Historical Origins of Statistical Distributions

This page documents the fascinating historical origins of some of the most important statistical distributions used in biostatistics and data analysis.

Poisson Distribution

The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile (1837).

Normal Distribution

“I have been trying to track it down, with partial success, by starting with the writings of Sir Francis Galton. He wrote about the normal distribution at great length, and with an amazing range of elegant variation. Among the terms he used were”the exponential distribution,” “the law of error,” “the law of deviation,” “the Gaussian Law,” “the law of statistical constancy,” and, of course, “the normal law,” the last first used by Galton in 1877 as far as we can find.

Karl Pearson claimed at a later time that he himself had coined the term “normal” as a neutral word to avoid the chauvinistic, competing, nationalistic “Gaussian” and “Laplacian,” but it seems to us that his memory was wrong. Our current hunch is based on the observation that Galton and his contemporaries, when writing of the normal distribution, not only used terms like “the exponential distribution” but, to avoid monotony, brought in vaguer expressions: “the usual distribution,” “the commonly encountered distribution,” and, of course, “the normal distribution.” (Indeed, “normal” in that sense was used still earlier, in 1873, by Charles Sanders Peirce, the great American philosopher and statistician.)

We think that somehow “normal” won out among the synonyms. Why should it have won? I speculate that “normal” is a word with powerful, positive connotations because of the ambiguity between its two meanings: (1) something desirable and (2) something commonly found. A major theme in our culture, after all, is the desirability of what is commonly found, so the two senses reinforce each other.”

F Distribution

The “F” of Snedecor’s F distribution is named in honor of Sir Ronald Fisher.

Chi-Square Distribution

‘Then: X² = constant, is the equation which the frequency of the system values which X must be …’

Student’s t-Distribution

“Student” was the pseudonym of William Sealy Gosset (1876-1937). Gosset once wrote to R. A. Fisher, “I am sending you a copy of Student’s Tables as you are the only man that’s ever likely to use them!” The letter appears in Letters from W. S. Gosset to R. A. Fisher, 1915-1936 (1970). Student’s tables became very important in statistics but not in the form he first constructed them.

In his 1908 paper, “The Probable Error of a Mean,” Biometrika 6, 1-25, Gosset introduced the statistic, z, for testing hypotheses on the mean of the normal distribution. Gosset used the divisor n, not the modern (n - 1), when he estimated s and his z is proportional to t with t = z√(n - 1).

Fisher introduced the t form for it fitted in with his theory of degrees of freedom. Fisher’s treatment of the distributions based on the normal distribution and the role of degrees of freedom was given in “On a Distribution Yielding the Error Functions of Several well Known Statistics,” Proceedings of the International Congress of Mathematics, Toronto, 2, 805-813. The t symbol appears in this paper but although the paper was presented in 1924, it was not published until 1928 (Tankard, page 103; David, 1995). According to the OED2, the letter t was chosen arbitrarily. A new symbol suited Fisher for he was already using z for a statistic of his own (see entry for F).

Student’s distribution (without “t”) appears in 1925 in R. A. Fisher, “Applications of ‘Student’s’ Distribution,” Metron 5, 90-104 and in Statistical Methods for Research Workers (1925). The book made Student’s distribution famous; it presented new uses for the tables and made the tables generally available.

“Student’s” t-distribution appears in 1929 in Nature (OED2).

t-distribution appears (without Student) in A. T. McKay, “Distribution of the coefficient of variation and the extended ‘t’ distribution,” J. Roy. Stat. Soc., n. Ser. 95 (1932).

t-test is found in 1932 in R. A. Fisher, Statistical Methods for Research Workers: “The validity of the t-test, as a test of this hypothesis, is therefore absolute” (OED2).

Eisenhart (1979) is the best reference for the evolution of t, although Tankard and Hald also discuss it.

These historical notes highlight the evolution of statistical terminology and the contributions of pioneering statisticians.

