Weldon, Walter Frank Raphael

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This article Walter Frank Raphael Weldon was adapted from an original article by Eileen Magnello, which appeared in StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. The original article ([ StatProb Source], Local Files: pdf | tex) is copyrighted by the author(s), the article has been donated to Encyclopedia of Mathematics, and its further issues are under Creative Commons Attribution Share-Alike License'. All pages from StatProb are contained in the Category StatProb.

Walter Frank Raphael WELDON [1] and published by kind permission of John Wiley & Sons, Ltd.

b. 15 March 1860 - d. 13 April 1906

Summary. Zoologist W.F.R. Weldon's efforts to make biology more rigorous and quantitative provided strong stimulus to K. Pearson, and through him the Biometric movement and the laying of the foundations of much of modern statistics.

W.F.R. Weldon, born in Highgate (London), was the second child of the journalist and industrial chemist, Walter Weldon and his wife Anne Cotton. His father changed residences so frequently that Weldon's early education was desultory until he became a boarder in 1873 at Caversham near Reading. Weldon matriculated at University College London (UCL) in the autumn of 1876 with the intention of pursuing a medical career. During his time at UCL, he acquired a respectable knowledge of mathematics from the Danish mathematician, Olaus Henrici, and attended the lectures of the zoologist, E. Ray Lankester.. In the following year he transferred to Kings College, London and on 6 April 1878, he entered St. John's College, Cambridge as a bye-term student.

Once at Cambridge, he met the zoologist, Francis Maitland Balfour, and subsequently gave up his medical studies for zoology. In 1881, he gained a first-class degree in the natural science Tripos: in the autumn he left for the Naples Zoological Station to begin the first of his studies in marine biological organisms. Upon returning to Cambridge in 1882, he was appointed university lecturer in Invertebrate Morphology, and in the following year he married Florence Tebb. He became a founding member of the Marine Biological Station in Plymouth in 1884 and resided there until 1887. From 1887 until his death in 1906, Weldon's work was centred around the development of a fuller understanding of marine biological phenomena and, in particular, the examination of the relationship between various organs of crabs and shrimps to determine selective death rates in relation to the laws of growth. During his first five years at the Marine Biological Station, Weldon's investigations were directed to the study of classification, morphology and the development of Decapod crustacea. His only work on invertebrate morphology contained an account of the early stage of segmentation and the building of the layers of shrimp. Weldon was both a master of histological techniques and a powerful and accurate draughtsman. In 1889 he succeeded E. Ray Lankester in the Joddrell Chair of Zoology at University College London.

During this time Weldon read Francis Galton's Natural Inheritance. In this book Galton had shown that the frequency distributions of the average size of certain organs in man, plants and in moths were normally distributed. Similar investigations had been pursued by the Belgian statistician, Adolphe Quetelet, whose work was confined to ``civilised man". Weldon was interested in investigating the variations in organs in a species living in a wild state which acted upon natural selection and other destructive influences.

When Galton was writing on heredity in 1889, he had predicted that selection would not change the shape of the normal distribution; he expected that his frequency distributions would remain normally distributed in all cases whether or not animals were under the action of natural selection. Around this time, Weldon began to study the variation of four organs in the common shrimp (Crangon vulgaris) and he collected five samples from waters fairly distant from Plymouth. His statistical analysis which he published in 1890, confirmed Galton's prediction. Shortly after the paper was published, Weldon was elected a Fellow of the Royal Society.

During the Easter vacation of 1892, Weldon and his wife collected 23 measurements from 1000 adult female shore crabs (Carcinus m{\oe}nas) from Malta and the Bay of Naples. Weldon discovered that all but one of the 23 characters he measured in the Naples group were normally distributed; he found that this one character (the frontal breadth of the carapace) was instead a `double-humped' (i.e., bi-modal) curve. His first attempt to interpret the data involved breaking up the curve into two normal distributions as Galton had advocated. Weldon then approached Karl Pearson (who had been appointed as Professor of Mechanism and Applied Mathematics at UCL in 1884) for assistance with interpreting his data. At that time, Pearson was teaching applied mathematics to engineering students at UCL and was also giving his Gresham lectures of Geometry at Gresham College. (This ancient educational foundation, located in the City of London, offered lectures to members of the public on an annual basis.)

