.2732 7.1700

7; m: 7.2706-.2167

-1.0910 1.2348.9558 1.1954

2.2822 2.13671.6621 1.80812.2507 2.22772.2598 2.3750

2.4041 2.38482.2343 2.35842.3399 2.38722.4376 2.50142.5307 2.5607

2.41522.68342.31742.3205

2.44232.44682.38392.3421

2.32932.34372.35362.3700

2.39802.40552.42802.48792.5747

18.516421.3402

-31.71547.8382

8.1040-1.0433

2.69551.6008

1.78542.15242.12622.3389

2.36812.34312.38072.50262.5621

2.39902.66022.28792.3074

2.45422.47572.39202.3375

2.32332.33702.35212.3820

-L2.41452.43562.46242.52422.6051

2.36642.65372.29542.3189

2.46872.48632.38662.3244

2.77402.32872.34272.3725

2.40542.42722.45692.52232:6061

16.73989.5389

-15.5642.1092

4.96901.76925.9535

-1.8298

4.0718.4126

2.93712.5563

2.49712.42282.48142.53002.5478

17.8984 18.669824.9787 22.5984

-22.0673 -18.045518.0825 19.9746

5.8762 6.3873.3806 -1.3368

-3.9384 3.0035.7362 1.1471

2.3874 1.04061.4901 2.17932.3393 2.08342.0481 1.9335

2.43132.02012.27932.36452.4998

nlx -nLxmnmx =

dnx

2.50442.70822.24752.2390

2.38952.43772.40392.3909

2.36982.37032.39462.4442

2.49122.51752.53132.56072.5985

2.38632.02042.23742.35952.4987

2.43542.71762.36342.3373

2.41782.40352.37352.3507

2.34112.35632.36092.3569

2.37312.36402.38762.44992.5449

2.42122.69962.40462.3663

2,42522.40062.36782.3427

2.33182.34362.34362.3455

2.33922.35692.34882.37802.4453

15.764221.2327

-4:.;;n;.

3.53982.22912.79423.3718

5.9329-.00912.97722.3614

2.69881.98442.73132.49642.5590

2.46712.81272.19102.2543

2.38512.40642.37922.3711

2.36582.38342.40242.4361

2.47292.49742.50642.53212.5693

3

The values of the average remaining lifetime

was then obtained by division .8X= TX + IX .

Formulas (1) and (3) respectively were not

used to compute the 9Xand Lx functions for the

first year of life and the final age group 85 years

and over. Rather, the special treatment of theseage groups used in the construction of U.S.abridged life tables for the years 1945 to 1953inclusive was contj,nued. The following explana-

tion has been adapted and extracted from a reportthat describes the method used to construct theseearlier tables.1

For the age group 85 years and over formula(2) shows that

:M85 is infinite since n= CO.

Hence the assumption that the value of amM85

is the same in the life table under constructionas in the standard table is not useful, and some

other assumption must be made. Instead, theratio ~,defined as the quotient of the value of

-Ma based on the actual data by the corre-

sponding value ~rr18~ for the stationary popu-

lation of the life table was assumed to be the same

in the table under construction as in the standardtable. But ~mg~ is the reciprocal of

average remaining lifetime. Thus, the

g 85 can be computed by the formula

885=JL~M85

86, the

value of

According to the standard tables (1949-51),z=. 9487119 for the total population. The valuesof < for the 4 subdivisions of the population bycolor and sex are shown below:

Subdivision ofthe population A

White males .9610759White females .9554947Nonwhite males .8534401Nonwhite females .8072982

The abridged life table for 1959 can then becomputed since

The value of qOthe proportion of liveborn

infants dying before reaching age 1, is computedfrom birth and death statistics, being taken asequal to the adjusted infant death rate. A lmethod

of adjusting the infant death rate for the changing

number of births is described in a previous publi-cation. 5 The adjustment is made by allocating thedeaths of infants occurring during a given yearto the year in which the infants were born. The

infant deaths so allocated are then related to thebirths occurring in the respective year of birth.The expression for computing the adjusted infantmortality rate per 1,000 live births may bewritten:

Adjusted rate =[ 1y +% x 1,000

where

D . number of infant deaths occurring inthe given year.

f = ratio of deaths occurring in the givenyear among infants born in the precedingyear to the total infant deaths of the

given year. This is referred to as the“separation factor. ”

E . number of births occurring in the givenyear.

