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# Multilingual Demographic Dictionary, second unified edition, English volume

# Instantaneous death rate

**Instantaneous death rate {{{CompleteIndexTerm}}}**

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**Probabilities of dying** ^{1} or **death probabilities** ^{1} are used to study in detail the mortality of a period or of a cohort. They are the probabilities that an individual of exact age *x* will die before exact age *x + n*, and are represented by the symbol _{n}q_{x}. If *n* = 1, we talk about **annual death probabilities** ^{2}; if *n* = 5, about **quinquennial death probabilities** ^{3}. The **instantaneous death rate**^{4}, or as it is occasionally called the **force of mortality** ^{4}, is the limit of the _{n}q_{x} value as *n* tends to zero. The **projective mortality probability** ^{5★} is the probability that individuals of the same cohort or group of cohorts died between two January 1^{st}. The name of this probability comes from its use in the calculation of population projections. It is also equal to 1-L_{x+n}/L_{x}, where L_{x} is the person-years lived by the stationary population from exact age x to exact age x+n. The complement to one of the probability of dying from exact age *x* to exact age *x + n* is the **probability of survival** ^{6} over this interval. In the preparation of population projections, we use **survival ratios** ^{7}; they represent the probability that individuals of the same birth cohort or group of cohorts will still be alive n years later.

- 1. The probability of death between age
*x*and*x + n*is defined as the ratio of deaths between ages*x*and*x+n*to the number of survivors at exact age*x*. It is not to be confused with the**central death rate**, the ratio of deaths between ages*x*and*x+n*to the mean population alive at that age. The central death rate is written_{n}m_{x}. - 6. The probability of survival from age
*x*to age*x+n*is written_{n}p_{x}. - 7. A
**survival ratio**is the complement to one of the projective mortality probability. Individuals in the cohorts do not have the same age and therefore are not at the same risk of dying.