Two New Mathematical Equalities in the Life Table

There is a rich body of literature on equalities in the period life table, which also can be interpreted as a stationary population, and a smaller, but no less rich, body on inequalities. The latter is important because it provides information on health disparities and, like the equality literature, serves as a foundation for formal mortality analysis. We straddle both of these bodies by reconciling a known inequality such that a new mathematical equality emerges in the period life table. We then show that this new equality links life expectancy at birth (mean age at death) directly to the average lifespan of the living members of a stationary population. This linkage represents a second newly identified equality in this paper that has the potential to yield useful insights because it links the average lifespan of the living members of a stationary population directly to elements that are central to the dynamics and structure of a stationary population, life expectancy, and variance in age at death.

Following brief background material, we first present a new equality hitherto not identified in the period life table by showing the factor that reconciles the inequality just described is the product of the life table crude death rate (d) and variance in age at death (σ 2 ). In the discussion that follows this reconciliation, we present a second new identity: when life expectancy at birth (e 0 ) is added to the product of the life table crude death rate (d) and variance in age at death (σ 2 ), the result is average lifespan (A) of the living members of a stationary population: A = e 0 + dσ 2 . Kim and Aron (1981) provide a proof that the mean age in a stationary population, μ a , is equal to the mean expected years remaining, μ r . Vaupel (2009) demonstrated that the mean number of years lived in a stationary population, μ l , is equal to the mean expected years remaining, μ r . Thus, as shown by Swanson and Tedrow (2021: 226-227), we see that the mean age of a stationary population is equal to its mean years remaining and, therefore, equal to its mean years lived:

Background
where: μ a = mean age in the stationary population μ r = mean years remaining in the stationary population μ l = mean years lived in the stationary population

A New Mathematical Equality
Following Swanson and Tedrow (2021: 227), we begin with work by Pressat (1972: 479-480), who found: where: μ a = mean age of the stationary population e 0 = life expectancy at birth in a stationary population σ 2 = variance in age at death in a stationary population As shown in Swanson and Tedrow (2021: 233), we can re-arrange the terms in Eq.
(2) to solve for σ 2 : In verbal terms, Eq. (3) states that variance in age at death is found by subtracting e 0 2 from the product, [e 0 (2μ a )]. Using the fact that 2μ a = μ r + μ l , we re-arrange the terms in Eq. (3): And noting that 1/e 0 is equal to the crude death rate in a stationary population, which we display as d, we can see that Thus, Eq. (4) reconciles the inequality identified by Swanson and Tedrow (2021), (μ r + μ l ) > e 0 , by showing that the difference between (μ r + μ l ) and e 0 is dσ 2 . Of course, d is equal to b, the crude birth rate in the period life table (stationary population) of interest, but as a concept, d is better suited as the scalar applied to variance in age at death, σ 2 . Equation (4) represents a hitherto un-identified mathematical equality in the period life table.