--- title: "Historical Origins of Statistical Distributions" --- This page documents the fascinating historical origins of some of the most important statistical distributions used in biostatistics and data analysis. ## [Poisson Distribution](https://en.wikipedia.org/wiki/Poisson_distribution#History) The distribution was first introduced by [Siméon Denis Poisson](https://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson) (1781–1840) and published together with his probability theory in his work *Recherches sur la probabilité des jugements en matière criminelle et en matière civile* (1837). ## [Normal Distribution](https://www.jstor.org/stable/41210378?seq=2) "I have been trying to track it down, with partial success, by starting with the writings of Sir Francis Galton. He wrote about the normal distribution at great length, and with an amazing range of elegant variation. Among the terms he used were "the exponential distribution," "the law of error," "the law of deviation," "the Gaussian Law," "the law of statistical constancy," and, of course, "the normal law," the last first used by Galton in 1877 as far as we can find. Karl Pearson claimed at a later time that he himself had coined the term "normal" as a neutral word to avoid the chauvinistic, competing, nationalistic "Gaussian" and "Laplacian," but it seems to us that his memory was wrong. Our current hunch is based on the observation that Galton and his contemporaries, when writing of the normal distribution, not only used terms like "the exponential distribution" but, to avoid monotony, brought in vaguer expressions: "the usual distribution," "the commonly encountered distribution," and, of course, "the normal distribution." (Indeed, "normal" in that sense was used still earlier, in 1873, by Charles Sanders Peirce, the great American philosopher and statistician.) We think that somehow "normal" won out among the synonyms. Why should it have won? I speculate that "normal" is a word with powerful, positive connotations because of the ambiguity between its two meanings: (1) something desirable and (2) something commonly found. A major theme in our culture, after all, is the desirability of what is commonly found, so the two senses reinforce each other." ## [F Distribution](https://en.wikipedia.org/wiki/F-distribution) The "F" of [Snedecor's F distribution](https://en.wikipedia.org/wiki/Snedecor%27s_F_distribution) is named in honor of Sir [Ronald Fisher](https://en.wikipedia.org/wiki/Ronald_Fisher). ## [Chi-Square Distribution](https://www.economics.soton.ac.uk/staff/aldrich/1900.pdf) 'Then: X² = constant, is the equation which the frequency of the system values which X must be ...' ## [Student's t-Distribution](https://web.universiteitleiden.nl/fsw/verduin/stathist/1stword.htm) **"Student"** was the pseudonym of **William Sealy Gosset** (1876-1937). Gosset once wrote to R. A. Fisher, "I am sending you a copy of Student's Tables as you are the only man that's ever likely to use them!" The letter appears in *Letters from W. S. Gosset to R. A. Fisher, 1915-1936* (1970). Student's tables became very important in statistics but not in the form he first constructed them. In his 1908 paper, "The Probable Error of a Mean," *Biometrika* 6, 1-25, Gosset introduced the statistic, z, for testing hypotheses on the mean of the normal distribution. Gosset used the divisor n, not the modern (n - 1), when he estimated s and his z is proportional to t with t = z√(n - 1). Fisher introduced the t form for it fitted in with his theory of degrees of freedom. Fisher's treatment of the distributions based on the normal distribution and the role of degrees of freedom was given in "On a Distribution Yielding the Error Functions of Several well Known Statistics," *Proceedings of the International Congress of Mathematics, Toronto*, 2, 805-813. The t symbol appears in this paper but although the paper was presented in 1924, it was not published until 1928 (Tankard, page 103; David, 1995). According to the OED2, the letter t was chosen arbitrarily. A new symbol suited Fisher for he was already using z for a statistic of his own (see entry for F). Student's distribution (without "t") appears in 1925 in R. A. Fisher, "Applications of 'Student's' Distribution," *Metron* 5, 90-104 and in *Statistical Methods for Research Workers* (1925). The book made Student's distribution famous; it presented new uses for the tables and made the tables generally available. "Student's" t-distribution appears in 1929 in *Nature* (OED2). t-distribution appears (without Student) in A. T. McKay, "Distribution of the coefficient of variation and the extended 't' distribution," *J. Roy. Stat. Soc.*, n. Ser. 95 (1932). t-test is found in 1932 in R. A. Fisher, *Statistical Methods for Research Workers*: "The validity of the t-test, as a test of this hypothesis, is therefore absolute" (OED2). Eisenhart (1979) is the best reference for the evolution of t, although Tankard and Hald also discuss it. --- *These historical notes highlight the evolution of statistical terminology and the contributions of pioneering statisticians.*