Comparing population distributions from bin-aggregated sample data: An application to historical height data from France

Abstract

We develop a methodology to estimate underlying (continuous) population distributions from bin-aggregated sample data through the estimation of the parameters of mixtures of distributions that allow for maximal parametric flexibility. The statistical approach we develop enables comparisons of the full distributions of height data from potential army conscripts across France's 88 departments for most of the nineteenth century. These comparisons are made by testing for differences-of-means stochastic dominance. Corrections for possible measurement errors are also devised by taking advantage of the richness of the data sets. Our methodology is of interest to researchers working on bin-aggregated or histogram-type data, something that is still widely done since much of the information that is publicly available is in that form, often due to restrictions based on confidentiality concerns.

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“…Nevertheless, one must bear in mind that here we are comparing our data set (the individual heights of all the young men born in 1848 and still alive in 1869) with that of Weir (the median height of all of those deemed fit for service). It seems only logical that the median height of those conscripts deemed fit for service would have been greater than the average height of the entire birth cohort; one must not forget that Weir had to correct his data because of the minimum-height requirement, which may have made for underestimation, and above all that the aggregated data used by Weir are apparently unreliable: ''the observed distribution diverges, sometimes to a ridiculous extent, from the bell curve (Bassino and Dormois 2009, p. 113; see also Duclos et al 2011), whereas in our data, the distribution of the individual…”