Pearson's correlation coefficient and the chi-square test

1896–1900 Number and measure method

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A formula for Galton's correlation: the product-moment coefficient r, and a whole system that grew up around it, the standard deviation, the method of moments, a family of skew curves, and in 1900 the chi-square test, which shows how well observations agree with a hypothesis.

The formula and the system were given in 1896 by the Englishman Karl Pearson, professor of applied mathematics at University College London. The 'best value' of r he derived by what would now be called the method of maximum likelihood. All of it was done for Darwinian questions: to measure variation, selection and inheritance in samples of thousands of individuals.

The lane is math because this is a method, and life because it was born of a biological problem. Historians argue over who the main source was: Galton with 'Natural Inheritance' or the zoologist Weldon with his questions.

Why this date

The date follows K. Pearson, 'Mathematical Contributions to the Theory of Evolution. III. Regression, Heredity, and Panmixia', Philosophical Transactions of the Royal Society A 187 (1896), pp. 253-318 (checked against Wikidata Q56143683, DOI 10.1098/rsta.1896.0007). The upper bound of the label is the chi-square test, printed in the Philosophical Magazine in July 1900. Magnello stresses that the impulse came not only from Galton but from Weldon's crab data of 1892, and that the standard deviation and the method of moments came before 1896; the axis takes the paper in which the coefficient itself was first derived.

What it grew from

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

  influenced · Confidence: medium

  Pearson built his statistics in direct argument with the conviction that every distribution must be normal, and that conviction Magnello traces to Quetelet's determinism. Hence the family of skew curves and the goodness-of-fit test: Pearson looked for a way of describing data without forcing them into the normal curve, 'as Galton and Quetelet had done'.

  Dispute: This is influence by opposition: Magnello describes Quetelet as the source of a view that Pearson was overcoming, not as the source of his method. The text read does not assert that Pearson read Quetelet directly, hence the medium confidence.

  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

    The Belgian statistician, Adolphe Quetelet (1796-1874) attached so much significance to the normal curve because of his belief in determinism, which meant that there was an ideal statistical mean value and that the normal curve was the ideal curve, since it followed the law of errors; hence, all the variation around the mean had to conform to this curve. That Pearson would go on to create a new type of statistics was to a large extent in response to the unshakable conviction, held by so many vital statisticians, mathematicians and philosophers, that the normal distribution was the only feasible distribution for the analysis and interpretation of statistical data.

  • 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

    Pearson wanted to find another way to interpret the data without trying to normalise it, as Galton and Quetelet had done.

  "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.

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

  influenced · Confidence: medium

  Magnello asserts that Pearsonian mathematical statistics was built on Darwin's understanding of a species as a set of 'statistical populations' with individual variation, and that Pearson saw in every Darwinian notion, from variation to inheritance, a demand for statistical analysis.

  Dispute: In Magnello the Darwinian influence on Pearson runs not directly from the book but through the zoologist Weldon, who brought Pearson the Darwinian questions and the Naples crab data. A direct reading of the Origin of Species as the impulse to the method is not asserted here, hence the medium confidence.

  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

    Pearsonian mathematical statistics was thus built upon Charles Darwin’s recognition that species comprised different sets of ‘statistical populations’ underpinned by individual variation.

  • 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

    Pearson recognized the fundamental statistical concepts in Darwin’s work, as ‘every idea of Darwin, from variation, natural selection, inheritance to reversion, seemed to demand statistical analyses’.

  "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.

• Regression to the mean and correlation: heredity as statistics

  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.

• The method of least squares

  influenced · Confidence: medium

  Pearson fitted regression curves by least squares, but his problem was different from Legendre's and Gauss's. Eisenhart puts it as an extension beyond the original formulation: in the classical problem one variable was measured with negligible error and the other with error, whereas in the Galton-Pearson situation both traits are measurable with negligible error in a single individual but have a joint distribution in a population.

  Dispute: Eisenhart stresses the difference of the two formulations rather than a borrowing: Pearson himself preferred the method of moments and, in Eisenhart's words, 'Pearson clearly favored his method of moments; but the method of least squares has prevailed'. The link is therefore drawn as "influenced" with medium confidence: least squares gave the language and the computational technique but was not a prerequisite for correlation.

