The origins of both probability and statistics are usually traced to the 17

**Histories of many of terms used in probability and statistics can be found on the companion
Words of Mathematics page. The list of such words that are discussed
on that page can be found
here.**

**Combinatorial analysis**: Many of the symbols of elementary combinatorial analysis found in modern probability and statistics books were created in the 19^{th}century. In 19^{th}century Britain probability was most visible in algebra textbooks as an application of combinatorial analysis.**The normal distribution**, also known as the Gaussian distribution, the second law of Laplace, the law of error ... , has been studied since the 18^{th}century and many people have left their tracks on the notation.**Probability**: At the turn of the 20^{th}century there was a revival of interest in probability in continental Europe. The central limit theorem was one of their main concerns. The main contributors were Russian and French and they created much of the modern terminology and notation around 1930.**Statistics**: None of the notation used by Laplace and Gauss and their followers has survived into modern statistics. The oldest notation still in use comes from the period 1890-1940 when the British biometrician/statisticians Karl Pearson and R. A. Fisher introduced many of the basic symbols and many of the principles for constructing new ones.

The languages of English statistics and continental European probability came together in the 1940s.

For a sketch of the history of probability and statistics and notes on some of the key people see Figures from the History of Probability and Statistics (405 Kb).

The notation *n*! was introduced by Christian Kramp (1760-1826)
in 1808.
In his *Élémens d'arithmétique universelle*
(1808), Kramp wrote:

Je me sers de la notation trés simpleIn "Mémoire sur les facultés numériques," published in J. D. Gergonne'sn! pour désigner le produit de nombres décroissans depuisnjusqu'à l'unité, savoirn(n- 1)(n- 2) ... 3.2.1. L'emploi continuel de l'analyse combinatoire que je fais dans la plupart de mes démonstrations, a rendu cette notation indispensable.

1. [...] Je donne le nom deFacultésaux produits dont les facteurs constituent une progression arithmétique, tels que

a(a+r)(a+ 2r)...[a+ (m-1)r];et, pour désigner un pareil produit, j'ai proposé la notation

a^{m}^{|r}.Les facultés forment une classe de fontions très-élementaires, tant que leur exposant est un nombre entier, soit positif soit négatif; mais, dans tous les autres cas, ces mêmes fonctions deviennent absolument transcendantes. [page 1]

2. J'observe que toute faculté numérique quelconque est constamment réductible ô la forme trés-simple

1 ^{m|1}= 1 . 2 . 3 ...mou à cette autre forme plus simple [page 2]

m!,si l'on veut adopter la notation dont j'ai fait usage dans mes

Éléments d'arithmétique universelle,no. 289. [page 3]

[Julio González Cabillón; Cajori vol. 2, p. 72]

In *The Elliptic Functions As They Should Be* (1958),
Albert Eagle advocated writing !*n* rather than *n*!, so
that the operator would precede the argument, as it does in
most cases [Daren Scot Wilson].

In his article "Symbols" in the *Penny Cyclopaedia* (1842) De Morgan complained:
"Among the worst of barabarisms is that of introducing symbols which are quite new
in mathematical, but perfectly understood in common, language. Writers have borrowed
from the Germans the abbreviation *n*! to signify 1.2.3.(*n* - 1).*n,*
which gives their pages the appearance of expressing surprise and admiration that 2, 3, 4,
&c. should be found in mathematical results" [Cajori vol. 2, p. 328].

**Combinations and permutations.** Leonhard Euler (1707-1783)
designated the binomial coefficients by *n* over *r* within
parentheses and using a horizontal fraction bar in a paper written in
1778 but not published until 1806. He used used the same device
except with brackets in a paper written in 1781 and published in 1784
(Cajori vol. 2, page 62).

The modern notation, using parentheses and no fraction bar, appears in
1826 in *Die Combinatorische Analyse* by Andreas von Ettingshausen
[Henry W. Gould]. According to Cajori (vol. 2, page 63) this notation
was introduced in 1827 by Andreas von Ettingshausen in *Vorlesungen
über höhere Mathematik,* Vol. I.

