thumb|right|2003 US mortality table, Table 1, Page 1

In actuarial science and demography, a life table (also called a mortality table or actuarial table) is a table which shows, for each age, the probability that a person of that age will die before their next birthday ("probability of death"). In other words, it represents the survivorship of people from a certain population. They can also be explained as a long-term mathematical way to measure a population's longevity. Tables have been created by demographers including John Graunt, Reed and Merrell, Nathan Keyfitz, and Thomas N. E. Greville.

Other life tables in historical demography may be based on historical records, although these often undercount infants and understate infant mortality, on comparison with other regions with better records, and on mathematical adjustments for varying mortality levels and life expectancies at birth.

From this starting point, a number of inferences can be derived.

  • The probability of surviving any particular year of age
  • The remaining life expectancy for people at different ages

Life tables are also used extensively in biology and epidemiology. An area that uses this tool is Social Security. It examines the mortality rates of all the people who have Social Security to decide which actions to take. allows study of life expectancy as a function of age already achieved.

| image2=20200101 Remaining life expectancy - US.svg |caption2= SSA life table data, "Life table" primarily refers to period life tables, as cohort life tables can only be constructed using data up to the current point, and distant projections for future mortality.

Life tables can be constructed using projections of future mortality rates, but more often they are a snapshot of age-specific mortality rates in the recent past, and do not necessarily purport to be projections. For these reasons, the older ages represented in a life table may have a greater chance of not being representative of what lives at these ages may experience in future, as it is predicated on current advances in medicine, public health, and safety standards that did not exist in the early years of this cohort. A life table is created by mortality rates and census figures from a certain population, ideally under a closed demographic system. This means that immigration and emigration do not exist when analyzing a cohort. A closed demographic system assumes that migration flows are random and not significant, and that immigrants from other populations have the same risk of death as an individual from the new population. Another benefit from mortality tables is that they can be used to make predictions on demographics or different populations.

However, there are also weaknesses of the information displayed on life tables. One being that they do not state the overall health of the population. There is more than one disease present in the world, and a person can have more than one disease at different stages simultaneously, introducing the term comorbidity. Therefore, life tables also do not show the direct correlation of mortality and morbidity.

The life table observes the mortality experience of a single generation, consisting of 100,000 births, at every age number they can live through.

Insurance applications

In order to price insurance products, and ensure the solvency of insurance companies through adequate reserves, actuaries must develop projections of future insured events (such as death, sickness, and disability). To do this, actuaries develop mathematical models of the rates and timing of the events. They do this by studying the incidence of these events in the recent past, and sometimes developing expectations of how these past events will change over time (for example, whether the progressive reductions in mortality rates in the past will continue) and deriving expected rates of such events in the future, usually based on the age or other relevant characteristics of the population. An actuary's job is to form a comparison between people at risk of death and people who actually died to come up with a probability of death for a person at each age number, defined as qx in an equation. or starting point, of <math>\,\ell_0</math> lives, typically taken as 100,000

:: <math>\,\ell_{x + 1} = \ell_x \cdot (1-q_x) = \ell_x \cdot p_x</math>

:: <math>\,{\ell_{x + 1} \over \ell_x} = p_x</math>

  • <math>\,d_x</math>: the number of people who die aged <math>\,x</math> last birthday

:: <math>\,d_x = \ell_x-\ell_{x+1} = \ell_x \cdot (1-p_x) = \ell_x \cdot q_x</math>

  • <math>\,{}_tp_x</math>: the probability that someone aged exactly <math>\,x</math> will survive for <math>\,t</math> more years, i.e. live up to at least age <math>\,x+t</math> years

::<math>\,{}_tp_x = {\ell_{x+t} \over \ell_x}</math>

  • <math>\,{}_{t\mid k}q_x</math>: the probability that someone aged exactly <math>\,x</math> will survive for <math>\,t</math> more years, then die within the following <math>\,k</math> years

::<math>\,{}_{t\mid k}q_x = {}_t p_x \cdot {}_k q_{x+t} = {\ell_{x+t} - \ell_{x+t+k} \over \ell_x}</math>

  • μ<sub>x</sub> : the force of mortality, i.e. the instantaneous mortality rate at age x, i.e. the number of people dying in a short interval starting at age x, divided by ℓ<sub>x</sub> and also divided by the length of the interval.

Another common variable is

  • <math>\,m_x</math>

This symbol refers to central rate of mortality. It is approximately equal to the average force of mortality, averaged over the year of age.

Further descriptions: The variable dx stands for the number of deaths that would occur within two consecutive age numbers. An example of this is the number of deaths in a cohort that were recorded between the age of seven and the age of eight. The variable ℓx, which stands for the opposite of dx, represents the number of people who lived between two consecutive age numbers. ℓ of zero is equal to 100,000. The variable Tx stands for the years lived beyond each age number x by all members in the generation. Ėx represents the life expectancy for members already at a specific age number.

  • The Forced Method: Select an ultimate age and set the mortality rate at that age equal to 1.000 without any changes to other mortality rates. This creates a discontinuity at the ultimate age compared to the penultimate and prior ages.
  • The Blended Method: Select an ultimate age and blend the rates from some earlier age to dovetail smoothly into 1.000 at the ultimate age.
  • The Pattern Method: Let the pattern of mortality continue until the rate approaches or hits 1.000 and set that as the ultimate age.
  • The Less-Than-One Method: This is a variation on the Forced Method. The ultimate mortality rate is set equal to the expected mortality at a selected ultimate age, rather 1.000 as in the Forced Method. This rate will be less than 1.000.

Epidemiology

In epidemiology and public health, both standard life tables (used to calculate life expectancy), as well as the Sullivan and multi-state life tables (used to calculate health expectancy), are the most commonly mathematical used devices. The latter includes information on health in addition to mortality. By watching over the life expectancy of any year(s) being studied, epidemiologists can see if diseases are contributing to the overall increase in mortality rates. Epidemiologists are able to help demographers understand the sudden decline of life expectancy by linking it to the health problems that are arising in certain populations.