E’ . number of births occurring in the pre-

ceding year.

The stationary population in the first year oflife was obtained by the formula Lo= 10– ( I– f )do,

EVALUATION OF THEABRIDGED LIFE TABLE METHODS

A set of U.S. abridged life tables, 1949-51for subdivisions of the population by color and sexwas constructed by the revised method of con-struction by reference t6 a standard table. Values

4

of the constants“w. and %x ‘eededin ‘e

construction of these tables were derived fromthe complete U.S. life tables, 1939-41, whichserved as the standard tables. The decennial U.S.life tables 1949-51 were the criterion tables forthe evaluation of the precision of the abridgedlife tables.

The basic data used in the preparation of theU.S. abridged life tables 1949-51 were essentiallythe same as those which had been used in the pre-paration of the complete U.S. life tables 1949-51.These included mortality data by age, sex, andcolor for the 3-year period 1949-51, extractedfrom the annual issues of the Vital Statistics ofthe United States published by the National VitalStatistics Division, and population data by age,sex, and color enumerated in the 1950 Census andpublished by the Bureau of the Census in U.S.Census of Population, Volume II,’ ‘Characteristicsof the Population. ”

There is close agreement (table 2) betweenthe values of the expectation of life based on thecomplete life tables and those based on the re-vised abridged life table method. The abridgedlife table values exceed the decennial life tablevalues at virtually all ages but the differencesare small. For example, the difference betweenthe values of the expectation of life at birth wasonly .01 years for the total population; it was lessthan .03 years for white males, white females, andnonwhite males; and .15 years for nonwhite fe-males. For each of these population groups, thereis a tendency for the differences between thevalues of the expectation of life to increase withadvancing age. At virtuall y all ages the differencesare greater for nonwhite than for white persons,and within each color group, the differences aregreater for females than for males.

Using the same basic data, that is the popu-lation data from the 1950 Census and the mor-tality data for the 3-year period 1949-51, anotherset of abridged U.S. life tabIes 1949-5 I were pre-pared by the original abridged life table method.This is the method of construction by reference toa standard table, that had been used to constructthe annual abridged U.S. J.ife tables, 1945-53.

The assumptions underlying the originalmethod are that in each age interval x to x+ n, ~

defined as the ratio ,qX + JWX and the values

of the ratio “jX = “LX + (IX+iX+~ ) were as-

sumed to have the same value in the life tableunder constructions as in the standard table.Values of the cons’tihlk “hXand ,jX iieeded in the

construction of the abridged U.S. life tables1949-51 by the original method were available 1for the decennial U.S. life tables 1939-41 whichserved as the standard tables.

The values of expectation of life based on theoriginal method exceed those of the decennial lifetable at every age (table 2). At virtually everyage, these differences are greater than theamounts by which values of expectation of lifebased on the revised method exceed those basedon the decennial life table. Thus, for the totalpopulation the value of expectation of life at birthaccording to the decennial life table is exceededby .01 years according to the revised method andit is exceeded by .15 years according to theoriginal method. It is noteworthy that both methodsof constructing life tables by reference to astandard table slightly overstate the values of theexpectation of life at every age, although the over-statement is consistently less for the revisedthan for the original life table method.

The absolute values of difference between~qX values based on the decennial life tables

and on the abridged life table are virtually alwayssmaller for the revised than for the originalabridged life table method (table 3). Furthermore,the original method in most age groups under-states the values of “9X,a tendency which is not

evident for the revised method.

