Life expectancy

In actuarial notation, the probability of surviving from age ${\displaystyle x}$ to age ${\displaystyle x+n}$ is denoted ${\displaystyle \,_{n}p_{x}\!}$ and the probability of dying during age ${\displaystyle x}$ (i.e. between ages ${\displaystyle x}$ and ${\displaystyle x+1}$) is denoted ${\displaystyle q_{x}\!}$ . For example, if 10% of a group of people alive at their 90th birthday die before their 91st birthday, the age-specific death probability at 90 would be 10%. This probability describes the likelihood of dying at that age, and is not the rate at which people of that age die.^[c] It can be shown that


The curtate future lifetime, denoted ${\displaystyle K(x)}$, is a discrete random variable representing the remaining lifetime at age ${\displaystyle x}$, rounded down to whole years. Life expectancy, more technically called the curtate expected lifetime and denoted ${\displaystyle \,e_{x}\!}$ ,^[a] is the mean of ${\displaystyle K(x)}$—that is to say, the expected number of whole years of life remaining, assuming survival to age ${\displaystyle x}$.^[146] So,


Substituting (1) into the sum and simplifying gives the final result ^[147]


If the assumption is made that, on average, people live a half year on the year of their death, the complete life expectancy at age ${\displaystyle x}$ would be ${\displaystyle e_{x}+1/2}$, which is denoted by e̊_x, and is the intuitive definition of life expectancy.


By definition, life expectancy is an arithmetic mean. It can also be calculated by integrating the survival curve from 0 to positive infinity (or equivalently to the maximum lifespan, sometimes called 'omega'). For an extinct or completed cohort (all people born in the year 1850, for example), it can of course simply be calculated by averaging the ages at death. For cohorts with some survivors, it is estimated by using mortality experience in recent years. The estimates are called period cohort life expectancies.


The starting point for calculating life expectancy is the age-specific death rates of the population members. If a large amount of data is available, a statistical population can be created that allow the age-specific death rates to be simply taken as the mortality rates actually experienced at each age (the number of deaths divided by the number of years "exposed to risk" in each data cell). However, it is customary to apply smoothing to remove (as much as possible) the random statistical fluctuations from one year of age to the next. In the past, a very simple model used for this purpose was the Gompertz function, but more sophisticated methods are now used.^[148] The most common modern methods include:


The age-specific death rates are calculated separately for separate groups of data that are believed to have different mortality rates (such as males and females, or smokers and non-smokers) and are then used to calculate a life table from which one can calculate the probability of surviving to each age. While the data required are easily identified in the case of humans, the computation of life expectancy of industrial products and wild animals involves more indirect techniques. The life expectancy and demography of wild animals are often estimated by capturing, marking, and recapturing them.^[149] The life of a product, more often termed shelf life, is also computed using similar methods. In the case of long-lived components, such as those used in critical applications (e.g. aircraft), methods like accelerated aging are used to model the life expectancy of a component.^[9]


The life expectancy statistic is usually based on past mortality experience and assumes that the same age-specific mortality rates will continue. Thus, such life expectancy figures need to be adjusted for temporal trends before calculating how long a currently living individual of a particular age is expected to live. Period life expectancy remains a commonly used statistic to summarize the current health status of a population. However, for some purposes, such as pensions calculations, it is usual to adjust the life table used by assuming that age-specific death rates will continue to decrease over the years, as they have usually done in the past. That is often done by simply extrapolating past trends, but some models exist to account for the evolution of mortality, like the Lee–Carter model.^[150]


As discussed above, on an individual basis, some factors correlate with longer life. Factors that are associated with variations in life expectancy include family history, marital status, economic status, physique, exercise, diet, drug use (including smoking and alcohol consumption), disposition, education, environment, sleep, climate, and health care.^[12]