Force of Mortality Calculator
The force of mortality, mu subscript x, is the instantaneous death rate at exact age x, defined as the negative derivative of the log survival function. It is the hazard rate underlying every life table. This calculator estimates it with a central difference from the survivors one year below and one year above the target age, and also reports the one-year survival probability p_x, the mortality rate q_x, and the constant-force estimate from p_x. The survivor counts are user-editable inputs taken from whatever mortality table fits your population, so nothing is assumed.
Force of mortality formula
mu(x) = -d/dx ln l(x)
Central difference: mu_x = 0.5 * ln(l(x-1) / l(x+1))
p_x = l(x+1) / l(x)
q_x = 1 - p_x
Constant-force estimate: mu = -ln(p_x)
The central difference approximates the slope of the log survival curve around age x. The constant-force estimate assumes a single force across the year, giving p_x = exp(-mu).
Force of mortality context
- mu_x is a rate, not a probability, and can exceed 1 at very old ages.
- q_x and p_x are probabilities bounded between 0 and 1.
- Under a constant force over a year, p_x = exp(-mu) and q_x = 1 - exp(-mu).
- The central difference needs survivors at the ages immediately below and above x.
- Use a single consistent mortality table for all three survivor counts.
Force of mortality: frequently asked questions
What is the force of mortality?
The force of mortality, written mu subscript x, is the instantaneous rate of death at exact age x. It is the hazard rate of the survival function: mu(x) = -d/dx ln l(x). Unlike the annual probability q_x, it is a rate per unit time, not a probability, and can exceed 1.
How is it estimated from a life table?
A central-difference approximation uses the survivors at ages x-1 and x+1: mu_x is approximately one half times the natural log of l(x-1) divided by l(x+1). This calculator also gives the constant-force estimate from one-year survival, mu is approximately minus ln(p_x).
What is the difference between mu_x and q_x?
q_x is the probability that a life aged x dies within one year, a number between 0 and 1. mu_x is an instantaneous rate. Under a constant force over the year, p_x = exp(-mu) and q_x = 1 - exp(-mu), so the two are related but not equal.
Where do the survivor counts come from?
l(x) values come from a mortality table such as the SSA period life table or a Society of Actuaries valuation table. Because the right table depends on the population, the survivor counts are user-editable inputs here, so nothing is assumed.
Why use the central difference?
The central difference around age x cancels first-order error and gives a smoother estimate of the instantaneous rate than a one-sided difference. It requires the survivors one year below and one year above the target age.
Official sources
- U.S. Social Security Administration: SSA Office of the Chief Actuary period life tables.
- Society of Actuaries: SOA mortality tables and study notes.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.