How to Construct a Life Table: Step-by-Step Guide

A life table is one of the oldest and most powerful tools in demography. It takes the mortality experience of a population during a given period and converts it into a clear picture of how a hypothetical group of people would live and die over time. From this single table, we can read off the probability of dying at each age, the number of years a newborn can expect to live, and the size of the surviving population at every birthday. Constructing one looks intimidating at first, but the logic is straightforward once you understand the assumptions, the columns, and how each column feeds into the next.

What a life table actually does 🔗

A life table follows a hypothetical group of people, called a cohort, from birth until the death of the last member. It records, age interval by age interval, how many remain alive, how many die, and how many person-years of life the group collectively accumulates. The most common version used in population studies is the period life table, which applies the age-specific death rates observed in a particular year (or short period) to a synthetic cohort. As the demographic literature describes, this gives us a snapshot of mortality conditions as if a baby born today were to experience those rates throughout life.

In practice, two formats are used. A complete life table uses single-year age intervals. An abridged life table groups ages into five-year intervals, with two exceptions: infancy (0-1) and early childhood (1-4) are kept separate, and the final interval is open-ended (for example, 85+ or 100+). The Office of the Registrar General of India publishes SRS-based Abridged Life Tables using exactly this structure.

Assumptions behind the construction 🔗

Before any numbers are computed, the life table rests on a small set of assumptions that simplify reality enough to make the mathematics tractable.

A hypothetical cohort of 100,000 (the radix) 🔗

The life table begins with an arbitrary large number of simultaneous births, called the radix, and denoted l₀. By convention, this number is 100,000, though 1,000 or 1,000,000 are also acceptable. The radix is just a base for comparison; it has no real demographic meaning beyond making the columns easy to read.

A closed cohort with no migration 🔗

The cohort is treated as closed. No one enters by immigration and no one leaves by emigration. The only way an individual exits the table is by dying. This assumption is what allows the cohort to shrink in a predictable, rule-bound way from age to age.

A fixed mortality schedule 🔗

Every person in the cohort is assumed to face the same age-specific mortality rates, and those rates are fixed and unchanging across the life of the table. As a review of life table methodology explains, individuals die according to a predetermined schedule at each age, and that schedule is treated as constant.

Deaths distributed uniformly within an interval 🔗

Within any age interval, deaths are assumed to occur uniformly. This simplification works well for most age groups but breaks down at the two extremes: infancy, where deaths cluster in the first few weeks, and the open-ended terminal interval, where survivors die over a long tail. For these intervals, special separation factors (often written as ₙaₓ) are used to capture where, on average, deaths fall within the interval. A common convention is 0.1 for the infant interval, 0.4 for the early childhood interval, and 0.5 for most others.

The columns of a life table 🔗

An abridged life table typically has seven columns. The second column is the input; everything else is derived from it.

Age interval (x to x+n) 🔗

This column lists the age groups. For an abridged table, the intervals are 0-1, 1-4, 5-9, 10-14, and so on, ending in an open interval like 85+ or 100+.

Probability of dying, ₙqₓ 🔗

This is the probability that a person who has reached exact age x will die before reaching age x+n. It is the engine of the table: every other column is built from it. The age-specific death rate (ₙMₓ) is observed from vital statistics, and it is converted into the probability of dying using a relationship like:

ₙqₓ = (n × ₙMₓ) / (1 + n × (1 − ₙaₓ) × ₙMₓ)

For the open-ended terminal interval, ₙqₓ is set to 1, because everyone in that interval eventually dies.

Number surviving, lₓ 🔗

This is the number of the original cohort still alive at exact age x. It starts at the radix and shrinks at each step:

l_(x+n) = lₓ − ₙdₓ, or equivalently, l_(x+n) = lₓ × (1 − ₙqₓ).

Number dying, ₙdₓ 🔗

This is the number of cohort members who die during the interval from age x to x+n:

ₙdₓ = lₓ × ₙqₓ

The sum of all ₙdₓ values equals the radix, since every member of the cohort eventually dies, as MEASURE Evaluation’s life table lesson points out.

Person-years lived, ₙLₓ 🔗

This is the total person-years lived by the cohort within the interval. For most age groups, deaths are assumed to occur on average at the midpoint of the interval, so:

ₙLₓ = n × l_(x+n) + (n × ₙaₓ) × ₙdₓ

For the open-ended final interval, ₙLₓ = lₓ / ₙMₓ, since the survivors will live, on average, the reciprocal of the death rate.

Total person-years remaining, Tₓ 🔗

This column tells us how many person-years remain to be lived by the cohort from age x onward. It is calculated by summing ₙLₓ from the bottom of the table upward:

Tₓ = ₙLₓ + ₙL_(x+n) + … + ₙL_w, where w is the final interval.

Life expectancy, eₓ 🔗

This is the average number of additional years a person can expect to live after reaching age x:

eₓ = Tₓ / lₓ

The value e₀, life expectancy at birth, is the single most cited summary measure from any life table.

Step-by-step construction 🔗

Putting these columns together follows a fixed sequence. Skip a step and the rest of the table breaks.

