Lost in translation

Actuaries have a long-standing habit of using different terminology to statisticians.  This page lists some common terms used by actuaries in mortality work and their "translation" for a non-actuarial audience.  The terms and notation are those used by actuaries in the UK, but in every country I have visited the local actuaries have used similar notation.

Table 1. Common actuarial terms and their definition for statisticians.

Actuarial term Actuarial notation
Statistical description
central exposed to riskEcxThe time exposed to risk of dying at age x.
curve of deathstpxmuxplustProbability density function for the future lifetime of an individual currently alive and aged exactly x.
force of mortalitymux

In models with a single mortality decrement, this refers to the instantaneous hazard rate for mortality at exact age x. In models with multiple states, this refers to the instantaneous transition intensity for mortality.

initial exposed to risk
ExThe number of individuals alive aged exactly x at the start of an investigation period.
mortality law variesA functional form for the instantaneous hazard rate or probability of death, e.g. the Gompertz Law is μx = exp(α+βx).
mortality rate
qxThe probability of death between ages x and x+1 in a Bernoulli model for the number of deaths, given that a life is alive at exact age x at outset.
survival rate
tpxThe probability of survival from age x to age x+t, given a life is alive at exact age x, i.e. the survival curve or survivor function.


It is interesting to note the different traditions within particular disciplines.  For example, actuaries almost always use μx to denote the force of mortality.  Engineers, however, call the same thing the failure rate and typically denote it by λ.  Statisticians, meanwhile, will call the self-same concept the hazard function.



Stephen Richards
(Sep 18, 2012)

A related translation table for non-statisticians can be found at [http://www.longevitas.co.uk/site/informationmatrix/lostintranslationreprise.html]


Find by key-word


Epidemics and pandemics are, by definition, fast-moving and difficult to ... Read more
Ever since the unhappy arrival of the SARS-COV-2 virus, COVID-19 ... Read more
The former UK prime minister Harold Wilson famously said that ... Read more
Stephen Richards
Stephen Richards is the Managing Director of Longevitas