Basic forms of statistics have been used since the beginning of civilization. Early empires often collated censuses of the population or recorded the trade in various commodities. The
Han dynasty and the
Roman Empire were some of the first states to extensively gather data on the size of the empire's population, geographical area and wealth. The use of statistical methods dates back to at least the 5th century BCE. The historian
Thucydides in his
History of the Peloponnesian War describes how the Athenians calculated the height of the wall of
Platea by counting the number of bricks in an unplastered section of the wall sufficiently near them to be able to count them. The count was repeated several times by a number of soldiers. The most frequent value (in modern terminology – the
mode) so determined was taken to be the most likely value of the number of bricks. Multiplying this value by the height of the bricks used in the wall allowed the Athenians to determine the height of the ladders necessary to scale the walls. The
Trial of the Pyx is a test of the purity of the coinage of the
Royal Mint which has been held on a regular basis since the 12th century. The Trial itself is based on statistical sampling methods. After minting a series of coins – originally from ten pounds of silver – a single coin was placed in the Pyx – a box in
Westminster Abbey. After a given period – now once a year – the coins are removed and weighed. A sample of coins removed from the box are then tested for purity. The
Nuova Cronica, a 14th-century
history of Florence by the Florentine banker and official
Giovanni Villani, includes much statistical information on population, ordinances, commerce and trade, education, and religious facilities and has been described as the first introduction of statistics as a positive element in history, though neither the term nor the concept of statistics as a specific field yet existed. The arithmetic
mean, although a concept known to the Greeks, was not generalised to more than two values until the 16th century. The invention of the decimal system by
Simon Stevin in 1585 seems likely to have facilitated these calculations. This method was first adopted in astronomy by
Tycho Brahe who was attempting to reduce the errors in his estimates of the locations of various celestial bodies. The idea of the
median originated in
Edward Wright's book on navigation (
Certaine Errors in Navigation) in 1599 in a section concerning the determination of location with a compass. Wright felt that this value was the most likely to be the correct value in a series of observations. The difference between the mean and the median was noticed in 1669 by Chistiaan Huygens in the context of using Graunt's tables. , a 17th-century economist who used early statistical methods to analyse demographic data The term 'statistic' was introduced by the Italian scholar
Girolamo Ghilini in 1589 with reference to this science. The birth of statistics is often dated to 1662, when
John Graunt, along with
William Petty, developed early human statistical and
census methods that provided a framework for modern
demography. He produced the first
life table, giving probabilities of survival to each age. His book
Natural and Political Observations Made upon the Bills of Mortality used analysis of the
mortality rolls to make the first statistically based estimation of the population of
London. He knew that there were around 13,000 funerals per year in London and that three people died per eleven families per year. He estimated from the parish records that the average family size was 8 and calculated that the population of London was about 384,000; this is the first known use of a
ratio estimator.
Laplace in 1802 estimated the population of France with a similar method; see for details. Although the original scope of statistics was limited to data useful for governance, the approach was extended to many fields of a scientific or commercial nature during the 19th century. The mathematical foundations for the subject heavily drew on the new
probability theory, pioneered in the 16th century by
Gerolamo Cardano,
Pierre de Fermat and
Blaise Pascal.
Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject.
Jakob Bernoulli's
Ars Conjectandi (posthumous, 1713) and
Abraham de Moivre's
The Doctrine of Chances (1718) treated the subject as a branch of mathematics. In his book Bernoulli introduced the idea of representing complete certainty as one and probability as a number between zero and one. In 1700,
Isaac Newton carried out the earliest known form of
linear regression, writing the first of the
ordinary least squares normal equations, averaging astronomical data, and summing the residuals to zero in his analysis of
Hipparchus’s equinox observations. He distinguished between two inhomogeneous sets of data and might have thought of an optimal solution in terms of bias, but not in effectiveness. A key early application of statistics in the 18th century was to the
human sex ratio at birth.