Harvey Goodwin used * _{n}P_{r}* for the number
of permutations of

G. Chrystal used * _{n}C_{r}* for the number of
combinations of

The normal distribution was first obtained as the limiting form
of the binomial distribution in the early 18^{th} century by
Abraham De Moivre
(the 20^{th} century editor provides the equation in footnote 2). From
the early 19^{th} century the normal distribution was the foundation
of the theory of errors, developed for use in astronomy and geodesy. The normal
distribution went by various names, including the law of error and the
probability curve. Although the most important early contributor
was Laplace, the most common way of writing the normal distribution--at least
in the English literature--came from Gauss.

The following equation appears on p. 244

Using modern conventions for brackets and squares this would be written

The errors (Δ) are centred on *0*.
In the English literature the quantity *h,* or its reciprocal, was often
called the modulus. See MODULUS

A typical presentation of Gauss's ideas can be found in Chauvenet's
A Manual of Spherical and Practical Astronomy ... with an Appendix on the Method
of Least Squares (4^{th} edition, 1871). The section on "the probability curve"
(pp. 478-485) discusses Gauss's function which appears on p. 484.

where *N* is the total number of particles. "Illustrations
of the Dynamical Theory of Gases," *Philosophical Magazine*, **19**,
(1860), 19-32.

In most of the biometric literature the number of units was represented by
*N*. See e.g. the equation for the sample mean in Section III of Student’s
"The Probable
Error of a Mean", *Biometrika*, **6**, (1908), 1-25.

where *m* is the mean.

Fisher soon went over to the biometric notation (but without the *c* or
*N*). He wrote the bivariate density in his
1915
paper on correlation (p. 508). When he next needs the univariate form he writes "the
chance of any observation falling in the range *dx* is

from
A Mathematical
Examination of the Methods of Determining the Accuracy of an Observation by
the Mean Error, and by the Mean Square Error (1920, p. 758.) Fisher
generally used *df* to denoted this chance--see the expression on p. 508
of his 1915 paper.

Fisher wrote the normal density like
this (see section 12 of his
Statistical Methods
for Research Workers) until the mid-1930s when he replaced *m* with μ.
The new symbol appears in The Fiducial Argument in
Statistical Inference (1935) and it went into the 1936 (sixth) edition of
the *Statistical Methods for Research Workers.*

**i. The normal distribution**

Fisher usually wrote the density in the form *df = ... dx* but more recently
*f* as been reserved for the density function and *F* for the distribution
function and so a more "correct" way of writing would be *dF = ... dx. *(See
symbols in probability.) However the differential notation has gone out of fashion
and it is more usual to write some variation of

The subscript *X* is used if there a danger of confusion with other random
variables. (See symbols in probability.)

**ii. The standard normal**

Modern texts often write the density function for the standard normal as

in accordance with Halperin, Hartley & Hoel's
"Recommended Standards for Statistical Symbols and Notation. ...,
(*American Statistician*, **19**, (1965), p. 12). They recommend Φ for
the distribution function and the corresponding lower case letter for the density,
however "the use of the variable, z, as argument, is optional." Φ had been
used in influential probability works by Cramér (1937) and Feller (1950). (See
symbols in probability.) In recent decades *z* has come to be very widely
used, particularly in the expression "z score." In earlier decades *z*
was not available as it was established with a different meaning in the analysis
of variance and in correlation.

Information on the history of the term *standard normal cuve* is here.

**Probability.** Symbols for the probability of an event *A* on the pattern of
*P*(*A*) or *Pr*(*A*) are a relatively recent development
given that probability has been studied for centuries. A. N. Kolmogorov's *Grundbegriffe
der Wahrscheinlichkeitsrechnung* (1933) used the symbol **P**(*A*).
The use of upper-case letters for events was taken from set theory where they
referred to sets H. Cramér's *Random Variables and Probability Distributions*
(1937), "the first modern book on probability in English," used *P*(*A*).
In the same year J. V. Uspensky (*Introduction to Mathematical Probability*)
wrote simply (*A*), following A. A. Markov
*Wahrscheinlichkeitsrechnung*
(1912, p. 179) W. Feller's influential *An Introduction to Probability Theory
and its Applications volume 1* (1950) uses *Pr*{A} and **P**{*A*}in
later editions.

See also the entry PROBABILITY and the "Earliest Uses of Symbols of Set Theory and Logic" page of this website.

