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Force of Mortality Calculator

Enter the Gompertz parameters B, c and an age x.

Result

0.009134
0.91%
μ(x) = B·c^x = 0.000030·1.1000^60

How it is calculated

The tool evaluates the Gompertz hazard μ(x) = B × cx, where B is the baseline hazard at age 0 (per year), c is the growth factor per year of age and x is the age in years. It then converts the rate into an approximate one-year death probability with qx ≈ 1 − e−μ(x). Because the hazard grows by the factor c each year, it doubles every ln 2 ÷ ln c years, about 7.3 years when c = 1.1.

B and c must be greater than zero and the age cannot be negative. The age may include decimals and is used exactly in the calculation, although the summary line rounds it to a whole number. The tool does not fit B and c to data; you must supply parameters from a table or model of your own. Also note that μ is a rate, not a probability, and the q figure assumes the hazard stays constant over the year.

Worked example

Take B = 0.00003, c = 1.1 and age x = 60. First cx = 1.160 ≈ 304.48. Then μ(60) = 0.00003 × 304.48 = 0.009134 per year. The approximate chance of dying within the year is q = 1 − e−0.009134 = 0.91%. At age 100 the same parameters give μ = 0.413418 and q = 33.86%, which shows how quickly the exponential term takes over at high ages.

Try this example →

Common mistakes

  • Reading μ(x) as a probability. It is a rate per year; the probability of dying in the year is the separate q figure, which is always smaller for large rates.
  • Entering B as a percentage (0.003 for 0.003%). B is a plain fraction per year, so a baseline of 3 in 100,000 is 0.00003.
  • Using c = 10 or c = 0.1 because it looks like a percentage. For human adults c is just above 1, since the hazard grows by a few percent of itself each year; the tool also accepts values below 1, but those model falling mortality, which is not Gompertz ageing.
  • Applying parameters fitted to one population, sex or period to another. B and c change with each, so results are only as good as the parameters.
  • Using the model at young ages. Gompertz describes ageing mortality and ignores infant and accident mortality, so results at age 0 to 20 are not realistic.

Sources

About this calculator

The force of mortality, written μ(x), is the instantaneous rate at which a person aged exactly x is dying at that moment — the continuous-time analogue of the annual mortality rate q_x used in life tables. The Gompertz law, proposed in 1825, models this hazard as growing exponentially with age: μ(x) = B · c^x, where B sets the baseline hazard and c (typically around 1.08–1.11 for human adult mortality) is the factor by which the hazard multiplies with each additional year of age, so risk doubles every ln 2 / ln c years — roughly every 7 to 9 years for those values.

Gompertz's model remains a good fit for adult mortality from roughly age 30 onward and underlies much of actuarial mortality modeling, including extensions like the Gompertz–Makeham law that adds a constant term for age-independent causes of death. Because μ(x) is a rate rather than a probability, this calculator also converts it to an approximate annual death probability via q_x ≈ 1 − e^(−μ(x)), the standard bridge actuaries use between the continuous hazard and the discrete life-table figure.

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