  Link sources

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

    However, use of the method of least squares to fit polynomial regression curves in a bivariate correlation situation involves an extension beyond the original formulation and development of the method of least squares by Legendre, Gauss, Laplace, and their followers in the nineteenth century.

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

    In this classical development of the method of least squares, one of the variables-x, for example—was a quantity that could be measured with negligible error, and the other, y, a quantity of interest functionally related to x, the observed values of which for particular values of x, Yx, were, however, subject to nonnegligible measurement errors.

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

    In the Galton-Pearson correlation situation, in contrast, the traits X and Y may both be measurable with negligible error with respect to any single individual but in some population of individuals have a joint frequency or probability distribution.

  "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

• The journal Biometrika: the statistics of life gets a journal of its own

  influenced · Confidence: high

  The journal was founded so that statistical work of this kind could be published at all. In December 1900 Pearson wrote to Galton that Bateson's report applied not to that one memoir but to all his work and to any work 'on similar statistical lines', and called it a practical notice to quit; hence Weldon's proposal to start a journal and the title Pearson supplied.

  Link sources

  • Churchill Eisenhart, «Pearson, Karl», Complete Dictionary of Scientific Biography (2008) Verified 5 Oct 2026

    Pearson was enthusiastically in favor of the idea-on 13 December 1900 he wrote to Galton that Bateson’s adverse criticism “did not apply to this memoir only but to all my work,. . . if the R. S. people send my papers to Bateson, one cannot hope to get them printed. It is a practical notice to quit. This notice applies not only to my work, but to most work on similar statistical lines.”

  • Churchill Eisenhart, «Pearson, Karl», Complete Dictionary of Scientific Biography (2008) Verified 5 Oct 2026

    Pearson replied with the suggestion that “the science in future should be called Biometry and its official organ be Biometrika

  • M. Eileen Magnello, Karl Pearson and the Origins of Modern Statistics: An Elastician becomes a Statistician, The Rutherford Journal 1 (2005) Verified 5 Oct 2026

    Pearson also established the professional accoutrements necessary to establish and to institutionalise the new discipline of mathematical statistics: with Weldon he founded Biometrika in November 1900, he established the Drapers’ Biometric Laboratory in 1903, and set up the first-ever degree course in mathematical statistics in 1917.

  "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.

• The correlation between relatives on the supposition of Mendelian inheritance: biometry and Mendelism reconciled

  influenced · Confidence: high

  Fisher was computing precisely those correlations between relatives that were measured with Pearson's coefficients, and he built on models which Pearson, Weldon and Yule had been considering ten to twenty years before him. Visscher and Goddard say outright: 'Fisher built upon previous work by Yule and Pearson'.

  Dispute: The relations of Pearson and Fisher were hostile: according to Eisenhart, in 1920 Pearson returned to Fisher the manuscript on the distribution of the correlation coefficient, and the work appeared in another journal. Continuity of method and personal conflict do not coincide here, and that is worth remembering when reading the link.

  Link sources

  • Peter M. Visscher, Michael E. Goddard, From R.A. Fisher's 1918 Paper to GWAS a Century Later, Genetics 211 (4), 1125-1130 (2019)

    However, its purpose was clearly set out in the introductory paragraph, namely, to interpret empirical results from biometry, in particular the correlations between relatives, in accordance with Mendelian inheritance, to provide a more precise analysis of the causes of variation in human complex traits. Fisher built upon previous work by Yule and Pearson.

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

    he improved upon the kinds of models that Pearson, Weldon, and Yule had been considering 10–20 years before, and showed clearly that the correlations observed between human relatives not only could be interpreted on the supposition of Mendelian inheritance, but also that Mendelian inheritance must lead to precisely the kind of correlations observed.

  "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

• 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,

• 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

  He continued to work on improving this method throughout the 1890s until he devised his chi-square goodness of fit test in 1900.

• 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

  Pearson had just devised the standard deviation in 1892 when Weldon approached him for assistance because he found that one of his distributions of data was bimodal, while the rest of his data were normally distributed.

• K. Pearson, Mathematical Contributions to the Theory of Evolution. III. Regression, Heredity, and Panmixia, Philosophical Transactions of the Royal Society of London A 187, 253-318 (1896)

  Mathematical Contributions to the Theory of Evolution. III. Regression, Heredity, and Panmixia

"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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