Lesson 3: Life Tables
3.0 Overview
The goal of this lesson is to review elements of ordinary life tables that are essential to understanding multipledecrement life tables.
The focus of the first section (3.1) is on understanding what the columns of an ordinary life table reveal. The second section (3.2) shows how to construct a life table.
For more extensive coverage of ordinary life tables, see Population Analysis for Planners, another free, online course developed as part of the DAPR project.
Download a zip file containing data for Lesson 3 in Excel and CSV formats.
3.1 An Ordinary Life Table
An ordinary life table is a statistical tool that summarizes the mortality experience of a population and yields information about longevity and life expectation. Although it is generally used for studying mortality, the life table format can be used to summarize any duration variable, such as duration of marriage, duration of contraceptive use, etc.
An Example of a Life Table
A typical life table contains several columns, each with a unique interpretation. We will learn about these columns and their interpretations by examining an illustrative life table. First, an introduction to the notation:
Table 3.1.1: Life Table Column Notation


Column  Notation  Definition 

1 
(x, x+ n)

Age interval or period of life between two exact ages stated in years 
2 
_{n}q_{x}

Proportion of persons alive at the beginning of the age interval who die during the age interval 
3 
l_{x}

Of the starting number of newborns in the life table (called the radix of the life table, usually set at 100,000) the number living at the beginning of the age interval (or the number surviving to the beginning of the age interval) 
4 
_{n}d_{x}

The number of persons in the cohort who die in the age interval (x, x+ n) 
5 
_{n}L_{x}

Number of years of life lived by the cohort within the indicated age interval (x, x+ n) (or personyears of life in the age interval) 
6 
T_{x}

Total personyears of life contributed by the cohort after attaining age x 
7 
e_{x}^{0}

Average number of years of life remaining for a person alive at the beginning of age interval x 
The table below (Table 3.1.2) gives an ordinary life table of the 1997 United States population (adapted from NCHS; National Vital Statistics Reports Vol. 47, No. 19, June 30, 1999).
Table 3.1.2: Abridged Life Table for the Total United States Population, 1997


(1) Age Interval (x, x + n)  (2) _{n}q_{x}  (3) l_{x}  (4) _{n}d_{x}  (5) _{n}L_{x}  (6) T_{x}  (7) e_{x}^{0} 

