Arbuthnot and the Human Sex Ratio

R. B. Campbell

Department of Mathematics

University of Northern Iowa

Cedar Falls, IA 50614-0506



Abstract: John Arbuthnot's 1629-1710 data manifests more deviation in the human sex ratio than would occur by chance. This cannot be explained by the overall mortality rate, but there is a significant association with the birth rate, and it is conjectured that knowledge of the particular diseases affecting the birth rate during those years would provide a fuller explanation.


John Arbuthnot’s 1710 paper ‘An Argument for Divine Providence’, based on the Christening records in London for 1629-1710, is often cited as the first paper in inferential statistics (Bellhouse 1989, Eisenhart and Birnbaum 1967), because of his use of a sign test to show that more males are born than females. However, he also makes other conclusions which do not follow from the data including that there is less variation in the sex ratio than would occur by chance, and that "Polygamy is contrary to the Law of Nature and Justice". This investigation is concerned with the fact that his data shows more variation in the sex ratio than would occur by chance, which is known but not explained (Anscombe 1981, Hald 1990).


The data is the Christening records for 1629-1710 used by Arbuthnot (1710), supplemented by the mortality data for London for the same period (Creighton 1965). Sunspot data for the years 1629-1710 (Eddy 1976) is also used since sunspots explain almost everything, even though sunspot data for those years is sparse. The number of births ranges from 5612 in 1650 to 16145 in 1705. (The low numbers near 1650 may reflect failure to Christen children due to the political and religious climate in England at that time.) The number of deaths ranges from 8393 in 1633 to 24620 in 1710. The proportion of males rather than the sex ratio is used in our analysis.


All analyses were done with Minitab. The variance in the proportion of males was calculated and compared to the expected variance. The coefficient of determination (square of the correlation) was calculated for the proportion of males versus number of births, number of deaths, number of deaths due to plague, fever, smallpox, measles, and griping of the guts, and sunspot number, in order to explain the excess variation. The adjusted coefficient of determination, which corrects for random associations, is reported.






For the 82 year period 1629-1710, the variance of the proportion of males is .00005, which is twice what would occur by chance for a population of size 10000. The strongest association, as measured by the coefficient of determination, for these years is r˛=.11 between the proportion of males and number of births; the coefficient of determination with number of deaths is only .03; furthermore, there is a positive correlation between the number of births and number of deaths. The coefficient of determination between the proportion of males and number of sunspots is .03.

Because there might be a sex bias associated with the low numbers of Christenings in the two decades prior to and including 1660, the data was reanalyzed omitting those years. For the remaining 62 years, the variance of the proportion of males was reduced to .00004, but the coefficient of determination with the number of births was only reduced to .08.

For the 26 year span 1661-1686 mortality data is available for several diseases. The variance of the proportion of males is .00005 for that period. The non-zero coefficients of determination for that period are .22 with the number of births and .07 with mortality due to smallpox.


Genetic causes (Edwards 1962) cannot explain the temporal variation in the sex ratio, since the genetic composition of the population should not change that rapidly. Hald (1990) has attributed the excess variation to errors in the data, but that would require systematic error. Disease is a likely explanation for the variation, since epidemics were rampant at that point in time. The positive correlation of the birth and death rates suggests that diseases affect prenatal and adult mortality differently. The association with the number of births only accounts for one-fifth of the excess variation in the proportion of males (more for the years 1661-1686), but it is conjectured that if the prevalence of more specific diseases were known, more of the excess variation in the sex ratio could be explained.


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