Step 1: Gather age-specific death rates 🔗

You need ₙMₓ for each age group. In the Indian context, these come from the Sample Registration System, a large dual-record demographic survey that produces reliable state-level mortality estimates for rural and urban areas.

Step 2: Convert death rates to probabilities of dying 🔗

Apply the ₙMₓ-to-ₙqₓ formula. Choose appropriate separation factors for infancy and early childhood. Set ₙqₓ = 1 for the open-ended terminal interval.

Step 3: Build the lₓ column 🔗

Start with l₀ = 100,000 and apply l_(x+n) = lₓ × (1 − ₙqₓ) iteratively down the table.

Step 4: Compute the ₙdₓ column 🔗

Multiply lₓ × ₙqₓ for each row. The sum should equal the radix.

Step 5: Compute the ₙLₓ column 🔗

Use the standard n × l_(x+n) + (n × ₙaₓ) × ₙdₓ formula, with special treatment for the terminal age.

Step 6: Compute Tₓ and eₓ 🔗

Start from the oldest age group and work upward, accumulating Tₓ. Then divide each Tₓ by the corresponding lₓ to get eₓ.

An abridged life table example for Bihar 🔗

Bihar is a useful illustration because it has historically reported some of the highest mortality and fertility indicators among the larger states. According to recent SRS reporting, states like Bihar, Uttar Pradesh, and Madhya Pradesh continue to lag the national average on key mortality indicators. Researchers have also documented an unusual male-female life expectancy crossover in Bihar, where male life expectancy at birth surpassed female life expectancy in 2011-15 and the pattern has persisted since.

Suppose, for illustration, we have the following hypothetical age-specific death rates (ₙMₓ) for Bihar:

Age 0-1: 0.0350 · Age 1-4: 0.0030 · Age 5-9: 0.0010 · Age 10-14: 0.0008 · Age 15-19: 0.0014 · … Age 60-64: 0.0220 · Age 70+: 0.0900.

Applying the conversion to ₙqₓ with separation factors of 0.1 for infancy and 0.5 for most other intervals gives a probability of dying in infancy of roughly 0.034 (about 34 deaths per 1,000 live births). Starting with a radix of 100,000:

l₀ = 100,000. Roughly 3,400 die in the first year, so l₁ ≈ 96,600. In the 1-4 interval, the probability of dying is small but not negligible; perhaps 1,180 more children die, so l₅ ≈ 95,420. The cohort then loses very few members through the school and young adult ages, with l₂₅ still around 93,000. Mortality begins climbing again in middle age, accelerates after 60, and the survivors are wiped out in the open-ended terminal interval.

Adding up the ₙLₓ values gives T₀, the total person-years that the cohort will collectively live. Dividing by 100,000 gives e₀, the life expectancy at birth. For India as a whole, the SRS Abridged Life Tables for 2019-23 report life expectancy at birth at around 68.5 years for males and 72.5 years for females. State-specific tables show Bihar somewhat below the national average, with Chhattisgarh, Madhya Pradesh, Uttar Pradesh, and Assam typically reporting the lowest values.

What is striking about reading any abridged life table for an Indian state is how much of the difference in e₀ across states is driven by mortality in the youngest and oldest intervals. Cut the probability of dying in infancy by half, and life expectancy at birth jumps noticeably even if mortality at every other age stays the same. This is why infant and child mortality reduction has been such a powerful lever for raising life expectancy in India over the past five decades.

Why these numbers matter 🔗

Life tables are not academic curiosities. They feed into population projections, pension and insurance calculations, public health planning, and the district-level estimation of life expectancy in India using NFHS and SRS data. The ₙLₓ and Tₓ columns are also used to derive survival ratios for the cohort-component method of population projection, where today’s age distribution is rolled forward into tomorrow’s. Every demographic model that needs to know how a population ages over time eventually leans on a life table.

What do you think? If Bihar’s infant mortality rate were brought down to the national average, by how many years do you think its life expectancy at birth would rise? And which assumption of the standard life table – the closed cohort, the fixed mortality schedule, or the uniform distribution of deaths within an interval – strikes you as the most limiting when applied to a rapidly changing population?

References

1. https://www.sciencedirect.com/topics/mathematics/life-table
2. https://censusindia.gov.in/census.website/data/SRSALT
3. https://www.demographytextbook.com/chapter05.php
4. https://medcraveonline.com/BBIJ/a-review-of-life-table-construction.html
5. https://www.measureevaluation.org/resources/training/online-courses-and-resources/non-certificate-courses-and-mini-tutorials/multiple-decrement-life-tables/lesson-3.html
6. https://censusindia.gov.in/nada/index.php/catalog/45558
7. https://www.business-standard.com/health/india-s-birth-death-rates-halve-in-50-yrs-infant-mortality-at-record-low-125090500478_1.html
8. https://pmc.ncbi.nlm.nih.gov/articles/PMC8638908/
9. https://affairscloud.com/srs-2023-report-indias-birth-and-death-rates-drop-by-half-over-50-years-to-18-4-and-6-4/
10. https://pmc.ncbi.nlm.nih.gov/articles/PMC11021017/