@article{Vaupel_S8_15,
author = {Vaupel, James W. and Zhang, Zhen},
title={{The difference between alternative averages}},
journal = {Demographic Research},
volume = {S8},
number = {15},
pages = {419--428},
doi = {10.4054/DemRes.2012.27.15},
year = {2012},
abstract = {Background: Demographers have long been interested in how compositional change, e.g., change in age structure, affects population averages. 

Objective: We want to deepen understanding of how compositional change affects population averages. 

Results: The difference between two averages of a variable, calculated using alternative weighting functions, equals the covariance between the variable and the ratio of the weighting functions, divided by the average of the ratio. We compare weighted and unweighted averages and also provide examples of use of the relationship in analyses of fertility and mortality. 

Comments: Other uses of covariances in formal demography are worth exploring. },
URL = {https://www.demographic-research.org/special/8/15/},
eprint = {https://www.demographic-research.org/special/8/15/s8-15.pdf}
}