< 1  0.00723  100000  723  99371  7650789  76.5 
14  .00144  99277  143  396774  7551418  76.1 
59  .00092  99135  91  495432  7154644  72.2 
1014  .00116  99043  115  494997  6659212  67.2 
1519  .00374  98929  370  493801  6164215  62.3 
2024  .00492  98558  485  491596  5670414  57.5 
2529  .00509  98073  499  489137  5178818  52.8 
3034  .00630  97574  615  486397  4689680  48.1 
3539  .00840  96959  814  482862  4203284  43.4 
4044  .01196  96145  1149  478017  3720422  38.7 
4549  .01757  94996  1669  471055  3242404  34.1 
5054  .02618  93327  2443  460915  2771349  29.7 
5559  .04123  90884  3747  445708  2310434  25.4 
6064  .06457  87136  5627  422450  1864727  21.4 
6569  .09512  81510  7753  389159  1442277  17.7 
7074  .14365  73757  10595  343402  1053118  14.3 
7579  .20797  63162  13135  284018  709716  11.2 
8084  .31593  50026  15805  211466  425698  8.5 
8589  .46155  34221  15795  130736  214232  6.3 
9094  .62682  18427  11550  60800  83496  4.5 
9599  .77325  6876  5317  18825  22696  3.3 
100+  1.0000  1559  1559  3871  3871  2.5 
Interpretation of and Relationships among the Columns
The _{n}q_{x} column has a probabilistic interpretation:
_{n}q_{x} ~ Probability that a person of age x will die in the
age interval (x, x + n)
Also note that
Or with reference to the columns,  Column 2 = 
Example in Table 3.1.2: 
A graph of _{n}q_{x} across the life span is given in Figure 3.1.1. A graph of agespecific death rates would have a similar shape.
Figure 3.1.1: _{n}q_{x} across the Life Span for US 1997
The l_{x} column also has a probabilistic interpretation:
~  Proportion of the newborns surviving to that age (In Table 3.1.2, divide Column 3 by 100,000.) 
These proportions are called survival probabilities. A plot of the survival probabilities across the life span is given in Figure 3.1.2.
Figure 3.1.2: l_{x} across the Life Span for US 1997
There are simple relationships between the _{n}d_{x} column and the l_{x} and the _{n}q_{x} columns in the life table:
(a) (In Table 3.1.2 multiply Column 3 by Column 2.)
Example From Table 3.1.2: 
(b) Because everybody eventually dies, the sum of the number of deaths in all the age intervals will be equal to the radix of the life table, i.e.:
Example In Table 3.1.2: The sum of the _{n}d_{x} column is equal to 100,000 = l_{0} 
(c) The relationships in (b) can be extended as follows:
Because everybody who survives to age x will eventually die, the sum of deaths from that age to the end of the table will be equal to the number surviving to that age, i.e.:
Example In Table 3.1.2: 
(d) Number of persons dying before a specified age x is the sum of deaths from the beginning of the table to that specified age:
Number dying before age x =
Example In Table 3.1.2, the number of 
(e) From (d) above, the proportion (probability) that a newborn will die before reaching age x is calculated as:
Example In Table 3.1.2, the probability 
(f) Although not shown in the life table, one useful quantity to calculate from the table is the proportion surviving each age interval. This proportion is denoted as _{n}p_{x.} Note that:
Therefore,
Also note that one can write:
Or the l_{x} column is related to the _{n}p_{x} column by the relation:
Thus, one can compute the cumulative survival function as the product of survival probabilities of each interval:
Example In Table 3.1.2: 
Exercise 6
Question 1
The radix of the life table is usually 100,000 but may be a different number. Where in an ordinary life table can you always look to find out what the radix is?
 In the first row of Column 7
 In the first row of Column 3
 In the last row of Column 7
 In the last row of Column 1
Question 2
According to Column 7 of Table 3.1.2, a newborn in the US in 1997 may expect to reach age 76.5. Once that child gets to age 50, what age would he/she expect to reach?
 No change  76.5
 29.7
 79.7
 63.8
Question 3
According to Table 3.1.2, of those born in the US in 1997 who make it to age 70, what percentage are expected to die before they reach age 75?
 14%
 20%
 6%
 9%
Question 4
According to Table 3.1.2, what is the probability of a newborn in the US in 1997 surviving to age 20?
 .992
 .950
 .986
 .917
3.2 Construction of an Ordinary Life Table
Knowledge of ordinary life table construction is essential in the construction of a multipledecrement life table. There are a number of methods available to construct an ordinary life table using data on agespecific death rates. The most common methods are those of Reed Merrell, Greville, Keyfitz, Frauenthal, and Chiang (for a discussion of these methods see Namboodiri and Suchindran, 1987).
In this section we construct an ordinary life table with data on agespecific death rates based on a simple method suggested by Fergany (1971. "On the Human Survivorship Function and Life Table Construction," Demography8(3):331334). In this method the agespecific death rate (_{n}m_{x}) will be converted into the proportion dying in the age interval (_{n}q_{x} ) using a simple formula:
Formula (1)
where e is the symbol for the base number of a natural log (a constant equal to 2.71828182...) and n is the length of the age interval. (Note: do not confuse the symbol e here with the e_{x}^{0} used in "expected life" notation.)
Once _{n}q_{x} is calculated with agespecific death rates, the remaining columns of the life table are easily calculated using the following relationships:
(As in Table 3.1.2, multiply Column 3 by Column 2.)  
(As in Table 3.1.2, subtract Column 4 from Column 3.)  
(Divide Column 4 in Table 3.1.2 by the corresponding agespecific death rate. Note: Table 3.1.2 did not use the Fergany method.) 

