Timeline of Probability and Statistics
The mathematics of chance and the science of data over more than 360 years, from Pascal and Fermat's letters of 1654 and Cardano's dice treatise through Bayes, Gauss and Fisher to computer-age methods and the American Statistical Association's 2016 statement on p-values.
- Category
- Natural Sciences
- Tag
- mathematics
- Events
- 29 events 1654–2016
- Generated
- by claude-opus-5-5
1654: Pascal and Fermat solve the problem of points
Prompted by the French writer Antoine Gombaud, Chevalier de Méré, Blaise Pascal and Pierre de Fermat worked out in a series of letters how to divide fairly the stakes of a game of chance interrupted before its end. Reasoning from the ways the game could have continued rather than from its history, they reached the same answer by different methods; historians date the beginning of modern probability theory to their correspondence.
Wikipedia1657: Huygens publishes the first printed treatise on probability
Having heard in Paris in 1655 of the problems Blaise Pascal and Pierre de Fermat had discussed, Christiaan Huygens worked out his own solutions in "De ratiociniis in ludo aleae" ("On Reasoning in Games of Chance"), which his former tutor Frans van Schooten translated into Latin and printed. It gave rules for valuing a chance, the idea now called expected value, and Jacob Bernoulli later reprinted it with commentary in "Ars Conjectandi".
Wikipedia1662: Graunt analyzes London's bills of mortality
John Graunt, a London haberdasher, tabulated the city's weekly bills of mortality in "Natural and Political Observations Made upon the Bills of Mortality", estimating London's population, comparing causes of death and drawing up what is credited as the first life table. Regarded as a founding work of demography, it was followed in 1693 by Edmond Halley's life table from Breslau records supplied by the pastor Caspar Neumann.
Wikipedia1663: Cardano's 16th-century dice treatise published posthumously
Gerolamo Cardano's "Liber de ludo aleae" ("Book on Games of Chance"), written in the 1560s, was printed about a century later, after his death in 1576. Analyzing dice, it expressed chances as ratios of favorable to unfavorable outcomes, the earliest known attempt at a systematic treatment of probability, but it appeared only after Blaise Pascal, Pierre de Fermat and Christiaan Huygens had worked out the basic theory, so its influence was limited.
Wikipedia1713: Jacob Bernoulli's law of large numbers published
Published by Nicolaus Bernoulli eight years after the death of his uncle Jacob Bernoulli, "Ars Conjectandi" ("The Art of Conjecturing") proved the first version of the law of large numbers: in repeated independent trials, the observed frequency of an event comes arbitrarily close to its true probability, with a probability that can be made as near to 1 as desired. Bernoulli called it his "golden theorem".
Wikipedia1733: De Moivre finds the normal approximation to the binomial
In a privately printed Latin pamphlet, Abraham de Moivre, a French Huguenot mathematician living in London, showed that the probabilities for the number of heads in many coin tosses follow approximately the bell-shaped curve now called the normal distribution, the first version of the central limit theorem. He added the result to the 1738 second edition of his textbook "The Doctrine of Chances".
Wikipedia1763: Price presents Bayes's essay to the Royal Society
Thomas Bayes, a Presbyterian minister, died in 1761, his essay unpublished; Richard Price edited it and had it read to the Royal Society, which printed it in 1764. Taking all values of an unknown success probability as equally likely a priori, it showed how observed successes and failures give the chance that the value lies in any range, a special case of Bayes' theorem. Pierre-Simon Laplace, apparently unaware of it, extended the result in 1774.
Wikipedia1809: Gauss ties least squares to the normal distribution
In "Theoria motus corporum coelestium", his treatise on orbits, Carl Friedrich Gauss justified least squares by deriving the law of errors under which the arithmetic mean is the most probable value: the normal, or Gaussian, distribution. Adrien-Marie Legendre had published the method first, in 1805; Gauss's claim to have used it since 1795 began a priority dispute, and Robert Adrain in the United States derived the error law independently.
Wikipedia1810: Laplace derives a general central limit theorem
Pierre-Simon Laplace showed that the sum of a large number of independent random errors is approximately normally distributed under broad conditions, extending Abraham de Moivre's result for coin tosses; rigorous proofs came only around 1900, notably from Aleksandr Lyapunov. After reading Carl Friedrich Gauss, he used the theorem to justify least squares for large samples and gathered these results in "Théorie analytique des probabilités" (1812).
