vital statistics, primarily records of the number of births and deaths in a population. Other factors, such as number of marriages and causes of death, by age groups, are regularly included. From these records can be computed birthrates and death (or mortality) rates from which trends are determined. The earliest known system of vital statistics was in China. In England the clergy was required as early as the 16th cent. to keep records of christenings, marriages, and burials; during the 17th cent. the clergy in France, Italy, and Spain began to keep similar records. The oldest continuous national records system is that of Sweden (since 1741). The clergy and government officials in the colonies of North America began to record vital statistics in the 17th cent.; on a national level, the U.S. government started publishing annual records of deaths in 1900 and of births in 1915. The most striking trend shown by recent vital statistics is the rapid increase of the populations of nonindustrial countries due to a sharp decline in the mortality rate and an acceleration of the birthrate.

statistics, science of collecting and classifying a group of facts according to their relative number and determining certain values that represent characteristics of the group. The most familiar statistical measure is the arithmetic mean, which is an average value for a group of numerical observations. A second important statistic or statistical measure is the standard deviation, which is a measure of how much the individual observations are scattered about the mean. The chi-square test is a method of determining the odds for or against a given deviation from expected statistical distribution. Other statistics indicate other characteristics of the group of observations. In addition to the problem of computing certain statistics for a particular group of observations, there is the problem of sampling. This is an attempt to determine for what larger group (called the population) of individuals or characteristics the statistics for this particular group (called the sample) would be a representative figure and how representative a figure it would be for a given larger group. This second problem of sampling can be solved only by resorting to the theory of probability and higher mathematics. In most applications of statistics to scientific and social research, insurance, and finance, the statistician is interested not only in the characteristics of the sample but also in those of some much larger population. Consequently, the theory of sampling is the most important part of statistical theory.

Fermi-Dirac statistics, class of statistics that applies to particles called fermions. Fermions have half-integral values of the quantum mechanical property called spin and are "antisocial" in the sense that two fermions cannot exist in the same state. Protons, neutrons, electrons, and many other elementary particles are fermions. See Bose-Einstein statistics; elementary particles; statistical mechanics.

Bose-Einstein statistics, class of statistics that applies to elementary particles called bosons, which include the photon, pion, and the W and Z particles. Bosons have integral values of the quantum mechanical property called spin and are "gregarious" in the sense that an unlimited number of bosons can be placed in the same state. All of the particles that mediate the fundamental forces of nature are bosons. See elementary particles; Fermi-Dirac statistics; statistical mechanics.

Branch of mathematics dealing with gathering, analyzing, and making inferences from data. Originally associated with government data (e.g., census data), the subject now has applications in all the sciences. Statistical tools not only summarize past data through such indicators as the mean (see mean, median, and mode) and the standard deviation but can predict future events using frequency distribution functions. Statistics provides ways to design efficient experiments that eliminate time-consuming trial and error. Double-blind tests for polls, intelligence and aptitude tests, and medical, biological, and industrial experiments all benefit from statistical methods and theories. The results of all of them serve as predictors of future performance, though reliability varies. Seealso estimation, hypothesis testing, least squares method, probability theory, regression.

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In quantum mechanics, one of two possible ways (the other being Bose-Einstein statistics) in which a system of indistinguishable particles can be distributed among a set of energy states. Each available discrete state can be occupied by only one particle. This exclusiveness accounts for the structure of atoms, in which electrons remain in separate states rather than collapsing into a common state. It also accounts for some aspects of electrical conductivity. This theory of statistical behaviour was developed first by Enrico Fermi and then by P.A.M. Dirac (1926–27). The statistics apply only to particles such as electrons that have half-integer values of spin; the particles are called fermions.

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One of two possible ways (the other is Fermi-Dirac statistics) in which a collection of indistinguishable particles may occupy a set of available discrete energy states. The gathering of particles in the same state, which is characteristic of particles that obey Bose-Einstein statistics, accounts for the cohesive streaming of laser light and the frictionless creeping of superfluid helium (see superfluidity). The theory of this behaviour was developed in 1924–25 by Satyendra Nath Bose (1894–1974) and Albert Einstein. Bose-Einstein statistics apply only to those particles, called bosons, which have integer values of spin and so do not obey the Pauli exclusion principle.

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