(Obtain cumulative sums of Column 5 in Table 3.1.2.)  
(In Table 3.1.2, divide Column 6 by Column 3.) 
Example Converting the AgeSpecific Death Rate into the Proportion Dying in the Age Interval Table 2.5.2 of Lesson 2.5 shows that the agespecific death rate for age group 14 (_{4}m_{1}) for Costa Rican males in 1960 is .00701 per person. (Keep in mind that tables presenting agespecific death rates will usually present the rate as "number of deaths per 1000 people," but in the calculations used in constructing an ordinary life table, the agespecific death rate is "number of deaths per person.") Using formula (1) from above,

Fergany Method, Step by Step
In this example we use the agespecific death rates from Table 2.5.2 of Lesson 2.5 to complete the construction of a life table for 1960 Costa Rican males. We will follow the Fergany Method.
Step 1
Obtain agespecific death rates. Note that agespecific death rates are per person (Column 2 of Table 2.5.2).
Step 2
Convert agespecific death rates (nMx) to the proportion dying in the age interval (_{n}q_{x}) values using the following formula (formula (1) from above):
, where n is the length of the age interval
Table 3.2.1: Life Table Construction: 1960 Costa Rican Males


(1)  (2)  (3)  (4)  (5)  (6)  (7)  

Age Interval  _{n}m_{x}  _{n}q_{x} 
l_{x}
 _{n}d_{x}  _{n}L_{x} 
T_{x}

e_{x}^{0}

<1 year  0.07505  0.07230  100,000  7,230  96,340  6,297,331  62.97331 
14  0.00701  0.02765  92,770  2,566  365,924  6,200,991  66.84287 
59  0.00171  0.00851  90,204  768  449,098  5,835,067  64.68736 
1014  0.00128  0.00636  89,436  569  445,757  5,385,970  60.22141 
1519  0.00129  0.00641  88,867  570  442,912  4,940,212  55.59081 
2024  0.00181  0.00899  88,298  793  439,502  4,497,301  50.93332 
2529  0.00163  0.00814  87,504  712  435,739  4,057,798  46.37254 
3034  0.00198  0.00984  86,792  854  431,822  3,622,059  41.73255 
3539  0.00302  0.01497  85,938  1,287  426,465  3,190,237  37.12251 
4044  0.00442  0.02184  84,651  1,849  418,616  2,763,772  32.64896 
4549  0.00645  0.03175  82,802  2,629  407,402  2,345,156  28.32242 
5054  0.00923  0.04509  80,173  3,615  391,758  1,937,754  24.16966 
5559  0.01344  0.06498  76,558  4,975  370,214  1,545,996  20.19379 
6064  0.02364  0.11146  71,583  7,979  337,577  1,175,782  16.42535 
6569  0.03633  0.16609  63,605  10,564  290,813  838,205  13.17838 
7074  0.05182  0.22826  53,040  12,107  233,629  547,391  10.32027 
7579  0.07644  0.31765  40,933  13,002  170,095  313,763  7.665223 
8084  0.13520  0.49135  27,931  13,724  101,508  143,668  5.143692 
85+  0.33698  1.00000  14,207  14,207  42,160  42,160  2.967532 
Step 2 Examples For age interval 01:
For age interval 14:
For age interval 59: For age interval 85+: 
Step 3
Use _{n}q_{x} to compute l_{x} values in Column 3.
First set = 100,000
Then =
Or in general...
(Note: This computational formula is easy to implement on the spreadsheet. First compute l_{1} and copy to the remaining cells of the column.)
Step 3 Examples (Round to integer) 
Step 4
Calculate the number of deaths in age intervals () in Column 4 as:
(Column 3 * Column 2)
Note: Sometimes it is easy to implement Steps 3 and 4 simultaneously:
First write = 100,000
Then calculate:
(Round to integer)
Then:
Step 5
In Column 5, compute the personyears of life in the indicated age interval () as:
(Column 4 / agespecific death rate)
Step 5 Example (Round to integer) 
Step 6
In Column 6, compute the cumulative personyears of life after a specified age (T_{x}):
(Sum values in Column 6 from a specified age to the end of the table.) 
Step 6 Examples 
Step 7
The final column of the life table (Column 7) is the expectation of life at specified ages. This column is computed as:
(Column 6 / Column 3)

