Regression to the mean and correlation: heredity as statistics

1886–1889 Number and measure method

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A device in which the trait in children is on average closer to the population mean than in their parents: children of tall parents are shorter than the parents, and children of short parents are taller. The slope of that line is measurable and can serve as a measure of the association of two quantities (co-relation).

The device was proposed in 1886 by the Englishman Francis Galton, Darwin's cousin, who measured the heights of parents and adult children. In 1888 he realised that the same procedure measures the association of any two quantities, for instance forearm length and height. 'Natural Inheritance' of 1889 gathered this into a book on which the zoologist Raphael Weldon and the mathematician Karl Pearson were raised.

The lane is math, since this is a statistical method; life is added because it was devised for the question of inheritance. Galton did not use least squares and reckoned with medians and quartiles; the mathematical form of his coefficient was given by Pearson.

Why this date

The date follows F. Galton, 'Regression towards mediocrity in hereditary stature', Journal of the Anthropological Institute 15 (1886), pp. 246-263 (Wikidata Q55933668, DOI 10.2307/2841583; the paper was read at Aberdeen in 1885). The upper bound of the label: 'Co-relations and their measurement, chiefly from anthropometric data', Proceedings of the Royal Society of London 45 (1888), pp. 135-145 (Wikidata Q56083620), and the book 'Natural Inheritance', London, Macmillan, 1889, where it was all brought together. The sweet-pea experiments of about 1875 are not taken for the axis: according to MacTutor, Galton did not yet understand that he had found a general property of any two measurements.

What it grew from

• 'Sur l'homme': the astronomers' law of error applied to people

  influenced · Confidence: high

  Galton learned of the application of the 'Law of Error' to anthropometric measurements, height and chest circumference, precisely from Quetelet's work, which reached him through the writings of Spottiswoode and Herschel: so Chatterjee writes. Stigler specifies the mechanism: Spottiswoode used Quetelet's own procedure to fit a normal curve to the orientation of mountain ranges, and it was this work that first turned Galton to the normal law of error.

  Dispute: In his autobiography of 1908 Galton himself traced his acquaintance with the law of error to Spottiswoode rather than to Quetelet and, in Tankard's words, 'tended to minimize Quetelet's influence on him'. Stigler, quoting the same autobiographical reference, remarks that it speaks more for the keenness of Galton's eye than for the quality of Spottiswoode's work. Moreover Galton departed from Quetelet in substance: Quetelet's 'average man' is a type from which deviations are errors, whereas for Galton variation itself is the object of measurement.

  Link sources

  • Shoutir Kishore Chatterjee, Statistical Thought: A Perspective and History, Oxford University Press, 286 (2003) Verified 25 Sep 2026, quotation assembled from fragments

    Galton became acquainted with Quetelet's use of the 'Law of Error' for describing the variation of anthropometric measurements like those on height and chest circumference...through the writings of Spottiswoode and Herschel

  • Stephen M. Stigler, The History of Statistics: The Measurement of Uncertainty before 1900, Harvard University Press, 219 (1986) Verified 25 Sep 2026

    Spottiswoode attempted to use Quetelet's own procedure to fit a normal curve to the distribution of direction of orientation of eleven mountain ranges to see whether they corresponded to a common "type."

  • Stephen M. Stigler, The History of Statistics: The Measurement of Uncertainty before 1900, Harvard University Press, 219 (1986) Verified 25 Sep 2026

    this work of Spottiswoode that caught his eye and first interested Galton in the normal law of error (Galton, 1908, p. 304)

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.

• 'On the Origin of Species': natural selection as the mechanism of species formation

  influenced · Confidence: high

  Magnello asserts the link directly: as soon as Darwin introduced continuous variation into biology, his cousin Galton set about devising statistical methods to measure that variation. Stigler describes the same dependence more precisely in substance: Galton's Aberdeen address of 1885 solved a puzzle left in the Origin of Species, and multivariate analysis came out of that solution.

  Link sources

  • M. Eileen Magnello, Karl Pearson and the Origins of Modern Statistics: An Elastician becomes a Statistician, The Rutherford Journal 1 (2005) Verified 20 Sep 2026

    Once Darwin introduced the idea of continuous variation into biological discourse, his cousin, Francis Galton, began to devise statistical methods to measure this variation.

  • Stephen M. Stigler, Darwin, Galton and the Statistical Enlightenment, Journal of the Royal Statistical Society A 173 (3), 469-482 (2010)

    On September 10th, 1885, Francis Galton ushered in a new era of Statistical Enlightenment with an address to the British Association for the Advancement of Science in Aberdeen. In the process of solving a puzzle that had lain dormant in Darwin's Origin of Species, Galton introduced multivariate analysis and paved the way towards modern Bayesian statistics.

