Averaging Life Expectancy

Andreev, E., Lutz, W. ORCID: https://orcid.org/0000-0001-7975-8145, & Scherbov, S. ORCID: https://orcid.org/0000-0002-0881-1073 (1989). Averaging Life Expectancy. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-89-035

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Abstract

More than 50 years ago a peculiar feature of averages was pointed out by Cohen and Nagel (1934, p. 449) in their well-known book on logic. Their example happened to be mortality from tuberculosis; Blacks in Richmond, Virginia, had a lower rate than Blacks in New York; Whites in Richmond had a lower rate than Whites in New York; yet the overall rate for Richmond was higher than that in New York. Many other examples of the paradox have subsequently been pointed out, and some general theory has been presented by Colin Blyth.

If this were only an arithmetical curiosity no one would care much about it, but in fact its very possibility is a troubling consideration for all numerical comparisons. If recognizing Blacks and Whites reverses the standing of New York and Richmond, how do we know that recognizing some further breakdown will not reverse the standing once again?

This paper presents a different paradox, but one that is also threatening to the drawing of conclusions from numerical data. To follow Cohen and Nagel's example but disregarding the distinction between Blacks and Whites, the arithmetic average mortality (expressed as a death rate) of Richmond and New York combined will always fall between the rate for New York and that for Richmond. The result of a linear averaging process cannot fall outside the units averaged.

This is no longer true when a non-linear form of average is used -- the present paper shows a hypothetical example for a harmonic mean. There are many questions that require non-linear averaging. One such is life expectancy, that is a weighted function of the usual (age-specific) rates, but the weighting is nonlinear. The authors came on this paradox in studying life expectancy for women in the Soviet Union as projected to the year 2020. All of the republics fall between 77.713 and 78.026, but the figure for the USSR comes out to 77.632.

This is not an error due to rounding; it is not due to Simpson's paradox that would result from internal heterogeneity in the several republics; it is due to the nonlinear weighting implicit in the calculation of the life expectancy.

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