The life table construction is complete with the implementation of Step 7.
Selected Features of the Life Table
We will examine some features of the constructed life table that are relevant to the construction and interpretation of a multipledecrement life table:
1. Sum of the values in Column 4 will be equal to 100,000 (= ).
2. Sum of the values in Column 4 from a specified age will be equal to the value at this age as shown in Column 3.
For example:
Thus, one can interpret as the cumulative number of deaths after a specified age.
3. Age at which people in the life table cohort die is also important in our understanding of the age pattern of death. The column (Column 4 of the life table) gives the frequency distribution of the age at death in the population.
A graph of this frequency distribution will show the age pattern of death in the population. Unfortunately, this frequency distribution is given in age intervals of unequal length (and an openended interval at the end). Therefore a graph adjusting for the unequal age intervals is more appropriate for this life table.
Figure 3.2.1 shows the pattern of the age distribution of deaths from the life table above (Table 3.2.1). Note that in this example the openended age interval 85+ is closed at 85100. The proportion of deaths in each age group is divided by the length of the age interval. The graph is drawn by connecting the values at the midpoint of each interval.
Figure 3.2.1: Age Distribution of Deaths for 1960 Costa Rican Males
The graph shows that a high proportion of the cohort dies in infancy. The deaths decrease until early adulthood, rise until age 80, and then begin to decrease again at the extreme ages. Note that the sharp decrease at the far right is due to the small number of extremely old survivors in this population.
4. The cumulative number of deaths from the beginning of life also can be calculated by summing the appropriate numbers in Column 4. For example, the number of persons in the cohort dying before reaching age 15 is:
Note that this number also can be calculated as:
Thus, proportion dying before reaching age 15 is:
Exercise 7
Note to students: This longer exercise will require the use of spreadsheet software. Good luck!
Use the data on agespecific deaths of the 1960 Costa Rican females from Exercise 5 to construct a life table using Fergany's Method as described above. (You downloaded the data file you need here as part of Exercise 5.)
Then use your constructed life table to do the following:
 Draw graphs of the and columns. Briefly describe these graphs.
 Draw a graph of the age distribution of deaths (adjusting for the unequal age intervals) using the column in the life table. Comment on the age pattern of mortality depicted in this graph.
 Verify that is the sum of the _{n}d_{x}_{}column from age 65 to the end of the table.
Once you have finished your work, compare your results to the answer key below.
Answers To Exercises
Exercise 6
Question 1
The radix of the life table is usually 100,000 but may be a different number. Where in an ordinary life table can you always look to find out what the radix is?
B. In the first row of Column 3. The radix is simply the starting number of newborns for the life table. Since Column 3 gives the starting number of people at each age interval, the first row gives the number of people starting at age 0. In this case, it is 100,000, as usual.
Question 2
According to Column 7 of Table 3.1.2, a newborn in the US in 1997 may expect to reach age 76.5. Once that child gets to age 50, what age would he/she expect to reach?
C. 79.7. Column 7 tells, on average, how many more years of life are expected for people who made it to the start of the age interval. So a 50yearold would expect another 29.7 years to live on average (50 + 29.7 = 79.7).
Question 3
According to Table 3.1.2, of those born in the US in 1997 who make it to age 70, what percentage are expected to die before they reach age 75?
A. 14%. Column 2 gives the proportion of persons alive at the beginning of the age interval who die during the age interval. So a 70yearold has a .14365 (rounded to 14%) chance of dying during the 7075 age interval.
Question 4
According to Table 3.1.2, what is the probability of a newborn in the US in 1997 surviving to age 20?
C. .986. Since Column 3 gives the number of people surviving to the beginning of the age interval (98,558 made it to age 20) and you know the number of people that started (100,000), the probability of making it to age 20 is 98,558/100,000 = .98558.
Exercise 7
Use the data on agespecific deaths of the 1960 Costa Rican females from Exercise 5 to construct a life table using Fergany's Method as described above. (You downloaded the data file you need here as part of Exercise 5.)
Then use your constructed life table to do the following:
 Draw graphs of the and columns. Briefly describe these graphs.
 Draw a graph of the age distribution of deaths (adjusting for the unequal age intervals) using the _{n}d_{x} column in the life table. Comment on the age pattern of mortality depicted in this graph.
 Verify that is the sum of the _{n}d_{x}_{}column from age 65 to the end of the table.
Exercise 7, Answer: Life Table: 1960 Costa Rican Females


