What is the difference between l_x and L_x?

Lowercase l_x counts survivors at an exact age or interval boundary; uppercase L_x measures person-time lived within an interval.

Why do tables often begin with 100,000?

It is a convenient standard radix that makes the survival schedule easy to compare.

Does higher life expectancy mean everyone lives longer?

No. It summarizes an average under the specified schedule and can conceal different changes across ages or groups.

AI Glossary / Statistics and probability / Intermediate

Life table

Definition

A life table summarizes survival and mortality across successive ages or time intervals. It links interval death probabilities with numbers surviving, numbers dying, time lived and expected remaining lifetime. In demography, a period life table applies the age-specific mortality conditions of a particular period to a hypothetical cohort, while a cohort life table follows or models an actual birth cohort's experience across time. Related life-table methods are also used in survival and reliability analysis, with appropriate treatment of censoring and the event being studied.

What it means

A single death rate cannot describe how survival changes with age. A life table organizes those age-specific patterns into a coherent sequence. Starting with a convenient hypothetical number of births, called the radix, it applies each interval's conditional death probability to those still alive. The initial count is a scaling device, not necessarily the size of an observed population. Life expectancy is an average derived from the resulting survival schedule; it is neither a promised lifespan for an individual nor a statement about the most common age at death.

How it works

Estimate age- or interval-specific mortality using suitable population and death data, or construct interval survival estimates from follow-up data with an appropriate censoring method. Let l_x represent survivors at the start of an interval and q_x the conditional probability of dying during it. Then deaths are d_x = l_x q_x and survivors advance to the next interval. Calculate person-time L_x using assumptions about timing within intervals, accumulate future person-time T_x, and divide T_x by l_x to obtain expected remaining lifetime e_x. Open-ended age groups and incomplete follow-up require explicit methods rather than arbitrary truncation.

Three practical examples

Population analysis: summarize how mortality conditions differ across periods or populations.
Long-term planning: use survival schedules as one input to projections, while separately specifying future mortality and population assumptions.
Reliability studies: organize interval survival of products or components, adapting the method to failures and censored observations. A table for one population or operating condition should not be transferred automatically to another.

Think of it this way

Think of a cohort moving through a sequence of gates. At each gate, a proportion of those who reached it leave the cohort. The life table records both how many continue and how much time the cohort collectively spends between gates. Expected remaining lifetime divides the future accumulated time by the number currently present, so it is conditional on having reached that age.

Worked example

Consider an artificial three-interval survival schedule, with intervals each one time unit long and a starting cohort of 1,000. Set conditional death probabilities to 0.1, 0.2 and 1. Survivors are therefore 1,000, 900, 720 and 0, while interval deaths are 100, 180 and 720. Assume deaths are spread uniformly within each interval. Person-time is then the average of starting and ending survivors: 950, 810 and 360. Total time lived is 2,120, giving expected lifetime at entry of 2.12 time units. Among the 900 survivors at time one, remaining person-time is 810 + 360 = 1,170, so expected remaining lifetime is 1.3 units. It is not 2.12 minus 1. This original toy schedule forces all remaining deaths into the third interval and is not a realistic human mortality table or a forecast.

Illustrative case study

In a fictional case, Willow Planning compares published period life tables when studying population change. Its report initially describes life expectancy as the predicted lifespan of babies born that year. Analysts correct the wording: the period measure summarizes what would occur under the period's age-specific mortality schedule, without automatically forecasting future changes. For a separate projection, they specify assumptions about future mortality and show how results vary. They also avoid presenting population averages as individual predictions.

Common misconceptions and mistakes

A period life table is not the same as observing one cohort throughout its actual lifetime. Life expectancy at a later age is conditional on survival to that age and is not generally life expectancy at birth minus elapsed years. The death probability q_x differs from a mortality rate expressed per unit of exposure time. The radix does not determine the implied probabilities. Treating censored individuals as deaths or as known permanent survivors biases survival estimates. Interval-width assumptions and the final open-ended age group can materially affect the table.

Business use

Businesses and public organizations should identify the table's population, period or cohort basis, age intervals and mortality assumptions before using it. Match the table to the application and assess uncertainty and transferability. For projections, distinguish observed historical mortality from assumed future changes. For product survival, record operating conditions and censoring rules. When communicating results, explain that an average remaining lifetime summarizes a distribution and does not replace individual information or scenario analysis.

Check your understanding

Why is remaining lifetime at time one 1.3 rather than 1.12 in the example?

Answer: The calculation conditions on the 900 survivors and divides their future person-time, 1,170, by 900. Those who died in the first interval are no longer part of that conditional population, so simple subtraction from the entry average is inappropriate.

Related terms

Survival function
Cohort
Censoring
Kaplan-Meier estimator
Restricted mean survival time
mortality rate
conditional death probability
period life expectancy
person-years

Learn next

Study the conversion between mortality rates and interval death probabilities, then examine person-time assumptions and open-ended age groups. Compare period and cohort tables and learn how censoring changes the construction of actuarial survival tables.

Sources

CDC/NCHS, U.S. Decennial Life Tables methodology: https://ftp.cdc.gov/pub/health_statistics/nchs/publications/DVD/DVD_3/National_Vital_Statistics_Reports/nvsr57/nvsr57_04.pdf . CDC/NCHS, Life Expectancy: https://www.cdc.gov/nchs/nvss/life-expectancy.htm . The short artificial schedule and fictional planning case above are original illustrations.

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