**Conditional probability.** Kolmogorov's
(1933) symbol for conditional probability ("die bedingte Wahrscheinlichkeit")
was **P**_{B}(*A*). Cramér (1937) wrote P* _{B}*
(

See also the entries CONDITIONAL PROBABILITY and POSTERIOR PROBABILITY and the "Earliest Uses of Symbols of Set Theory and Logic" page of this website.

**Expectation.** A large script E was used for the expectation in W. A. Whitworth's well-known textbook
*Choice and Chance* (fifth edition) of 1901 but neither the symbol nor the calculus of expectations
became established in the *English* literature until much later. For example, Rietz *Mathematical
Statistics* (1927) used the symbol *E* and commented that "the expected value of the variable is
a concept that has been much used by various continental European writers..." For the continental European
writers *E* signified "Erwartung" or "'éspérance."

**Random variable.** The use of upper and lower case letters to distinguish a random variable from
the value it takes, as in *Pr*{*X* = * x _{j}* }, became popular
around 1950. The convention is used in Feller's

**Distribution function and density function.** The use of *F* for the generic distribution function has been established
in the probabillity literature since the 1920s. Paul Lévy *Calcul des
Probabilités* (1925) (p. 136), conforming to the usual notation for the Stieltjes
integral.

Lévy uses *f* for the density function but its use in that
role was not automatic--thus Cramér (1937) uses *f* for the characteristic
function corresponding to *F*. Since the 1940s the *F* for distribution
function and *f* for density convention (within the broader convention
of using the upper-case and corresponding lower-case letters in these roles)
has been widely adopted, particularly by statisticians, following the treatises
by M. G. Kendall *The Advanced Theory of Statistics *(1943) and
S. S. Wilks *Mathematical Statistics *(1944).

*F* and *f* are often adorned with affixes to
register the random variable concerned. Kolmogorov (1933) wrote *F ^{x}*
but now

The **convergence in probability** symbol plim was introduced by H. B. Mann and A Wald
"On Stochastic Limit and Order Relationships,"
*Annals of Mathematical Statistics*,
**14**, (1943), 217-226. The **stochastic order symbols** *O _{p}*
and

** for the sample mean**
is a relic of a convention that has otherwise vanished from
probability and statistics. It derives from the practice of applied mathematicians
of representing *any* kind of average by a bar. J. Clerk Maxwell's
"On the
Dynamical Theory of Gases (*Philosophical Transactions of the Royal Society*,
**157**, (1867) p. 64) uses
for the "mean velocity" of molecules while W. Thomson & P. G. Tait's *Treatise
on Natural Philosophy* (1879) uses
for the centre of inertia,
( = *wx* / *x*)
Karl Pearson, the leading statistician of the early 20th century, had such a physics background.
Pearson and his contemporaries used the bar for sample averages *and* for
expected values but eventually *E* replaced it in the latter role. The
survival of for the
sample mean is probably due to the influential example of R. A. Fisher who used
it in all his works; the first of these was
"On an Absolute
Criterion for Fitting Frequency Curves" (1912). See Expectation in *Symbols in Probability*
above and also AVERAGE,
MEAN
and EXPECTATION
on the Math Words page.

**Standard deviation and variance.** (See STANDARD DEVIATION and
VARIANCE
on the Math Words page.) The use of σ for standard deviation first occurs in Karl Pearson's 1894 paper,
"Contributions
to the Mathematical Theory of Evolution," *Philosophical Transactions
of the Royal Society of London, Ser. A,* **185**, 71-110. On page 80,
he wrote, " Then σ will be
termed its standard-deviation (error of mean square)" (David, 1995). When
Fisher introduced variance in 1918
he did not introduce a new symbol but instead used σ^{2}.

Pearson's notation did not distinguish between parameter and
estimate. Student (W. S. Gosset) in
"The Probable Error of a Mean",
*Biometrika,* **6**, (1908), 1-25, used *s* for an estimate of σ,
though contrary to modern practice his divisor was *n*, not (*n* - 1). Fisher
eventually adopted Student's *s*^{2} (with adjusted *n*) as
an estimate of σ^{2} beginning with his 1922 paper,
"The goodness
of fit of regression formulae, and the distribution of regression coefficients"
(*J. Royal Statist. Soc.*, **85**, 597-612).

