On the emergence of the correlation between life expectancy and the variance in the age at death

Recent empirical studies have found various patterns in the correlations between lifespan inequality and life expectancy in modern human populations. However, it is unclear how general these regularities are. Here we establish three theorems that provide theoretical foundations for such regularities. We show that for populations with a finite maximum lifespan ω, and under certain continuity assumptions, the variance in the age at death is bounded by a function of lifespan that has a maximum and tends to zero as life expectancy tends to zero and ω. We show how the change in said variance is determined by a particular interplay between the coefficient of variation and the mean age in the population. These results lead to three hypotheses—a three-phased pattern of change for the correlation between the variance and life expectancy, a particular shape of the associated variance function, and that survival curve Type is one driver of the pattern. We illustrate those hypotheses empirically via a study of the 10 countries in the Human Mortality Database with the oldest available data. Our results elucidate the emergence of the aforementioned correlation patterns and provide demographically meaningful conditions under which those correlations reverse.