DEMOGRAPHIC METHODS

and those contributed by individuals who die with-
in that interval:

nLx = [n.^ x+n] + [nax. ndx] (^7 )

Tx equals the number of person-years lived
beyond exact age x:

Tx = ∑ (^) nLa = Tx+n + nLx
∞
a=x,n
( 8 )
ex equals the expected number of years of life
remaining for an individual who has already sur-
vived to exact age x. It is the total number of
person-years experienced by the cohort above that
age divided by the number of individuals starting
out at that age:
ex = Tx
x
( 9 )
The nLx and Tx columns are generated from
the oldest age to the youngest. If the last age
category is, for example, eighty-five and above (it is
typically ‘‘open-ended’’ in this way), we must have
an initial value for T 85 in order to begin the proc-
ess. This value is derived in the following fashion:
Since for this oldest age group, l 85 =∞d 85 (due to the
fact that the number of individuals in a cohort who
will die at age eighty-five or beyond is simply
the number surviving to age eighty-five) and
T 85 =∞L 85 , we have:
( 10 )
e 85 = =^1
85 / T 85

11
∞d 85 / ∞L 85
T 85
85
≈
∞M 85
From the life table, we can obtain mortality
information in a variety of ways. In table 2, we see,
for example, that the expectation of life at birth, e 0 ,
is 76.1 years. If an individual in this population
survives to age eighty, then he or she might expect
to live 8.4 years longer. We might also note that the
probability of surviving from birth to one’s tenth
birthday is l 10 /l 0 , or 0.99022. Given that one has
already lived eighty years, the probability that one
survives five additional years is l 85 /l 80 , or 33,629/
49,276=0.68246.
POPULATION PROJECTION
The life table, in addition, is often used to project
either total population size or the size of specific
age groups. In so doing, we must invoke a different
interpretation of the nLx’s and the Tx’s in the life
table. We treat them as representing the age distri-
bution of a stationary population—that is, a popula-
tion having long been subject to zero growth.
Thus, 5 L 20 , for example, represents the number of
twenty- to twenty-four-year-olds in the life table
‘‘population,’’ into which l 0 , or 100,000, individu-
als are born each year. (One will note by summing
the ndx column that 100,000 die every year, thus
giving rise to stationarity of the life table population.)
If we were to assume that the United States is a
closed population—that is, a population whose net
migration is zero—and, furthermore, that the mor-
tality levels obtaining in 1996 were to remain
constant for the following ten years, then we would
be able to project the size of any U.S. cohort up to
ten years into the future. Thus, if we wished to
know the number of fifty- to fifty-four-year-olds in
2006, we would take advantage of the following
relation that is assumed to hold approximately:
nPx+t ≈
τ+t
nPx
τ
nLx+t
τ+t
nLx
τ (^11 )
where
is the base year of the projection (e.g.,
1996) and t is the number of years one is projecting
the population forward. This equation implies
that the fifty- to fifty-four-year-olds in 2006, 5 P 502006 ,
is simply the number of forty-to forty-four-year-
olds ten years earlier, 5 P 401996 , multiplied by the
proportion of forty- to forty-four-year-olds in the
life table surviving ten years, 5 L 50 / 5 L 40.
In practice, it is appropriate to use the above
relation in population projection only if the width
of the age interval under consideration, n, is suffi-
ciently narrow. If the age interval is very broad—
for example, in the extreme case in which we are
attempting to project the number of people aged
ten and above in 2006 from the number zero and
above (i.e, the entire population) in 1996—we
cannot be assured that the life table age distribu-
tion within that interval resembles closely enough
the age distribution of the actual population. In
other words, if the actual population’s age distri-
bution within a broad age interval is significantly