Wikipedia1835: Quetelet publishes his treatise on "social physics"
In "Sur l'homme et le développement de ses facultés, ou Essai de physique sociale", the Belgian astronomer Adolphe Quetelet developed the "average man", defined by mean values, that he had proposed in 1831, and sought statistical laws behind rates of crime, suicide and marriage. From 1844 he argued that traits such as height cluster around the mean along the normal curve. His "social physics" influenced Florence Nightingale and Francis Galton.
Wikipedia1885: Galton describes regression toward the mean
In an address printed in Nature, Francis Galton reported that adult children of very tall or short parents were on average closer to the population mean, a pattern he called "regression towards mediocrity" and had first reported for sweet-pea seeds in 1877. Fitting a line to it in an 1886 paper, he gave regression analysis its name. Galton, who coined the word "eugenics", introduced correlation in 1888, partly anticipated by Auguste Bravais.
Wikipedia1900: Pearson introduces the chi-squared test
Karl Pearson's chi-squared statistic measured how far observed counts depart from the counts a hypothesis predicts, and he computed the probability, labeled P, of a discrepancy at least that large arising by chance, the first formal p-value. A protégé of Francis Galton, Pearson co-founded the journal Biometrika in 1901 and in 1911 founded the first university statistics department, at University College London.
Wikipedia1906: Markov begins the study of Markov chains
To refute Pavel Nekrasov's claim that the law of large numbers requires independent events, Andrey Markov analyzed sequences in which each outcome depends only on the one before and proved that, under certain conditions, their averages still converge. In 1913 he applied such chains to vowels and consonants in Alexander Pushkin's "Eugene Onegin"; they became standard models in fields from physics and biology to finance and speech processing.
Wikipedia1908: Gosset publishes the t-distribution as "Student"
William Sealy Gosset, a chemist and brewer at Guinness in Dublin who worked with very small samples, published "The Probable Error of a Mean" in Biometrika under the pen name "Student", as Guinness let its scientists publish only without their own names or the company's. It gave the distribution behind the t-test, later reworked and popularized by Ronald Fisher; Jacob Lüroth had derived the same distribution in 1876 as a Bayesian result.
Wikipedia1922: Fisher introduces maximum likelihood estimation
In "On the Mathematical Foundations of Theoretical Statistics", Ronald Fisher set out maximum likelihood, which estimates unknown parameters by the values that make the observed data most probable, with the criteria of consistency, efficiency and sufficiency. He had first presented the procedure in 1912 and now severed it from inverse (Bayesian) probability; Carl Friedrich Gauss and Francis Edgeworth had used similar ideas earlier.
Wikipedia1925: Fisher publishes "Statistical Methods for Research Workers"
Ronald Fisher's "Statistical Methods for Research Workers", drawn from his work at the Rothamsted agricultural research station, gave experimenters worked examples and tables for significance tests, small-sample methods and the analysis of variance, and urged that field plots be arranged at random. Fisher called it "convenient" to take P = .05 as a limit for judging significance; the 5 percent level later hardened into a widespread convention.
Wikipedia1933: Neyman and Pearson define most powerful tests
Jerzy Neyman and Egon Pearson, Karl Pearson's son, who since 1928 had cast hypothesis testing as a choice between a null and an alternative hypothesis with two kinds of error, proved that for simple hypotheses the likelihood-ratio test is the most powerful test with a given rate of false rejection. Their frequentist, decision-oriented theory differed from Ronald Fisher's significance tests, which weigh evidence against a single hypothesis.
Wikipedia1933: Kolmogorov axiomatizes probability theory
In "Grundbegriffe der Wahrscheinlichkeitsrechnung" ("Foundations of the Theory of Probability"), Andrey Kolmogorov defined probability as a measure on a field of events, with six axioms now usually condensed into three: probabilities are non-negative, the certain event has probability 1, and probabilities of countably many mutually exclusive events add. Building on Henri Lebesgue's measure theory, it became the subject's standard foundation.
Wikipedia1934: Neyman introduces confidence intervals
In a paper on stratified sampling in surveys, read to the Royal Statistical Society, Jerzy Neyman introduced confidence intervals, ranges given by a procedure that captures an unknown quantity's true value in a stated proportion of repeated samples, crediting Ronald Fisher's fiducial argument with showing that such intervals need no prior probabilities. The paper helped make random sampling the basis of surveys; a general theory followed in 1937.
