Dynamics of the coefficient of variation of the age at death distribution

Indicators of lifespan inequality, such the life table entropy, provide a measure of inequality in the timing of death. A range of measures of relative and absolute inequality exist and their evolution over time and sensitivity to changes in age-specific mortality have been studied. A common result found for the indicators the variance, the years of life lost, the life table entropy, and the Gini coefficient is the existence of a “threshold age” above which age-specific mortality improvements translate to greater lifespan inequality. However, the coefficient of variation, a relative measure defined as the standard deviation divided by the mean of the age at death distribution has yet to be studied and the existence and form of the threshold age has not been determined. We prove that changes in the coefficient of variation can be written as a weighted sum of changes in age-specific mortality rates, and a unique threshold age exists. The threshold age for the coefficient of variation occurs later than that of its absolute counterpart, the standard deviation, a result verified for other pairs of relative and absolute measures. We apply these mathematical findings empirically using data from the Human Mortality Database as Spanish life tables segregated by educational level. The results reveal that the ages most important for reducing lifespan inequality depend on both the context as well as the choice of indicator used to measure lifespan inequality. 

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