REFERENCESll-homa~N.E. ~r&lle and @sCaV A. c~Ison) “~hhod of

Constructing rbe Abridged Life Tables for tbc United States, 1949, ”VifaI Statistics--Specid Reports, Vol. 33, No. 15, June 30, 1953.

2, ~onroe G. Sirk=n ad ‘.iOtier %@drnanl Wethod of Ccm-strucring the 1949-5 I Narionsl, Divisional, nnd Srate Life Tables, ”Vital Statistics--Specia[ Reports, Vol. 41, No. 5, July 31, 1959.

%kmroe G. Sirken, “United States Life TabIes, 1949-51,”Vital Statistics--Special Reports, Vol. 41, No. 1, November 23, 1954.

4Thomas N. E. Greville, “On rbe Formula for rhe L-Function

io a Special ‘,iortaliry Table Eliminating a Given Cause of Death, ”Transactions oj the Society of Actuaries, Vol. VI, Meeting No. 14,April 19:4.

51W=0 hf. ],foriyaa and Thomas N. E. Greville, “Effect of

Changing Birrb Rates Upon Infsnt ?.forrality Rates,” Vital Statisfics--Special Reports, VOL 19, No. 21> November 10, 1944.

%omas N. E. Grevilk, “United States Life TabIes and Acru-arial TabIes, 1939-41, ” U.S. Ciovernmenr Printing Office, 1947.

5

Table 2. Differences between values of expectation of life in the complete life tableand in abridged life tables, by color, sex and age: United States, 1949-51

Sex andage

U

o-1--------- ---- --- -

L?o-------1o-15------

15-20------20-25------25-30------30-35------

35-40------40-45------45-50------50-55------

55-60------60-65------65-70------70-75------

75-80------80-85------85+--------

FEMALE

o-1-------------- ---L?o-------1o-15------

15-20------20-25------25-30------30-35------

35-40------40-45------45-50------50-55------

55-60------60-65------65-70------70-75------

75-80------80-85------85+--------

~ basedX oncomplete

lifetables

66.3167.4163.7758.98

54.1849.5244.9340.29

35.6831.1726.8722.83

19.1115.7612.7510.07

7.775.884.35

72.0372.7769.0964.26

59.3954.5649.7745.00

40.2835.6431.1226.76

22.5818.6415.0011.68

8.876.594.83

White

Abridged life table valuesminus

complete lLfe table values

Originalabridgedmethod

.10

.10.11.12

.12

.12

.11

.12

.11

.12

.11

.11

.12

.12

.14

.15

.15

.21

.05

.20

.20

.22

.21

.21

.22

.22

.22

.22

.22

.22

.22

.23

.23

.22

.22

.22

.19

.11

Revisedabridgedmethod

.,00-.01

.00

.00

.01

.00

.00

.00

.00

.00

.00

.00

.00

.00

.01

.01

.01

.02

.05

.02

.01

.02

.02

.02

.03

.03

.03

.03

.03

.03

.03

.03

.03

.03

.03

.05

.06

.11

~ basedx oncomplete

lifetables

58.9161.0657.6952.96

48.2343.7339.4935.31

31.2127.2923.5920.25

17.3614.9112.7510.74

8.837.075.38

62.7064.3760.9356.17

51.3646.7742.3538.02

33.8229.8226.0722.67

19.6216.9514.5412.29

10.158.156.15

Nonwhite


complete life table values

.16

.17

.22

.23

.22

.23

.23

.23

.24

.22

.24

.22

.19

.21

.27

.35

.44

.54

.61

.30

.31

.37

.36

.37

.37

.37

.37

.36

.35

.38

.40

.37

.39

.45

.51

.36

.741.18


——.