John Arbuthnot studied this question in 1710. Arbuthnot examined birth records in London for each of the 82 years from 1629 to 1710. In every year, the number of males born in London exceeded the number of females. Considering more male or more female births as equally likely, the probability of the observed outcome is 0.5^82, or about 1 in 4,8360,0000,0000,0000,0000,0000; in modern terms, the
p-value. This is vanishingly small, leading Arbuthnot that this was not due to chance, but to divine providence: "From whence it follows, that it is Art, not Chance, that governs." This is and other work by Arbuthnot is credited as "the first use of
significance tests" the first example of reasoning about
statistical significance and moral certainty, and "... perhaps the first published report of a
nonparametric test ...", de Moivre was studying the number of heads that occurred when a 'fair' coin was tossed. In 1763 Richard Price transmitted to the Royal Society
Thomas Bayes proof of a rule for using a binomial distribution to calculate a posterior probability on a prior event. In 1765
Joseph Priestley invented the first
timeline charts.
Johann Heinrich Lambert in his 1765 book
Anlage zur Architectonic proposed the
semicircle as a distribution of errors: : f(x) = \frac{ 1 }{ 2 } \sqrt{ ( 1 - x^2 ) } with -1 r, the "probable error" of a single observation was widely used and inspired early
robust statistics (resistant to
outliers: see
Peirce's criterion). In the 19th century authors on
statistical theory included Laplace,
S. Lacroix (1816), Littrow (1833),
Dedekind (1860), Helmert (1872),
Laurent (1873), Liagre, Didion,
De Morgan and
Boole.
Gustav Theodor Fechner used the median (
Centralwerth) in sociological and psychological phenomena. It had earlier been used only in astronomy and related fields.
Francis Galton used the English term
median for the first time in 1881 having earlier used the terms
middle-most value in 1869 and the
medium in 1880.
Adolphe Quetelet (1796–1874), another important founder of statistics, introduced the notion of the "average man" (''l'homme moyen'') as a means of understanding complex social phenomena such as
crime rates,
marriage rates, and
suicide rates. The first tests of the normal distribution were invented by the German statistician
Wilhelm Lexis in the 1870s. The only data sets available to him that he was able to show were normally distributed were birth rates.
Development of modern statistics Although the origins of statistical theory lie in the 18th-century advances in probability, the modern field of statistics only emerged in the late-19th and early-20th century in three stages. The first wave, at the turn of the century, was led by the work of
Francis Galton and
Karl Pearson, who transformed statistics into a rigorous mathematical discipline used for analysis, not just in science, but in industry and politics as well. The second wave of the 1910s and 20s was initiated by
William Sealy Gosset, and reached its culmination in the insights of
Ronald Fisher. This involved the development of better
design of experiments models, hypothesis testing and techniques for use with small data samples. The final wave, which mainly saw the refinement and expansion of earlier developments, emerged from the collaborative work between
Egon Pearson and
Jerzy Neyman in the 1930s. Today, statistical methods are applied in all fields that involve decision making, for making accurate inferences from a collated body of data and for making decisions in the face of uncertainty based on statistical methodology. , founded in 1834 The first statistical bodies were established in the early 19th century. The
Royal Statistical Society was founded in 1834 and
Florence Nightingale, its first female member, pioneered the application of statistical analysis to health problems for the furtherance of epidemiological understanding and public health practice. However, the methods then used would not be considered as modern statistics today. The
Oxford scholar
Francis Ysidro Edgeworth's book,
Metretike: or The Method of Measuring Probability and Utility (1887) dealt with probability as the basis of inductive reasoning, and his later works focused on the 'philosophy of chance'. His first paper on statistics (1883) explored the law of error (
normal distribution), and his
Methods of Statistics (1885) introduced an early version of the
t distribution, the
Edgeworth expansion, the
Edgeworth series, the method of variate transformation and the asymptotic theory of maximum likelihood estimates. The Norwegian
Anders Nicolai Kiær introduced the concept of
stratified sampling in 1895.
Arthur Lyon Bowley introduced new methods of data sampling in 1906 when working on social statistics. Although statistical surveys of social conditions had started with
Charles Booth's "Life and Labour of the People in London" (1889–1903) and
Seebohm Rowntree's "Poverty, A Study of Town Life" (1901), Bowley's key innovation consisted of the use of
random sampling techniques. His efforts culminated in his
New Survey of London Life and Labour.
