## 2. Mode-based Measures

As compression and rectangularization advance, the mode becomes the central indicator of the length of life and - if they reach their ultimate, postulated form - its sole indicator. The dispersion of the lives lived beyond the mode is therefore a relevant measure of compression. Historical and recent values of the mode and the standard deviation above the mode are given in Table 1 (col. 2-3), based on life tables from four countries with reliable data. The development has been roughly parallel in all four countries, with the mode steadily increasing and the standard deviation above it steadily declining. This means that the increase in life expectancy has not resulted in the mode simply sliding to the right, maintaining its form intact, but undergoing at the same time a considerable transformation in which the right hand slope has become steeper. Results of less systematic historical comparisons made for Australia, Austria, Denmark, France, Italy, Japan, Sweden and the United States are fully in line with these findings. The mode appears to have been moving against an invisible wall and having been flattened vertically in the process [Kannisto, in press]. In spite of many attempts, an absolute limit to human life has never been conclusively proven and recent extensive studies of the very oldest have not found evidence of any limit [Thatcher, Kannisto and Vaupel 1998]. Nevertheless, as the probability of dying keeps rising in modern populations at least till age 110, it reduces the dispersion progressively. In the words of Keyfitz [Keyfitz 1978], there is "an uphill road ahead" for extending human life.

(Table 1 here)

When we plot the standard deviation above mode against life expectancy at mode, we find an extremely tight correlation in Figure 1. In 16 countries and for all periods combined r = +0.995. The ratio of standard deviation above mode to life expectancy at mode varied generally within the narrow limits of 1.22 and 1.25. It is interesting to recall that in a normal curve, the ratio of the standard deviation to the mean deviation equals:

The observed empirical distributions - which may include observation errors - conform thus very well to normal curves, giving strong support to the views of Lexis.

(Figure 1 here)

Next, we give in Table 1 (col. 4-5) the mean and standard deviation of the age at death in the highest quartile. The mean of this quartile has been historically several years higher than the mode, and increasing over time. Its standard deviation has declined relatively even more than that above mode: 39-48% vs. 21-31%. These findings support those made in relation to the mode that relative compression increases at high ages. However, the greatest absolute concentration is always situated around the mode.

 Measuring the Compression of Mortality Väinö Kannisto © 2000 Max-Planck-Gesellschaft ISSN 1435-9871 http://www.demographic-research.org/Volumes/Vol3/6