J);

.06

.07

.06

.07

.07

.07

.09

.07

.10

.09

.08

.11

.15

.20

.27

.40

.61

.15

.15

.19

.18

.18

.19

.19

.20

.20

.19

.22

.24

;;:

.35

.41

.57

.771.18

6

Table 3. Differences between values of the probability of dying in the complete lifetable and in abridged life tables, by color, sex, and age: United States, 1949-51

Sex andage

interval

MALE

- -----.L.?o -----1o-15----15-20----

20-25----25-30----30-35----35-40----

40-45----45-50----50-55----55-60----

60-65----65-70----70-75----75-80----80-85----

FEMALE

- ------L;o-----1o-15----15-20----

20-25----25-30----30-35----35-40----

40-45----45-50----50-55----55-60----

60-65----65-70----70-75----75-80----80-85----

~qx basedon

completelife

~ables

.00544

.00347

.00354

.00652

.00852

.00853

.01013

.01480

.02381

.03821

.05963

.09098

.13163

.18580

.26348

.37002

.49946

.00457

.00246

.00210

.00312

.00396

.00485

.00657

.00945

.01440

.02200

.03294

.05039

.07812● 12021.19465.30096.43860

White




.00016

.00001-.00005.00002

.00002-.00003-.00002-.00008

-.00009-.00021-.00009-.00032

-.00064-.00142-.00287-.00838-.02093

.00011

.00013

.00000

.00001

-.00001-.00001-.00001-.00006

- ●00005-.00012-.00016-●00041

-.00080-.00219-.00401-.01071-.02056


.00015

.00002-.00006.00007

.00012-.00001.00004

-.00004

-.00003-.00008.00008.00000

-.00008-.00018.00021.00025.00330

.00011

.00002-.00001.00002

.00001

.00000

.00000-.00003

-.00002-.00003-.00001-.00008

-.00001-.00050-.00016-.00114-.00257

nqx basedon

completelifetables

.01043

.00498

.00522

.01102

.01801

.02168

.02703

.03616

.05005

.07365

.10658

.14721

.18614

.22524

.27260

.33636

.41444

.00894

.00396

.00355

.00846

.01291

.01665

.02196

.03100

.04410

.06382

.08845

.12020

.15221

.18615

.22601

.28105

.34418

Nonwhite




.00080-.00002-.00005.00000

-.00007-.00018-.00013-.00077

.00031-.00198-.00301-.00073

.00028

.00094-.00325-.00581-.02013

.00059

.00002

.00001-.00008

-.00009-.00009-.00042-.00059

-.00008-.00028-.00289-.00165

-.00022-.00278-.00008-.00445-.00583


.00067

.00001-.00006.00000

.00001-.00003.00028

-.00066

.00086-.00130-.00110.00031

.00066

.00123-.00092.00149

-.00279

.00047

.00004-.00006.00004

.00003

.00001-.00012-.00061

.00067-.00020-.00101-.00013

.00119-.00145.00188.00041.00117

7

APPENDIX

EXPLANATION OF

Column 1—Age intewal (x to x + n ).-Theage interval shown in column 1 is the interval

between the two exact ages indicated. For in-stance, “20-25” means the 5-year interval be-

tween the 20th and the 25th birthdays.

Column 2—Popzzlation (, P, ) .—This column

shows the estimated midyear population for the

indicated age interval. Births for 1958 and 1959were used in computing qO.

Column 3—Deaths (n L)X). —This COhlInll

shows the number of deaths for the age interval

during 1959.Columns 4 and 5—Death Yates (, MxI .—The

age -specific death rate shown in column 4 is thecentral death rate for the age interval. In column5, these rates have been adjusted proportionatelyfor deaths for which age was not reported on the

death certificate.Column 6— Conve~sion facto~ ( u .—This

,Mx)

column is derived from a “standard” table, inthis instance, the life table for the total popu-lation of the United States, 1949-51. These con-version factors are shown in table 1.