By the end of 1892 Pearson began to devise a probability system of curve fitting for Weldon's data, and he used this material in his Gresham lectures in the following year. From his analysis of Weldon's data, Pearson concluded that two separate species had arisen. Up until the middle of the nineteenth century species were defined in terms of types or essences. Charles Darwin's recognition that species comprised different sets of `statistical' populations rather than types or essences, prompted a reconceptualization of statistical populations by Pearson and Weldon. Moreover, this required the use of new statistical methods. Following Pearson's work on curve fitting of asymmetrical distributions, in his Gresham lecture of 23 November 1893, he devised a goodness of fit test for asymmetrical distributions for Weldon's data. These statistical innovations formed much of the basis of Pearson's work in the nineteenth century, and in 1900 Pearson devised the chi-square $({\chi}^2 , P)$ goodness of fit test (his single most important contributions to modern statistical theory). When Pearson was working out the mathematical properties of simple correlation and regression in 1896, Weldon suggested to Pearson the idea of a negative correlation. (Galton had used positive correlations only.) Pearson regarded Weldon as `one of the closest friends he ever had'. Their relationship could be characterised by an emotional and intellectual intimacy that engendered a symbiotic alliance. It is thus not surprising that one of the most extensive sets of letters in Pearson's archives are those of Weldon and his wife Florence, which consist of nearly 1,000 pieces of correspondence.

Though not a statistician by training, it was Weldon's interest in finding statistical tools to demonstrate empirical evidence of Darwin's theory of natural selection in marine organisms that provided the impetus to Pearson's development of the modern theory of statistics. Weldon also provided Pearson with the basis of a programme that underpinned the construction of his statistical innovations in the 1890s which, in turn, provided the infrastructure for Pearson's statistical developments in the twentieth century. Weldon's influence, which exceeded that of any other person in the emergence and development of Pearsonian statistics, arose from the following factors: he provided the stimulus for new statistical methods by asking Pearson biological and statistical questions that could be answered only by devising a new statistical approach, he offered continual moral support as well as encouragement; and he promulgated the Pearsonian corpus of statistics to `serious students of animal evolution' throughout the 1890s and until his death in 1906. Though some of the statistical work of John Venn and Francis Ysidro Edgeworth played a role in Pearson's early work on probability, there seems to have been no other person whose influence on the emergence and development of Pearsonian statistics was as rapid and immediate in its impact as that of Weldon.

When Weldon went up to Oxford in 1899, to take up the Linacre Chair of Comparative Anatomy, he carried on the biometric tradition by gathering a number of students who began to look for empirical evidence of natural selection acting upon various animals and plants. Despite Weldon's move, he and Pearson made arrangements to be together every year during the Easter and Christmas vacations and throughout the summer months. They continued their collaborative work investigating biometrical problems such as natural selection, inheritance and, in particular, Mendelian inheritance. These joint biometrical projects were pursued until Weldon's untimely death in Oxford in 1906. Weldon's death was for Pearson the single greatest loss in his life.


[1] Bourne, Gilbert Charles (1906). Walter Frank Raphael Weldon. (1860-1906). Dictionary of National Biography. Oxford University Press, 629-630.
[2] Cowan, Ruth Schwartz (1981). Walter Frank Raphael Weldon. (1860-1906). Dictionary of Scientific Biography, ed. Charles C. Gillispie, Vol. XIV, Charles Scribner's Sons, New York, 251-252.
[3] Magnello, M. Eileen (1996). 'Karl Pearson's Gresham lectures: W.F.R. Weldon, speciation and the origins of Pearsonian Statistics', British Journal for the History of Science, 29, 43-63.
[4] Magnello, M. Eileen (Forthcoming). 'Karl Pearson's mathematisation of inheritance. From Galton's ancestral heredity to Mendelian Genetics (1895-1909)', Annals of Science.
[5] Pearson, Karl (1906). Walter Frank Raphael Weldon. 1860-1906. Biometrika, 5, 1-52. Also reprinted in E.S. Pearson and Maurice Kendall, Studies in the History of Statistics and Probability, Volume 1, Griffin, London, 1970.

  1. Abridged version of an article in the Encyclopedia of Statistical Sciences (update)

Reprinted with permission from Christopher Charles Heyde and Eugene William Seneta (Editors), Statisticians of the Centuries, Springer-Verlag Inc., New York, USA.

How to Cite This Entry:
Weldon, Walter Frank Raphael. Encyclopedia of Mathematics. URL:,_Walter_Frank_Raphael&oldid=53085