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A life table traces the mortality experience of a cohort — real or synthetic — from birth (or any starting age) to extinction, converting age-specific death rates into a comprehensive model of survivorship. Its columns include the probability of dying in each age interval (qx), the number surviving to each age (lx), person-years lived (Lx), cumulative person-years remaining (Tx), and life expectancy at each age (ex). Period life tables use current age-specific mortality rates applied to a hypothetical cohort; cohort life tables follow an actual birth cohort through time. Life tables are the central analytic tool in demography, actuarial science, and epidemiology, transforming a set of age-specific rates into interpretable summary measures like life expectancy at birth.
Build a complete abridged life table by hand from age-specific death rates for a real country. Working through each column — converting rates to probabilities, computing survivors, person-years, and finally life expectancy — makes the logic transparent. Then compare your table to one from a country with very different mortality patterns.
You already know how to compute age-specific death rates — deaths in an age group divided by the mid-year population in that group. The life table takes a full set of these rates and transforms them into a model of how a cohort lives and dies across the entire age range. It is, in essence, a bookkeeping device that converts observed mortality rates into the survival experience of a population.
The construction proceeds through a series of linked columns. Start with nMx, the age-specific death rate for the interval from age x to x+n. Convert this to nqx, the probability of dying in that interval, using a formula that accounts for the distribution of deaths within the interval. Apply nqx to the number of survivors at the start of the interval (lx, starting from a conventional radix of 100,000 at birth) to get the number dying (ndx) and the number surviving to the next age (lx+n). Compute person-years lived in the interval (nLx) by accounting for when within the interval deaths occur. Sum person-years from age x onward to get Tx. Finally, divide Tx by lx to get ex, life expectancy at age x — the average number of years remaining for someone who has survived to exact age x.
The most commonly cited output is e0, life expectancy at birth. But e0 can be profoundly misleading if interpreted as "the age at which most people die." In historical populations where infant mortality was 200-300 per 1,000, e0 might be 35 years, yet a person who survived to age 20 might expect to live to 55 or 60. The life table makes this transparent: e5 or e20 shows the conditional expectation of remaining life for those who survived the dangerous early years. In populations with concentrated early-age mortality, ex actually *increases* from birth to age 1 or 5 — a result that surprises students but follows directly from the mathematics.
A critical distinction separates period and cohort life tables. A period life table takes the age-specific mortality rates observed in a single calendar year and applies them to a hypothetical cohort, as if those rates would persist unchanged forever. A cohort life table follows an actual birth cohort through time, using the mortality rates they really experienced at each age. Period tables are available immediately; cohort tables can only be completed after the last member of the cohort has died. Since mortality generally improves over time, period life expectancy at birth typically *underestimates* how long people born that year will actually live — a systematic bias that matters for pension planning and policy projections.
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