Columns 7and S—Proportion dying ( ~qx) .—

The number shown in column 7 is the denominator

of the proportion of the cohort dying in the ageinterval according to formula(1), page 3. Column 8shows the proportion of the cohort who are alive at

the begiming of an indicated age interval who willdie before reaching the end of that age interval.For example, for the population in the age in-terval 20-25, the proportion dying is 0.0061 —out

of every 1,000 persons alive and exactly 20 yearsold at the beginning of the period, 6.1 will die

before reaching their 25th birthday. In otherwords, the “qXvalues represent probabilities that

persons who are alive at the beginning of a spe-

COLUMNS OF TABLE A

cific age interval will die before reaching thebeginning of the next age interval. The “propor-tion dying” column forms the basis of the lifetable: the life table is so constructed that allother columns are derived from it.

Column 9—NumbeY swviving ( ~x).—This

column shows the number of persons, starting

with a cohort of 100,000 Iive births, who sur-vive to the exact age marking the beginning ofeach age interval. The IXvalues are computed

from the ,qx values, which are successively ap-

plied to the remainder of the original 100,000 per-

sons still alive at the beginning of each age in-terval. Thus, out of 100,000 live born babies,

97,357 will complete the first year of life andenter the second, 96,948 will begin the sixth year;

96,051 will reach 20; and 17,877 will live to age 85.Column 10 —Numbev dying ( ,dX ) .—This col-

umn shows the number dying in each successiveage interval out of 100,000 live births. Out of100,000 persons born alive, 2,643 die in the firstyear of life, 409 in the succeeding 4 years, 584 inthe 5-year period between exact ages 20 and 25,

and 17,877 die after reaching age 85. Each figure

in column 10 is the difference between twosuccessive figures in column 9.

Column 11— Conversion facioy ( anm ). —x

This column is derived from a “standard” table,

in this instance, the life table for the total popu-lation of the United States, 1949-51. These con-version factors are shown in table 1.

Columns 12 and 13—Stationary population(,LX and ~) .–Suppose that a group of 100,000

individuals is born every year and that the pro-portions dying in each such group in each age in-terval throughout the lives of the members areexactly those shown in column 8. If there were no

8

migration and if the births were evenly distri-buted over the calendar year, the survivors ofthese births would make up what is called astationary population-stationary because in sucha population the number of persons living in anygiven age group would never change. Thus, acensus taken at any time in such a stationarycommunity would always show the same totalpopulation and the same numerical distribution ofthat population among the various age groups. Insuch a stationary population supported by 100,000annual births, column 9 shows the number of per-sons who, each year, reach the birthday whichmarks the beginning of the age interval indicatedin column 1, and column 10 shows the number ofpersons who die each year in the indicated ageinterval.

Column 12 shows the number of persons inthe stationary population in the indicated age in-terval. For example, the figure given in the ageinterval 20-25 is 478,829. This means that in astationary population supported by 100,000 annualbirths and with proportions dying in each agegroup always in accordance with column 8, acensus taken on any data would show 478,829 per-sons between exact ages 20 and 25.

Column 13 shows the number of persons inthe stationary population in the indicated age in-terval (column 12) and all subsequent age in-tervals. For example, in the stationary popu-lation referred to in the last illustration, column

13 shows that there would be at any given moment,a total of 5,030,781 persons who have passedtheir 20th birthday. The population at all ages Oand above (in Q@r words, the total population ofthe stationary community) would be 6,965,532.

Column 14--Average remaini~ lifetime(8X) .—The average remaining lifetime (also

called expectation of life) at any given age is theaverage number of years remaining to be livedby those surviving to that age on the basis of agiven set of age-specific rates of dying. In orderto arrive at this value, it is first necessary toobserve that the figures in column 12 can alsobe interpreted in terms of a single life tablecohort without introducing the concept of thestationary population. From this point of view,each figure in column 13 represents the totaltime (in years) lived between two indicatedbirthdays by all those reaching the earlier birth-day among the survivors of a cohort of 100,000live births. Thus, the figure 478,829 in the ageinterval 20-25 is the total number of years livedbetween the 20th and 25th birthdays by the 96,051persons (column 9) who reached the 20th birthdayout of 100,000 live born babies. The correspondingfigure (5,030,781) in column 13 is the total num-ber of years lived after attaining age 2(3by the96,051 persons reaching that age. This numberof years divided by the number of persons(5,030,781 divided by 96,051) gives 52.4 years asthe average remaining lifetime at age 20.

