The Gompertz–Makeham Law Behind the Death Clock
The Gompertz–Makeham law of mortality states that the death rate is the sum of an age-independent term and a term that rises exponentially with age.
The death clock applies the Gompertz–Makeham law of mortality as the equation behind every result. The equation is one part of the model behind the number, which also sets the country baseline and the lifestyle adjustments.
- Author
- Jacob V.
- Last reviewed
- Next review
Reviewed against the published methodology and UN World Population Prospects 2024.
The hazard equation and its three constants
The hazard equation is μ(x) = A + B·e^(b·x), where A is the age-independent Makeham term, B sets the level of age-dependent mortality, and b sets how fast mortality rises with age.
μ(x) = A + B·e^(b·x).
| Symbol | Meaning | Value on this site | Unit | Source |
|---|---|---|---|---|
| A, men | Age-independent Makeham term A | Equals 0.0016 | per year | Calculator script |
| A, women | Age-independent Makeham term A | Equals 0.0009 | per year | Calculator script |
| b | How fast mortality rises with age | Equals 0.0815 (= ln 2 ÷ 8.5) | per year | Calculator script |
| B | Level of the Gompertz term B·e^(b·x) | Is calibrated to each country and sex; no single value | per year | Calculator script |
S(x) = exp(−A·x − (B/b)·(e^(b·x) − 1)): the share of people still alive at age x under that hazard. (Calculator script (engine v11))
The Gompertz law is the exponential term alone, B·e^(b·x): the death rate rising exponentially with age.
What the mortality doubling period means
The mortality doubling period is the number of years over which the age-dependent death rate doubles.
This site uses 8.5 years, which gives b = ln(2) ÷ the doubling period = 0.0815 per year.
b is computed from the mortality doubling period: a shorter period gives a larger b and a steeper rise in the age-dependent Gompertz term B·e^(b·x), while a longer period gives a flatter rise.
The mortality doubling period is evidenced by the calculator script, which uses the same constants stated in the methodology.
How B is solved for each country and sex
B is solved separately for each country and sex in three steps: A and b are fixed, a trial B generates a survival curve, and B is adjusted until the curve reproduces the published life expectancy.
- Fix A and b so each equals the site’s constant shown in the table above.
- Generate the survival curve that a trial B returns.
- Adjust B by bisection so it is calibrated to the country-and-sex baseline, with the curve’s life expectancy at birth within 0.001 years of the United Nations (UN) figure at birth.
B is computed by this procedure separately for each country and sex across the calculator’s 96 countries, supplying the country-and-sex constant in its Gompertz–Makeham law of mortality.
The tables the curve must reproduce
The curve has to reproduce the life expectancy at birth in published life tables, and those tables are described on their own page.
B is solved so the curve gives the life expectancy at birth in the UN life tables for each country and sex.
Who Gompertz and Makeham were
Benjamin Gompertz published the exponential law of mortality in 1825, in the Philosophical Transactions of the Royal Society, and William Makeham added the age-independent term in 1860, in the Assurance Magazine and Journal of the Institute of Actuaries.
Benjamin Gompertz’s contribution to the Gompertz term B·e^(b·x) is evidenced by his 1825 paper in Philosophical Transactions of the Royal Society, volume 115, pages 513–583 (Gompertz–Makeham law of mortality (secondary overview; primary bibliography checked)).
William Makeham’s addition of the Makeham term A is evidenced by his 1860 paper in the Assurance Magazine and Journal of the Institute of Actuaries, volume 8, pages 301–310 (Gompertz–Makeham law of mortality (secondary overview; primary bibliography checked)).