Editing Francis Galton (section)

=== Model for population stability ===
[[File:Sir Francis Galton, 1890s.jpg|thumb|left|Sir Francis Galton, 1890s]]
Galton's formulation of regression and its link to the bivariate normal distribution can be traced to his attempts at developing a mathematical model for population stability. Although Galton's first attempt to study Darwinian questions, ''Hereditary Genius'', generated little enthusiasm at the time, the text led to his further studies in the 1870s concerning the inheritance of physical traits.{{sfn|Stigler|2010|pp=469–482}} This text contains some crude notions of the concept of regression, described in a qualitative matter. For example, he wrote of dogs: "If a man breeds from strong, well-shaped dogs, but of mixed pedigree, the puppies will be sometimes, but rarely, the equals of their parents. They will commonly be of a mongrel, nondescript type, because ancestral peculiarities are apt to crop out in the offspring."{{sfn|Galton|1914|p=57}}

This notion created a problem for Galton, as he could not reconcile the tendency of a population to maintain a normal distribution of traits from generation to generation with the notion of inheritance. It seemed that a large number of factors operated independently on offspring, leading to the normal distribution of a trait in each generation. However, this provided no explanation as to how a parent can have a significant impact on his offspring, which was the basis of inheritance.{{sfn|Stigler|1986|pp=265–299}}

Galton's solution to this problem was presented in his Presidential Address at the September 1885 meeting of the [[British Association for the Advancement of Science]], for he was serving at the time as President of Section H: Anthropology.<ref name=":3">{{cite journal |last=Galton |first=Francis |title=Opening address as President of the Anthropology Section of the British Association for the Advancement of Science, September 10th, 1885, at Aberdeen |journal=Nature |volume=32 |pages=507–510 |year=1885}}</ref> The address was published in ''[[Nature (journal)|Nature]]'', and Galton further developed the theory in "Regression toward mediocrity in hereditary stature" and "Hereditary Stature".{{sfn|Galton|1886|pp=246–263}}{{sfn|Galton|1886b|pp=295–298}} An elaboration of this theory was published in 1889 in ''Natural Inheritance''. There were three key developments that helped Galton develop this theory: the development of the law of error in 1874–1875, the formulation of an empirical law of reversion in 1877, and the development of a mathematical framework encompassing regression using human population data during 1885.{{sfn|Stigler|1986|pp=265–299}}

Galton's development of the law of regression to the mean, or reversion, was due to insights from the [[Galton board]] ('bean machine') and his studies of sweet peas. While Galton had previously invented the quincunx prior to February 1874, the 1877 version of the quincunx had a new feature that helped Galton demonstrate that a normal mixture of normal distributions is also normal.{{sfn|Galton|1877|pp=492–495, 512–514, 532–533}} Galton demonstrated this using a new version of quincunx, adding chutes to the apparatus to represent reversion. When the pellets passed through the curved chutes (representing reversion) and then the pins (representing family variability), the result was a stable population. On Friday 19 February 1877, Galton gave a lecture entitled ''Typical Laws of Heredity'' at the [[Royal Institution]] in London.{{sfn|Galton|1877|pp=492–495, 512–514, 532–533}} In this lecture, he posited that there must be a counteracting force to maintain population stability. However, this model required a much larger degree of intergenerational natural selection than was plausible.{{sfn|Stigler|2010|pp=469–482}}

In 1875, Galton began growing sweet peas, and addressed the Royal Institution on his findings on 9 February 1877.{{sfn|Galton|1877|pp=492–495, 512–514, 532–533}} He found that each group of progeny seeds followed a normal curve, and the curves were equally disperse. Each group was not centred on the parent's weight, but rather at a weight closer to the population average. Galton called this reversion, as every progeny group was distributed at a value that was closer to the population average than the parent. The deviation from the population average was in the same direction, but the magnitude of the deviation was only one-third as large. In doing so, he demonstrated that there was variability among each of the families, yet the families combined to produce a stable, normally distributed population. When he addressed the British Association for the Advancement of Science in 1885, he said of his investigation of sweet peas, "I was then blind to what I now perceive to be the simple explanation of the phenomenon."<ref name=":3" />

Galton was able to further his notion of regression by collecting and analysing data on human stature. Galton asked for help of mathematician J. Hamilton Dickson in investigating the geometric relationship of the data. He determined that the regression coefficient did not ensure population stability by chance, but rather that the regression coefficient, conditional variance, and population were interdependent quantities related by a simple equation.{{sfn|Stigler|1986|pp=265–299}} Thus Galton identified that the linearity of regression was not coincidental but rather was a necessary consequence of population stability.

The model for population stability resulted in Galton's formulation of the Law of Ancestral Heredity. This law, which was published in ''Natural Inheritance'', states that the two parents of an offspring jointly contribute one half of an offspring's heritage, while the other, more-removed ancestors constitute a smaller proportion of the offspring's heritage.{{sfn|Bulmer|1998|pp=579–585}} Galton viewed reversion as a spring, that when stretched, would return the distribution of traits back to the normal distribution. He concluded that evolution would have to occur via discontinuous steps, as reversion would neutralise any incremental steps.{{sfn|Gillham|2001b|pp=1383–1392}} When [[Mendelian inheritance|Mendel's]] principles were rediscovered in 1900, this resulted in a fierce battle between the followers of Galton's Law of Ancestral Heredity, the biometricians, and those who advocated Mendel's principles.{{sfn|Gillham|2013|pp=61–75}}