**Moments.** Pearson introduced the basic symbol *μ* to which numerical subscripts would be added
to indicate the order and a prime could be added to indicate about which value
the moment is taken. Originally the moment was given by an expression of the
form *αμ* where *α* is the "area of the entire system;" see e.g.
Contributions
to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material,
*Philosophical Transactions of the Royal Society A*, **186**, p. 347.
Eventually the area was normalised to unity and the moment coefficient became
the moment. Fisher applied the Graeco-Latin convention and twinned the *μ*'s
with *m*'s in his paper on cumulants
(1929).
See MOMENT
on the Math Words page.

**Correlation.** (See CORRELATION
on the Math Words page.) When Galton introduced correlation in
"Co-Relations and Their Measurement",
*Proc. R. Soc.,* 45, 135-145, 1888 (also on Galton
website) he chose the symbol *r* for the index of co-relation,
perhaps for its affinity with regression. The use of *ρ* for the population
linear correlation coefficient is found in 1892 in F. Y Edgeworth, "Correlated
Averages," *Philosophical Magazine, 5th Series,* **34**, 190-204.
The symbol appears on page 190 (David, 1995).

Karl Pearson, who dominated correlation research from the
mid-1890s, favoured *r* (for both parameter and estimate), using *ρ*
only if a second correlation symbol was required; thus both symbols appear on p. 302 of
Contributions
to the Mathematical Theory of Evolution.
Note on Reproductive Selection," *Proc. R. Soc.*, **59**,
(1895-6), 301-305. Student (W. S. Gosset) in "The Probable Error of the
Correlation Coefficient" (*Biometrika,* 6, 302-310 1908) had different
symbols for the parameter value (*R*) and for the estimate (*r*).
H. E. Soper (*Biometrika,* **9**, 91-115, 1913) used *ρ* and *r*
in these roles. R. A. Fisher used the Soper symbols from his first work in correlation
(1915).

G. Udny Yule introduced the notation *r*_{12.3}
for the **partial correlation** between *x*_{1} and *x*_{2}
holding *x*_{3} fixed in his 1907
"On the
Theory of Correlation for any Number of Variables,
Treated by a New System of Notation," *Proc. R. Soc. Series
A,* 79, pp. 182-193. The Greek forms, including *ρ*_{ 12.3},
followed in M. S. Bartlett's 1933 "On the theory of statistical regression,"
*Proc. Royal Soc. Edinburgh*, **53**, 260-283.

*R* has been used for the double,
triple, ..., *n*-fold or **multiple correlation** coefficient, at least
since Yule used it in 1896. *R* is now generally used for the *sample*
coefficient. This is awkward because the upper-case *ρ,* the natural
choice for the population coefficient, is the unappealing letter, *P.*

**Regression.** (See REGRESSION
and METHOD OF LEAST SQUARES on the Math Words page.)
Modern regression analysis has its roots in Gauss's work (1809/-23) on the use
of least squares for combining observations and in the work of Galton and Pearson
on heredity. Gauss's notation can be seen in Chauvenet's
Manual pp. 509ff with the special notation
for Gaussian elimination on pp. 530ff. Pearson’s correlation-based notation
can be seen in the equation for *H*_{1} on p. 241 of his
"Note on Regression
and Inheritance in the Case of Two Parents," *Proc. R. Soc.*,
**58**, (1895), 240-2. The notational highpoint of the correlation/regression development was Yule’s
"On the
Theory of Correlation for any Number of Variables,
Treated by a New System of Notation," *Proc. R. Soc.*,
*A*, **79**, (1907), 182-193 where b_{12..3} stands for
the partial regression of *x*_{1} on *x*_{2} holding
*x*_{3} fixed. (Cf. correlation notation above).

Yule’s regression notation is used sometimes in multivariate analysis but the
most familiar modern regression notation dates from the 1920s when R. A. Fisher
drew the Gauss and Pearson lines together. In his
*Statistical Methods for Research Workers*
(1925) Fisher presents regression using *y* and *x* and the terms
"dependent variable" and "independent variable." For the
population values of the intercept and slope Fisher uses
α and β, for the estimates he uses *a* and *b.* This textbook exposition was based on a 1922 paper,
"The goodness of fit of regression formulae, and
the distribution of regression coefficients" (*J. Royal Statist.
Soc.*, **85**, 597-612.