Wikipedia1945: Rao bounds how precisely parameters can be estimated
Calyampudi Radhakrishna Rao of the Indian Statistical Institute proved that, under regularity conditions, no unbiased estimator has variance below the reciprocal of the Fisher information, and that conditioning an estimator on a sufficient statistic never worsens it. Maurice Fréchet had published the bound in 1943, and Harald Cramér and David Blackwell found the results independently; the paper won Rao the 2023 International Prize in Statistics.
Wikipedia1948: Medical Research Council reports a randomized streptomycin trial
In the British Medical Journal, the Medical Research Council reported that the antibiotic streptomycin benefited patients with pulmonary tuberculosis in a trial that assigned them to the drug or to bed rest alone by random numbers kept in sealed envelopes, a scheme devised by the statistician Austin Bradford Hill. It is often cited as the first randomized controlled trial in medicine, a design whose modern form Hill is credited with conceiving.
Wikipedia1949: Metropolis and Ulam publish the Monte Carlo method
Nicholas Metropolis and Stanislaw Ulam published the first unclassified paper on the Monte Carlo method, which estimates quantities by repeated random sampling. Ulam conceived it in 1946 while working out the odds of winning at solitaire, and with John von Neumann applied it on early computers to neutron diffusion in nuclear weapons work at Los Alamos; Enrico Fermi had tried similar sampling in the 1930s but never published it.
Wikipedia1954: Savage grounds statistics in personal probability
In "The Foundations of Statistics", Leonard Jimmie Savage derived personal probabilities and utilities from axioms for consistent choices under uncertainty, extending Frank Ramsey's and Bruno de Finetti's view of probability as a degree of belief. Following Harold Jeffreys's "Theory of Probability" (1939), the book helped revive Bayesian statistics, a name dating from the 1950s, as a rival to the methods of Ronald Fisher and Jerzy Neyman.
Wikipedia1972: Cox introduces the proportional hazards model
In "Regression Models and Life-Tables", David Cox showed how to estimate the effects of treatments and other factors on the risk of an event such as death over time, even when some subjects are followed only part of the way, without specifying the baseline risk. With Edward Kaplan and Paul Meier's 1958 survival-curve estimator, the Cox model became a standard tool of medical research; it won Cox the first International Prize in Statistics.
Wikipedia1977: Tukey publishes "Exploratory Data Analysis"
In "Exploratory Data Analysis", John Tukey urged that data be explored with simple summaries and graphics, such as the box plot and the stem-and-leaf display, to suggest hypotheses before formal tests. His championing of exploratory analysis and robust summaries such as medians and quartiles encouraged the development of the S language at Bell Labs, which in turn inspired R.
Wikipedia1979: Efron introduces the bootstrap
In "Bootstrap Methods: Another Look at the Jackknife", first given as a lecture in 1977, Bradley Efron proposed estimating the uncertainty of almost any statistic by recomputing it on many samples drawn with replacement from the observed data. One of the first computer-intensive statistical techniques, the bootstrap replaced algebraic derivations with simulation, and it won Efron the 2019 International Prize in Statistics.
Wikipedia1990: Gelfand and Smith popularize Markov chain Monte Carlo
Alan Gelfand and Adrian Smith showed that Gibbs sampling, described for image analysis by Stuart and Donald Geman in 1984, could approximate by simulation the posterior distributions of Bayesian models too complex for exact calculation. Such Markov chain Monte Carlo (MCMC) methods, descended from the Metropolis algorithm devised at Los Alamos in 1953 and generalized by W. K. Hastings in 1970, then spread through mainstream statistics.
Wikipedia1995: Benjamini and Hochberg define the false discovery rate
Yoav Benjamini and Yosef Hochberg proposed controlling the false discovery rate, the expected proportion of false positives among the results declared significant, instead of the probability of even one false positive, with a simple procedure based on ranked p-values. Less conservative than older corrections, it was the first such alternative to gain broad acceptance, notably in genomics, where thousands of genes are tested at once.
Wikipedia2016: American Statistical Association issues statement on p-values
Amid reproducibility concerns, the American Statistical Association (ASA) issued six principles on p-values: they can indicate how incompatible data are with a specified model but do not measure the probability that a hypothesis is true or the size or importance of an effect, and conclusions should not rest only on passing a threshold. In 2021 an ASA presidential task force added that properly applied p-values increase the rigor of conclusions.
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