Life Table Simulator: Survival Curves & Life Expectancy Calculation

simulator intermediate ~10 min
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e₀ = 74 years — life expectancy at birth

With infant mortality of 30‰ and a Gompertz hazard slope of 0.085, the life table yields a life expectancy at birth of approximately 74 years, with 78% of the cohort surviving to age 65.

Formula

μ(x) = α × Math.exp(β × x) — Gompertz mortality hazard
e₀ = Σ lₓ / l₀ — life expectancy at birth
qₓ = 1 - Math.exp(-∫μ(t)dt) — age-specific death probability

Tracking a Cohort Through Life

A life table follows a hypothetical cohort — typically 100,000 individuals — from birth to the last survivor. At each age, it records the probability of dying, the number surviving, the person-years lived, and the expected remaining lifetime. This systematic accounting, first formalized by John Graunt in 1662 and refined by Edmund Halley in 1693, remains the cornerstone of actuarial science, public health planning, and demographic analysis.

The Gompertz Law of Aging

In 1825, Benjamin Gompertz discovered that adult mortality risk increases exponentially with age — a pattern so consistent it has been called a law of nature. After age 30, the annual probability of dying roughly doubles every 8-9 years. This exponential hazard function captures the fundamental biology of aging: cumulative damage to DNA, proteins, and cellular machinery progressively degrades the body's ability to maintain homeostasis and resist disease.

The Rectangularization of Survival

Pre-modern survival curves dropped steeply in infancy, then declined gradually throughout life. Modern curves have 'rectangularized' — remaining nearly flat until ages 60-70, then plunging steeply. This dramatic reshaping reflects the elimination of premature death from infection, malnutrition, and childbirth, concentrating mortality into a narrow age window near the biological lifespan limit. Whether this limit can be extended remains one of demography's most debated questions.

Life Expectancy: A Misunderstood Metric

Historical life expectancy of 35 years does not mean most people died at 35 — it means the average across all deaths, heavily weighted by infant mortality. A Roman who survived childhood could expect to live into their 60s. Today's life expectancy gains increasingly come from pushing back elderly mortality rather than reducing infant deaths, meaning each additional year of life expectancy requires progressively more medical intervention and expense.

FAQ

What is a life table?

A life table (or actuarial table) tracks a hypothetical cohort of individuals from birth, recording the probability of dying at each age, the number of survivors, and the expected remaining lifetime. Period life tables use current mortality rates; cohort life tables follow actual birth cohorts through time. Life tables are the foundation of life insurance, pension funding, and public health analysis.

What is the Gompertz law of mortality?

Benjamin Gompertz observed in 1825 that adult mortality rates increase exponentially with age — the risk of dying roughly doubles every 8-9 years after age 30. The Gompertz hazard function μ(x) = α × exp(βx) captures this exponential aging, where α is the baseline hazard and β is the rate of mortality acceleration. This law holds remarkably well from ages 30 to 90.

Why does infant mortality affect life expectancy so much?

Life expectancy at birth averages the ages at death across the entire cohort. Deaths in infancy pull the average down dramatically — a baby dying at age 0 subtracts a full lifetime from the numerator. Reducing infant mortality from 200‰ to 20‰ can raise life expectancy by 15+ years even if adult mortality stays constant. This is why historical life expectancies of 35-40 years did not mean most people died at 40.

What is the survival curve?

The survival curve S(x) = lₓ/l₀ plots the fraction of a cohort still alive at each age. It starts at 1.0 (everyone alive at birth) and decreases to 0. In pre-modern populations, the curve drops steeply in infancy then gradually. In modern populations, it remains nearly flat until age 60-70 before plunging — a 'rectangularization' reflecting concentrated mortality at old ages.

Sources

Embed

<iframe src="https://homo-deus.com/lab/demography/life-table/embed" width="100%" height="400" frameborder="0"></iframe>
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