Francis Galton is credited as one of the principal founders of statistical theory. His contributions to the field included introducing the concepts of
standard deviation,
correlation,
regression and the application of these methods to the study of the variety of human characteristics – height, weight, eyelash length among others. He found that many of these could be fitted to a normal curve distribution. Galton submitted a paper to
Nature in 1907 on the usefulness of the median. He examined the accuracy of 787 guesses of the weight of an ox at a country fair. The actual weight was 1208 pounds: the median guess was 1198. The guesses were markedly non-normally distributed (cf.
Wisdom of the Crowd). , the founder of
mathematical statistics Galton's publication of
Natural Inheritance in 1889 sparked the interest of a brilliant mathematician,
Karl Pearson, then working at
University College London, and he went on to found the discipline of mathematical statistics. He emphasised the statistical foundation of scientific laws and promoted its study and his laboratory attracted students from around the world attracted by his new methods of analysis, including
Udny Yule. His work grew to encompass the fields of
biology,
epidemiology, anthropometry,
medicine and social
history. In 1901, with
Walter Weldon, founder of
biometry, and Galton, he founded the journal
Biometrika as the first journal of mathematical statistics and biometry. His work, and that of Galton, underpins many of the 'classical' statistical methods which are in common use today, including the
Correlation coefficient, defined as a product-moment; the
method of moments for the fitting of distributions to samples;
Pearson's system of continuous curves that forms the basis of the now conventional continuous probability distributions;
Chi distance a precursor and special case of the
Mahalanobis distance and
P-value, defined as the probability measure of the complement of the
ball with the hypothesized value as center point and chi distance as radius. In 1911 he founded the world's first university statistics department at
University College London. The second wave of mathematical statistics was pioneered by
Ronald Fisher who wrote two textbooks,
Statistical Methods for Research Workers, published in 1925 and
The Design of Experiments in 1935, that were to define the academic discipline in universities around the world. He also systematized previous results, putting them on a firm mathematical footing. In his 1918 seminal paper
The Correlation between Relatives on the Supposition of Mendelian Inheritance, the first use to use the statistical term,
variance. In 1919, at
Rothamsted Experimental Station he started a major study of the extensive collections of data recorded over many years. This resulted in a series of reports under the general title
Studies in Crop Variation. In 1930 he published
The Genetical Theory of Natural Selection where he applied statistics to
evolution. Over the next seven years, he pioneered the principles of the
design of experiments (see below) and elaborated his studies of analysis of variance. He furthered his studies of the statistics of small samples. Perhaps even more important, he began his systematic approach of the analysis of real data as the springboard for the development of new statistical methods. He developed computational algorithms for analyzing data from his balanced experimental designs. In 1925, this work resulted in the publication of his first book,
Statistical Methods for Research Workers. This book went through many editions and translations in later years, and it became the standard reference work for scientists in many disciplines. In 1935, this book was followed by
The Design of Experiments, which was also widely used. In addition to analysis of variance, Fisher named and promoted the method of
maximum likelihood estimation. Fisher also originated the concepts of
sufficiency,
ancillary statistics,
Fisher's linear discriminator and
Fisher information. His article
On a distribution yielding the error functions of several well known statistics (1924) presented
Pearson's chi-squared test and
William Sealy Gosset's
t in the same framework as the
Gaussian distribution, and his own parameter in the analysis of variance
Fisher's z-distribution (more commonly used decades later in the form of the
F distribution). The 5% level of significance appears to have been introduced by Fisher in 1925. Fisher stated that deviations exceeding twice the standard deviation are regarded as significant. Before this deviations exceeding three times the probable error were considered significant. For a symmetrical distribution the probable error is half the interquartile range. For a normal distribution the probable error is approximately 2/3 the standard deviation. It appears that Fisher's 5% criterion was rooted in previous practice. Other important contributions at this time included
Charles Spearman's
rank correlation coefficient that was a useful extension of the Pearson correlation coefficient.
William Sealy Gosset, the English statistician better known under his pseudonym of
Student, introduced
Student's t-distribution, a continuous probability distribution useful in situations where the sample size is small and population standard deviation is unknown.
Egon Pearson (Karl's son) and
Jerzy Neyman introduced the concepts of "
Type II" error, power of a test and
confidence intervals.
Jerzy Neyman in 1934 showed that stratified random sampling was in general a better method of estimation than purposive (quota) sampling. ==Design of experiments==