9

Appendix Table A. Computation of abridged life table

AGE INTERVAL

Period of lifebetween twoexact agesstated inyears

x to x +n

(1)

o-11 ---------------1-5----------------5-10 ---------------1o-15--------------15-20--------------

20-25--------------25-30--------------30-35--------------35-40--------------40-45--------------

45-50--------------50-55--------------55-60--------------60-65--------------65-70--------------

70-75--------------75-80--------------80-85--------------85 and over~-------

Estimatedpopulation

July 1, 1959within ageinterval

Pnx

(2)

. . .16,000.18,i’0316,43512>850

10,86710,92211,92812,29911>382

10,9079,5758,2287,1335,752

4>2842,9711,520

860

Deathsin 1959withinage

interval

“D,

(3)

112,00817,116

9,0287>402

11,931

13,33714,08419,73428,47741,569

62,54487,521

114,895148,102191,536

214,256210,524177,601174,369

Deathrate un-adjusted

Col. 3

Col. 2

(4)

..*3.001069

.000483

.000450

.000928

.001227

.001290

.001654

.002315

.003652

.005734

.009141

.013964

.020763

.033299

.050013

.070860

.116843

.202753

Death rateadjustedfor age

not stated

Col. 4 x1.00054

. M,

(5)

. . .0.001070

.000483

.000450

.000929

.001228

.001290

.001655

.002316

.003654

.005737

.009145

.013970

.020773

.033315

.050037.070893.116898.202850

Conver-sion

factor

(Seetable 1)

““M,

(6)

. . .18.725317.1188

-27.768010.2732

7.53260.1806

-1.09100.95582.2822

1.66212.25072.25982.40412.2343

2.33992.43762.5307

. . .

Denomi-nator offormula(;:g;s;:

l+CO1.5 xco~. 6

(7)

. . .1.020041.008270.987491.00954

1.009251.000230.998191.002211.00834

1.009541.020581.031571.049941.07444

1.117081.172811.29583

. . .

lFor method of computing values at these ages, see text on Page LO

10

for the total population of the United States, 1959

Proportionof personsalive atbeginningof ageintervaldyingduringinterva1

n Colo 5

Col. 7

“qx

(8)

0.0264.0042.0024.0023.0046

.0061

.0064

.0083

.0116

.0181

.0284

.0448

.0677

.0989

.1550

.2240

.3022

.45111.0000

Number survivingto exact age xout of 100,000born ali~e

Col.Col.

9 (Line above)-10 (Line above)

Ix

(9)

100,00097,35796,94896,71696,495

96,05195,46794,85194,06592,978

91,29488,70084.72678;98971,175

60.14146;67232,56617,877

Numberdyingin ageinter-va1

:01. 8 X:01. 9

,dx

(10)

2,643409232221444

584616786

1,0871,684

2,5943,9745,7377,81411,034

13,46914,10614,68917,877

Conver-sionfactor

(Seetable 1

~m ~

(11)

. . .2.41522.68342.31742.3205

2.44232.44682.38392.34212.3293

2.34372.35362.37002.39802.4055

2.42802.48792.5747

...