(1)  (2)  (3)  (4)  (5)  (6)  (7)  

Age Interval  _{n}m_{x}  _{n}q_{x}  l_{x}  _{n}d_{x}  _{nL}_{x}  T_{x}  e_{x}^{0} 
<1 year  0.0640  0.0620  100000  6199  96,868  6,544,328  65.44 
14  0.0077  0.0303  93801  2840  369,496  6,447,460  68.74 
59  0.0017  0.0084  90961  761  452,900  6,077,964  66.82 
1014  0.0010  0.0048  90200  429  449,925  5,625,064  62.36 
1519  0.0008  0.0042  89771  377  447,912  5,175,139  57.65 
2024  0.0013  0.0063  89394  565  445,556  4,727,227  52.88 
2529  0.0018  0.0087  88829  776  442,202  4,281,672  48.20 
3034  0.0023  0.0113  88053  993  437,779  3,839,470  43.60 
3539  0.0028  0.0137  87060  1191  432,318  3,401,690  39.07 
4044  0.0029  0.0146  85870  1251  426,212  2,969,373  34.58 
4549  0.0046  0.0229  84618  1940  418,222  2,543,161  30.05 
5054  0.0070  0.0343  82678  2833  406,265  2,124,939  25.70 
5559  0.0105  0.0511  79845  4078  388,937  1,718,675  21.53 
6064  0.0188  0.0896  75766  6789  361,591  1,329,737  17.55 
6569  0.0288  0.1340  68977  9241  321,227  968,146  14.04 
7074  0.0462  0.2062  59736  12317  266,702  646,918  10.83 
7579  0.0678  0.2874  47419  13626  201,109  380,216  8.02 
8084  0.1329  0.4853  33793  16401  123,455  179,108  5.30 
85+  0.3125  1.0000  17392  17392  55,653  55,653  3.20 
Then use your constructed life table to do the following:
1. Draw graphs of the and columns. Briefly describe these graphs.
The proportion of people who die during the age interval is a little higher in the first two age intervals, low and flat until about age 45, and rises fairly steeply after that until it is 1.0 for the 85+ age group.
Naturally, the number of people alive at the start of each interval starts dropping more rapidly around age 45.
2. Draw a graph of the age distribution of deaths (adjusting for the unequal age intervals) using the _{n}d_{x} column in the life table. Comment on the age pattern of mortality depicted in this graph.
The greatest mortality rate is in the very first age interval. After the second age interval, mortality rates are low and flat before they start rising at around 47.5 (age interval midpoint), peaking at 82.5. The steep drop in the last age group is partly because of the small number of survivors and partly because it is an openended interval. If the table continued with fiveyear intervals, the drop would appear to be more gradual.
3. Verify that is the sum of the _{n}d_{x}_{}column from age 65 to the end of the table.
= 68977 = 9241 + 12317 + 13626 + 16401 + 17392