Discussion
Wrigley-Field and Feehan (2021: 5-6) note that the average lifespan (A) of the living members of a stationary population is twice the average age of its members. Using the notation employed here, along with the steps in going from Eq. (3) to (4), we can see that A = e 0 + dσ 2 = (μ r + μ l ) = 2μ a . Thus, A = e 0 + dσ 2 is a second new equality we have identified in this paper, one that has the potential to yield useful insights because it links A directly to e 0 and σ 2 , elements that are central to the dynamics and structure of a stationary population.
Wrigley-Field and Feehan (2021:216) demonstrate that the average age of the population, μ a , and the average remaining lifespan, A, are each half of ALL (average lifespan of the living members of the stationary population). Thus, ALL = A + μ r . Because we know from Eq. (1) that μ r = μ a , it is easy to show that Thus, as shown in Eq. (5), ALL can be expressed in five different ways, each of which provides a different path to exploring the finding by Wrigley-Field and Feehan (2021: 2018) that the lifespans of the living members of a stationary population form a length-biased sample of the cohort (and period) lifespan distribution. In turn, this connects demographic lifespan measures with a well-developed body of statistics and epidemiological applications, which offers the potential for new insights.
Applying the calculations needed to implement Eq. (4) using data extracted from Swanson and Tedrow (2021: 230) illustrates this finding in the form of Table 1. It shows the relationships among e 0 , σ 2 , d, dσ 2 , and (μ r + μ l ) as found in US period life tables for all sexes every 10 years from 1935 to 2005 (n = 8). Figure 1 graphically displays the relationships among e 0 , (μ r + μ l ), and σ 2 found in Table 1. As can be seen in Table 1, life expectancy at birth in the USA for all sexes in 1935 is 60.89 and the mean years remaining plus mean years lived is 70.94. Again as shown in Table 1, adding In looking at Fig. 1, we can see, as expected, that as e 0 increases (along the x axis) from 60.89 to 77.63, σ 2 declines at a relatively faster pace, from 611.94 to 277.1 (along the y axis), while (μ r + μ l ) tracks slightly above and closely to e 0 .
As shown in Table 1, it is clear that as e 0 increases, σ 2 declines, as does d, which, in turn, leads to a decline in the difference between e 0 and (μ r + μ l ). The inverse relationship between e 0 and σ 2 was not unexpected (Aburto et al., 2020;Swanson & Tedrow, 2021;Tuljapurkar, 2010;Tuljapurkar & Edwards, 2011;Wrigley-Field & Feehan, 2021); neither was the decline in the difference between e 0 and (μ r + μ l ) (Swanson & Tedrow, 2021). Keep in mind, however, that the difference between e 0 and (μ r + μ l ) has an empirical lower limit greater than zero because increases in e 0 will never lead to e 0 = (μ r + μ l ). This would only occur if σ 2 were zero, which is so unlikely as to be impossible because it would mean that all members of a species die at exactly the same age, and as observed by Swanson and Tedrow (2021: 232), there are no known species for which this occurs. Thus, 0 < σ 2 is an empirical inequality found in all period life tables for humans and other known species. Because variance in age at death can be defined as σ 2 = [(μ r + μ l )/d] − 1, we can also state that 0 < [(μ r + μ l )/d] − 1, which we hypothesize is a hitherto unknown empirical inequality found in all period life tables for humans and other known species.
While σ 2 will never equal zero, it is clear that it could equal and even be less than e 0 . It is also clear that e 0 would need to be much higher than the 2005 level of 77.63 shown for the USA in Table 1 for this to occur. We can estimate how high it would need to be using the data in Table 1. From the increase in e 0 between 1935 and 2005 shown in Table 1,  The relationships among e 0 , μ r + μ l , and σ 2 (Table 1) occur when e 0 is approximately 104 years, after which increases in e 0 would lead to σ 2 values becoming steadily less than e 0 , keeping in mind the empirical constraint, 0 < σ 2 . While this equality and subsequent crossover between e 0 and σ 2 may occur in a few very high e 0 countries that also have a high pace of e 0 improvement, it does not appear to in the cards for very many countries, including the USA. WHO (2020) estimates 78.5 years as the 2019 e 0 for all sexes combined in the USA. Given that less than a year of additional life expectancy at birth was gained over a 14-year period, going from 77.63 in 2005 to 78.5 in 2019, it appears that around 364 more years would be required by the USA to achieve an increase of the nearly 26 years in life expectancy at birth for all sexes combined to get to e 0 = 104. One of the reviewers of this paper pointed out that the term dσ 2 (which, recall is equivalent to σ 2 /e 0 , where σ 2 is the variance in age at death and e 0 is equivalent to the mean age at death) is very close to the definition of the coefficient of variation (CV), which is the ratio of the standard deviation in a set of data to the mean of the same set. It differs only in that the numerator is variance, the square of which serves as the CV's numerator, the standard deviation. This is of interest because the CV is one of several measures used to explore the relationship between life expectancy and lifespan inequality (Németh, 2017;Vaupel, 2010). Given the role of dσ 2 in reconciling the inequality, (μ r + μ l ) > e 0 , and its link to A and ALL, it may represent a path to a better understanding of why lifespan inequality tends to decrease as life expectancy increases.

Conclusion
We have identified two new equalities in this paper. The first reconciles the inequality (μ r + μ l > e 0 ) found in all period life tables for humans and other known species, namely, that when the product of the life table crude death rate (d) and variance in age at death (σ 2 ) is added to life expectancy at birth, the result is equal to the sum of mean years remaining and mean years lived: e 0 + dσ 2 = (μ r + μ l ). The second is that when life expectancy at birth (e 0 ) is added to the product of the life table crude death rate (d) and variance in age at death (σ 2 ), the result is the average lifespan (A) of the living members of a stationary population: A = e 0 + dσ 2 .