Life table: how to read an actuarial table

A life table is the starting point for almost every calculation on this site. It is part of our guides to how the maths works, and the how long will my money last calculator reads it on every keystroke.

The idea in plain English

Imagine 100,000 babies born on the same day, and follow them through life. At each birthday, count how many are still alive, and how many of them die before the next birthday. Write one row per age. That list is a life table.

Nobody actually follows a real group for a century or more. A period life table takes the death rates seen at every age in one calendar year and applies them to an imaginary group, as if those rates never changed. The table on this site is the US Social Security Administration’s 2023 period table (Period Life Table, 2023, as used in the 2026 Trustees Report). It is a snapshot of one year’s mortality, not a forecast. It doesn’t allow for any change in death rates, and over most of the last century US death rates fell, so period tables have tended to understate how long people went on to live.

“Life table”, “mortality table” and “actuarial table” are three names for the same thing. Actuaries use them to price pensions, annuities and life insurance; the SSA uses them to project Social Security.

The notation

Each row of a life table is labelled by an age Age xYour age today, in whole years.How it works: Life tables, in whole years. The columns are:

Survivors lₓIn a life table, the number of people still alive at exact age x out of a starting group (100,000 at birth in SSA tables).How it works: Life tables: the number still alive at exact age $x$ , out of the starting group (the “radix”). In the SSA table the radix is 100,000 at birth.
Deaths dₓIn a life table, the number of the lₓ survivors who die between ages x and x + 1.How it works: Life tables: the number of those who die before reaching $x+1$ .
Death probability qₓThe chance that someone aged exactly x dies before reaching x + 1, read from the life table.How it works: Life tables: the probability that someone aged exactly $x$ dies within a year.
One-year survival pₓThe chance that someone aged x survives one more year: 1 − qₓ.How it works: Life tables: the probability that they survive the year.
Limiting age ωThe age by which everyone in a life table is assumed to have died. Survival to ω or beyond is zero.How it works: Life tables: the limiting age, by which everyone is assumed to have died.

The subscript is always the age at the start of the year. So $q_{65}$ is the chance that someone who has just turned 65 dies before turning 66.

The formulas

The columns are tied together by three relations:

\begin{gathered} d_{x} = l_{x} - l_{x+1} \\ q_{x} = \frac{d_{x}}{l_{x}} \\ p_{x} = 1 - q_{x} = \frac{l_{x+1}}{l_{x}} \end{gathered}

Deaths in the year are the fall in survivors; the death probability is deaths as a share of those alive at the start of the year.

In practice the death probabilities come first and the survivor column is generated from them, one row at a time:

l_{x+1} = l_{x}\,\left(1 - q_{x}\right)

Start with the radix at age 0 and multiply down the table.

How a period table is built

In outline, an actuary starts from two counts for each age in the year: the number of deaths and the population at risk. Dividing one by the other gives a death rate. That rate is converted to a probability $q_{x}$ , smoothed so it moves sensibly from one age to the next, and extended to the oldest ages, where there are too few people for the raw rates to be reliable. Then the radix is run down the table with the formula above to fill in $l_{x}$ and $d_{x}$ , and the life expectancy column is calculated from the survivors.

Because $l_{x}$ is printed rounded to whole people, recomputing $q_{x}$ as $d_{x}/l_{x}$ from the printed columns gives a value that can differ from the printed $q_{x}$ in the last digits. The printed $q_{x}$ is the one to use.

Worked example: reading the SSA table at 65

In the SSA 2023 period table, 79,084 of the original 100,000 boys reach age 65. Of those, 1,301 die before turning 66, leaving 77,783. So

the death probability is $q_{65}$ = 1,301 ÷ 79,084 = 0.016451, against a printed value of 0.016455;
the survival probability is $p_{65}$ = 1 − 0.016455 = 0.983545.

That is about 16.5 deaths a year per 1,000 men aged 65. For women of the same age the printed rate is 0.010188, about 10.2 per 1,000.

Death rates climb steeply with age. For men, $q_{x}$ is 0.003115 at 40, 0.016455 at 65, 0.092680 at 85 (about 5.6 times the rate at 65) and 0.366568 at 100. The table prints single ages up to 119; the calculators hold the age-119 rate constant beyond that. The last column, life expectancy, is 18.12 years for a man aged 65; how life expectancy is calculated explains where it comes from.

Try it: read the table from any age

Type an age to see the next ten rows. Each row is one year: start with $l_{x}$ people, lose $d_{x}$ of them, and carry $l_{x+1}$ into the next row.

Read the life table

Table

MaleFemale

Start at age

Age $x$	$l_{x}$ alive	$d_{x}$ die	$q_{x}$	$p_{x}$
65	79,084	1,301	0.016455	0.983545
66	77,783	1,367	0.017574	0.982426
67	76,416	1,432	0.018735	0.981265
68	74,984	1,498	0.019981	0.980019
69	73,486	1,570	0.021366	0.978634
70	71,916	1,647	0.022903	0.977097
71	70,269	1,730	0.024615	0.975385
72	68,539	1,817	0.026504	0.973496
73	66,722	1,911	0.028648	0.971352
74	64,811	2,014	0.031071	0.968929

Of 79,084 people alive at 65 in this table, 1,301 die before 66. That is 1,301 ÷ 79,084 = 0.016451. The printed death probability is 0.016455; any gap comes from rounding the survivor counts to whole people.

Source: US Social Security Administration, 2023 period life table (Period Life Table, 2023, as used in the 2026 Trustees Report). Do My Numbers calculations.

Reading a life table well

A life table describes a population, not a person. The SSA table splits by sex and nothing else: it mixes smokers and non-smokers, healthy and ill, rich and poor. Your own chances may be better or worse than the average row.

Chaining the one-year probabilities gives the chance of surviving many years: see survival probability. The life expectancy by age table shows the results for every age, and mortality laws explains why the $q_{x}$ column rises so regularly through adult life.

Where this is used

Questions

What is an actuarial life table?

An actuarial life table lists, for every age, the chance that someone of that age dies before their next birthday. From those chances it also shows how many of a starting group of 100,000 people would still be alive at each age, and their average remaining lifetime. "Life table", "mortality table" and "actuarial table" usually mean the same thing.

Where does the Social Security life expectancy table come from?

The Social Security Administration's Office of the Chief Actuary publishes it. It is a period life table built from death rates in one calendar year for the Social Security area population, separately for men and women. The edition used here is the one that supports the annual Trustees Report.

What is the difference between a period and a cohort life table?

A period table applies one year's death rates at every age, as if they never changed. A cohort table follows people born in the same year and uses the death rates expected for them at each future age, including expected improvements. Cohort tables usually show longer lives, because death rates have mostly fallen over time.

What does qx mean in a life table?

qₓ is the probability that someone who has just reached age x dies before reaching x + 1. It is the basic number in the table; everything else, including survivors and life expectancy, is calculated from the column of qₓ values.

Life tables: how to read one