page_title: Life expectancy
section_title: Calculation
hierarchical_section_title: Life expectancy / Calculation
caption_attribution_description: A survival tree to explain the calculation of life-expectancies.
context_page_description: Life expectancy is a statistical measure of the average time an organism is expected to live, based on the year of its birth, its current age, and other demographic factors including gender. The most commonly used measure is life expectancy at birth, which can be defined in two ways. Cohort LEB is the mean length of life of an actual birth cohort and can be computed only for cohorts born many decades ago, so that all their members have died. Period LEB is the mean length of life of a hypothetical cohort assumed to be exposed, from birth through death, to the mortality rates observed at a given year.
  National LEB figures reported by national agencies and international organizations for human populations are indeed estimates of period LEB. In the Bronze Age and the Iron Age, human LEB was 26 years; the 2010 world LEB was 67.2 years. For recent years, LEB in Eswatini is about 49, while LEB in Japan is about 83. The combination of high infant mortality and deaths in young adulthood from accidents, epidemics, plagues, wars, and childbirth, particularly before modern medicine was widely available, significantly lowers LEB.
context_section_description: 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 iron out, 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.
  These are the most common methods now used for that purpose:
  to fit a mathematical formula, such as an extension of the Gompertz function, to the data.
  for relatively small amounts of data, to look at an established mortality table that was previously derived for a larger population and make a simple adjustment to it (as multiply by a constant factor) to fit the data.
  with a large number of data, one looks at the mortality rates actually experienced at each age, and applies smoothing (as by cubic splines).
  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. 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: in aircraft, methods like accelerated aging are used to model the life expectancy of a component.
  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, and perhaps smokers and non-smokers if data are available separately for those groups) and are then used to calculate a life table from which one can calculate the probability of surviving to each age. In actuarial notation, the probability of surviving from age  to age  is denoted  and the probability of dying during age  (between ages  and ) is denoted . 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%. That is a probability, not a mortality rate.
  The expected future lifetime of a life age  in whole years (the curtate expected lifetime of (x)) is denoted by the symbol . It is the conditional expected future lifetime (in whole years), assuming survival to age . If  denotes the curtate future lifetime at ,
  Substituting  in the sum and simplifying gives the equivalent formula: If the assumption is made that on average, people live a half year in the year of death, the complete expectation of future lifetime at age  is .
  Life expectancy is by definition 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 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.
  It is important to note that the statistic is usually based on past mortality experience and assumes that the same age-specific mortality rates will continue into the future. 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 decre
caption: A survival tree to explain the calculations of life-expectancy. Red numbers indicate chance of survival at a specific age, and blue ones indicate age-specific death rates.
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