  • J. J. O'Connor, E. F. Robertson, Francis Galton, MacTutor History of Mathematics Archive, University of St Andrews (2003) Verified 20 Sep 2026

    Perhaps it was the publication of Charles Darwin's Origin of the species in 1859 which marked a change in direction of Galton's interests.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

What grew out of it

• Pearson's correlation coefficient and the chi-square test

  improved · Confidence: high

  Galton obtained correlation and regression by rough procedures, from medians and quartiles, and called the coefficient an 'index of co-relation'; Pearson gave the same notion a rigorous form by deriving the product-moment coefficient. Eisenhart says outright that the concepts and techniques originated with Galton and that it was Pearson who established the best value of the coefficient. This is a direct improvement, hence "improved" rather than "influenced".

  Dispute: Historians differ over Galton's weight. Eisenhart and MacTutor describe 'Natural Inheritance' as the book that turned Pearson around, to the point of Eisenhart's phrase 'Pearson became quite charmed by the concept and implications of Galton's “correlation”'. Magnello disputes this: on her account the decisive person for Pearson was Weldon, Pearson's first reaction to the book in March 1889 was actually cautious, and the grateful account was written in 1934, half a century later. She also recalls that Galton never used Pearson's coefficient.

  Link sources

  • Churchill Eisenhart, «Pearson, Karl», Complete Dictionary of Scientific Biography (2008) Verified 20 Sep 2026

    As statistical concepts and techniques of correlation and regression originated with Galton, who devised rudimentary arithmetical and graphical procedures (utilizing certain medians and quartiles of the data in hand) to derive sample values for his “regression” coefficient, or “index of co-relation,” r.

  • Churchill Eisenhart, «Pearson, Karl», Complete Dictionary of Scientific Biography (2008) Verified 20 Sep 2026

    It was Pearson who established, by what would now be termed the method of maximum likelihood, that the “best value of the correlation coefficient” (?) of a bivariate normal distribution is given by the sample product-moment coefficient of correlation,

  • Churchill Eisenhart, «Pearson, Karl», Complete Dictionary of Scientific Biography (2008) Verified 20 Sep 2026

    Natural Inheritance summed up Galton’s work on correlation and regression, concepts and techniques that he had discovered and developed as tools for measuring the influence of heredity; presented all that he contributed to their theory; and clearly reflected his recognition of their applicability and value in studies of all living forms.

  • J. J. O'Connor, E. F. Robertson, Francis Galton, MacTutor History of Mathematics Archive, University of St Andrews (2003) Verified 20 Sep 2026

    Karl Pearson read Natural inheritance and it had a profound influence on his thinking:-

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

Node sources

• Stephen M. Stigler, Francis Galton's Account of the Invention of Correlation, Statistical Science 4 (2), 73-79 (1989) Verified 20 Sep 2026

  Francis Galton's invention of correlation dates from late in the year 1888, and it arose when he recognized a common thread in three different scientific problems he was studying.

• J. J. O'Connor, E. F. Robertson, Francis Galton, MacTutor History of Mathematics Archive, University of St Andrews (2003) Verified 20 Sep 2026

  He made two-way plots of heights of parents and the heights of their adult children. He was able to draw the plots in such a way that the coefficient of regression became the slope of the regression line. In 1888 he also examined the size of two different organs from the same person and applied the methods he had been developing to study the degree of association of the sizes. He defined an index of correlation as a measure of the degree to which the two were related.

• J. J. O'Connor, E. F. Robertson, Francis Galton, MacTutor History of Mathematics Archive, University of St Andrews (2003) Verified 20 Sep 2026

  In 1889 Galton published Natural inheritance in which presented as summary of the work he had done on correlation and regression.

• Churchill Eisenhart, «Pearson, Karl», Complete Dictionary of Scientific Biography (2008) Verified 20 Sep 2026

  As statistical concepts and techniques of correlation and regression originated with Galton, who devised rudimentary arithmetical and graphical procedures (utilizing certain medians and quartiles of the data in hand) to derive sample values for his “regression” coefficient, or “index of co-relation,” r.

• F. Galton, Regression towards mediocrity in hereditary stature, Journal of the Anthropological Institute of Great Britain and Ireland 15, 246-263 (1886)

  Regression towards mediocrity in hereditary stature

"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

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