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Life Expectancy


An l_x table is a tabulation of numbers which is used to calculate life expectancies.

xn_xd_xl_xq_xL_xT_xe_x
010002001.000.200.902.702.70
18001000.800.120.751.802.25
27002000.700.290.601.051.50
35003000.500.600.350.450.90
42002000.201.000.100.100.50
5000.00--0.000.00--
sum10002.70

x: Age category (x=0, 1, ..., k). These values can be in any convenient units, but must be chosen so that no observed lifespan extends past category k-1.

n_x: Census size, defined as the number of individuals in the study population who survive to the beginning of age category x. Therefore, n_0=N (the total population size) and n_k=0.

d_x: =n_x-n_(x+1); sum_(i=0)^(k)d_i=n_0. Crude death rate, which measures the number of individuals who die within age category x.

l_x: =n_x/n_0. Survivorship, which measures the proportion of individuals who survive to the beginning of age category x.

q_x: =d_x/n_x; q_(k-1)=1. Proportional death rate, or "risk," which measures the proportion of individuals surviving to the beginning of age category x who die within that category.

L_x: =(l_x+l_(x+1))/2. Midpoint survivorship, which measures the proportion of individuals surviving to the midpoint of age category x. Note that the simple averaging formula must be replaced by a more complicated expression if survivorship is nonlinear within age categories. The sum sum_(i=0)^(k)L_x gives the total number of age categories lived by the entire study population.

T_x: =T_(x-1)-L_(x-1); T_0=sum_(i=0)^(k)L_x. Measures the total number of age categories left to be lived by all individuals who survive to the beginning of age category x.

e_x: =T_x/l_x; e_(k-1)=1/2. Life expectancy, which is the mean number of age categories remaining until death for individuals surviving to the beginning of age category x.

For all x, e_(x+1)+1>e_x. This means that the total expected lifespan increases monotonically. For instance, in the table above, the one-year-olds have an average age at death of 2.25+1=3.25, compared to 2.70 for newborns. In effect, the age of death of older individuals is a distribution conditioned on the fact that they have survived to their present age.

It is common to study survivorship as a semilog plot of l_x vs. x, known as a survivorship curve. A so-called l_xm_x table can be used to calculate the mean generation time of a population. Two l_xm_x tables are illustrated below.

Population 1

xl_xm_xl_xm_xxl_xm_x
01.000.000.000.00
10.700.500.350.35
20.501.500.751.50
30.200.000.000.00
40.000.000.000.00
R_0=1.10sum=1.85
T=(sumxl_xm_x)/(suml_xm_x)=(1.85)/(1.10)=1.68
(1)
r=(lnR_0)/T=(ln1.10)/(1.68)=0.057.
(2)

Population 2

xl_xm_xl_xm_xxl_xm_x
01.000.000.000.00
10.700.000.000.00
20.502.001.002.00
30.200.500.100.30
40.000.000.000.00
R_0=1.10sum=2.30
T=(sumxl_xm_x)/(suml_xm_x)=(2.30)/(1.10)=2.09
(3)
r=(lnR_0)/T=(ln1.10)/(2.09)=0.046.
(4)

x: Age category (x=0, 1, ..., k). These values can be in any convenient units, but must be chosen so that no observed lifespan extends past category k-1 (as in an l_x table).

l_x: =n_x/n_0. Survivorship, which measures the proportion of individuals who survive to the beginning of age category x (as in an l_x table).

m_x: The average number of offspring produced by an individual in age category x while in that age category. sum_(i=0)^(k)m_x therefore represents the average lifetime number of offspring produced by an individual of maximum lifespan.

l_xm_x: The average number of offspring produced by an individual within age category x weighted by the probability of surviving to the beginning of that age category. sum_(i=0)^(k)l_xm_x therefore represents the average lifetime number of offspring produced by a member of the study population. It is called the net reproductive rate per generation and is often denoted R_0.

xl_xm_x: A column weighting the offspring counted in the previous column by their parents' age when they were born. Therefore, the ratio T=sum(xl_xm_x)/sum(l_xm_x) is the mean generation time of the population.

The Malthusian parameter r measures the reproductive rate per unit time and can be calculated as r=(lnR_0)/T. For an exponentially increasing population, the population size N(t) at time t is then given by

 N(t)=N_0e^(rt).
(5)

In the above two tables, the populations have identical reproductive rates of R_0=1.10. However, the shift toward later reproduction in population 2 increases the generation time, thus slowing the rate of population growth. Often, a slight delay of reproduction decreases population growth more strongly than does even a fairly large reduction in reproductive rate.


See also

Gompertz Curve, Logistic Map, Makeham Curve, Malthusian Parameter, Population Growth, Survivorship Curve

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References

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 294-295, 1999.

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Life Expectancy

Cite this as:

Weisstein, Eric W. "Life Expectancy." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LifeExpectancy.html

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