STATIONARY POPULATION

In ageinterval

n Col. 9-(10)x (11)

,Lx

( 12)

97,681388,440484,117483,068481,445

478,829475,828472,381467,779460,967

450,390434,147410,034376,207329,333

268,002198,265125,01083,609

In this andall sub-sequentintervals

sumof Col.12 for thisline andall below

T,

(13)

6,965,5326,867,8516,479,4115,995,2945,512,226

5,030,7814,551,9524;076;1243,603,7433,135,964

2,674,9972,224,6071.790.4601;380;4261,004,219

674,886406,884208,61983,609

Averageyears of

liferemain-ing tosurvi-vorsat agex

Col. 13Col. 9

0ex

(14)

69.770.566.862.057.1

52.447.743.038.333.7

29.325.121.117.514.1

11.2

:::4.7

11

U. J. National center /or HeaIfbSlatislics,

Comparison of two methods of constructing abridged life tables by reference to a“standard” table; comparison of the revised and the prior method of constructing

the abridged Ii fe tables for the United States. \Vashington, U.S. Department ofHealth, Education, and We]fare. Public Health Ser~ ice, 1964.

I Ip. tables. 27cm. (1[s Viral and Health Statistics, Series 2, no. 4)U.S. Public Health Service. Publication no. 1000, Series 2, no. 4.

1. U.S. - Statistics, Vital - Methodology. I. Title.Cataloged by Department of Health, Education, and Welfare Libraq.

U.S. GOVERNMENT PRINTING OFFICE : 1966 0 -204-963

911.200

OUTLINE OF REPORT SERIES FOR VITAL AND HEALTH STATISTICS

P~bl;c Health Service Publication No. IO(N

Sen”es 1.

Sen”es 2.

Series 3.

Series 4.

series 10.

Series 11.

Sen-es 12.

Series 20.

Wies 21.

series 22.

Programs and aJJectiort @ocedwes. -Reports which describe the general programs of the NationalCenter for Health Statistics and its offices and divisions, data collection methods used, definitions, andother material necessary for understanding tbe data.

Reporta number 1-4

&to ewdktilm & methods research. -Studies of new statistical methodology including: experimentaltests of new survey methods, studies of vital statistics collection methods, new analytical techniques,objective evaluations of reliability of collected data, contributions to statistical theory.

Reports number 1-15

Analytical stwties .-Reports presenting analytical or interpretive studies based on vital and health sta-tistics, carrying the analysis further than the expository types of reports in tbe other series.

Reports number 1-4

Docwnertts and committee re@7rts.—. FinaI reports of major committees concerned with vital and healthstatistics, and documents such as recommended model vital registration laws and revised birth anddeath certificates.

Reports number 1 and 2

Data Fvom the Health Interview Survey .-Statistics on illness, accidental injuries, disability, use ofhos@&r, medical, dental, and other services, and other health-related topics, based on data collected ina cmtinuing national household interview survey.

Reports number 1-27

Data Prom the HeaLtk Examinationmeasurement of nacicnal samples ofcific diseases, and distributions ofmeasurements.

Reports number 1-12

&&vey.-Statistics based on the direct examination, testing, andthe population, including the medically defined prevalence of spe -the population with respect to various physical and physiological

Data From the Health Records Survey. -Statistics from records of hospital discharges and statisticsrelating to the health characteristics of persons in institutions, and on hoapitai, medical, nursing, andpersonal care received, based on national samples of establishments providing these services andsamples of the residents or patients.

Reports number 1-4

Data m mortality .-Various statistics on mortality other than as included in annual or monthly reporte--special analyses by cause of death, age, and other demographic variables, also geographic and timeseries analyses.

Reporta number 1

Data oa natality, marriage, and divwce. —Various switistics on natality, marriage, and divorce otherthan as included in annual or monthly reports-special analyses by demographic variables, also geo-graphic and tfme series analyses, studies of fertility,

Reports number 1-8

Data From the National I?ataUty U@ Mwtaiity Surveys. —Statistics on characteristics of births anddeaths not available from the vital records, based on sample surveys stemming from these records,including such top~s as mortality by socioeconomic class, medical experience in the last year of life,characteristics of pregnancy, etc.

Reports number 1

For a list of titles of reports published in these series, write to Natfonal Center for Health StatisticsU.S. Public Health ServiceWashington, D.C. 20201