**Matrix notation** in regression was first used in the 1920s but only came into wide
use in the 1950s. The most noticed of the early contributions was a paper by
A. C. Aitken,
"On least squares and linear combinations of observations," *Proc. Royal Soc. Edinburgh*, **55**,
(1935), 42-48. This paper is also notable for its account of what has been called
"Aitken’s generalised least squares." Aitken appears not to have regarded this
work highly; it belonged with the "mere applications ... to standard problems."
The practice of writing an error term in the equation also became common around
1950. See the entry ERROR on the Math Words page.

**θ as the generic "unknown" parameter.** R. A. Fisher established the role and θ in it in
"On the Mathematical Foundations of Theoretical
Statistics" (*Phil. Trans. R. Soc.* 1922) and the papers
that followed. However Fisher had already used the notation in his first publication,
a paper he wrote as a third year undergraduate,
"On an Absolute
Criterion for Fitting Frequency Curves
" (*Messenger of Mathematics,* 1912, **41**: 155-160).

** κ for cumulants (cumulative moment functions) and the corresponding k-statistics.**
Fisher introduced this notation in his 1929 paper
"Moments and Product Moments
of Sampling Distributions",

** μ for the mean of the normal distribution.** (See

**Symbols associated with testing hypotheses.**

*P***-value**.
Please see the entry on the mathematical words page here.

** H_{0}**
was used to represent "the hypothesis in which we are particularly interested" in J. Neyman and E. S. Pearson’s
"On the Problem
of the Most Efficient Tests of Statistical Hypotheses,
"

**λ for the (maximised) likelihood ratio.** This symbol was introduced
by J. Neyman and E. S. Pearson in their "On the Use of Certain Test Criteria
for Purposes of Statistical Inference, Part II" *Biometrika,* (1928),
**20A,** 263-294. They called the quantity it denoted the *likelihood*
but later authors called it the likelihood ratio. See the entry LIKELIHOOD RATIO
on the mathematical
words page here.

** α for the size of the critical region** appears in J. Neyman and E. S. Pearson’s
"Contributions to the Theory of Testing Statistical Hypotheses,"

** β for the power function** was introduced
by J. Neyman "Tests of Statistical Hypotheses Which are Unbiased
in the Limit,"

** F distribution.** Please see the entry on the mathematical
words page here.

**χ^{2} (chi-squared).** Please see the entry on the mathematical words page
here.

**(Student's) t.** Please see the entry Student's

**Number of degrees of freedom**.
In 1921, when R. A. Fisher introduced the concept of degrees of freedom, he
used *n* for the number of degrees of freedom, following Pearson's (1900)
chi-square goodness of fit paper. Pearson derived the distribution of a quadratic
form in *n* jointly normal variables where this normal is the limiting
form of a multinomial with *n*' = *n*+1 cells. When Fisher used
*n* for the number of degrees of freedom in the *t*-distribution,
he could not use it for the number of observations and so he used *n'* for that number; see e.g.
chapter V
of his *Statistical Methods for Research Workers* (1925, with 13 further editions to 1970.) C. P.
Eisenhart (1979, p. 8n) relates in his "On the Transition from Student's
*z* to Student's *t*," *American Statistician*, **33**,
6-10 how there was an aversion among many "not Fisherian" statisticians to Fisher’s
use of *n* and how E. S. Pearson and some colleagues decided on the Greek
letter ν. This letter appears in M. G. Kendall’s *The Advanced Theory
of Statistics* (1943) and in Halperin, Hartley & Hoel's "Recommended
Standards for Statistical Symbols and Notation. ..., (*American Statistician*,
**19**, (1965), p. 12). For references and further details see the entries on
DEGREES OF FREEDOM,
CHI SQUARE and
STUDENT'S t DISTRIBUTION on the math words page.

** T^{2}** was introduced by Harold Hotelling in "The Generalization of Student's
Ratio,"

**z** has played several roles. Today it most often stands for the standard normal;
see *Symbols associated with the Normal distribution* above. R. A. Fisher
used *z* in the analysis of variance (see the entry z AND z DISTRIBUTION
here) and in transforming the correlation
coefficient (see the entry FISHER’S z TRANSFORMATION OF THE CORRELATION COEFFICIENT
here.)
Student had used originally z for the test statistic that was turned into "Student’s
*t*". (See the entry STUDENT’S *t*-DISTRIBUTION
here.)