Binomial Distribution

BINOMIAL DISTRIBUTION, such a distribution of values in a statistical series, in which the corresponding individual members follow the coefficients of a binomial raised to some power. This distribution is observed with multiple (and sufficiently large) measurements of any object (having a constant size under given conditions). If the readings obtained from measurements of the size of the object being studied are arranged in a series, with the first line listing in ascending order (from smallest to largest) or descending order all encountered sizes that differ from each other (called 'variants'), and the second line lists the numbers of repetitions of identical sizes, then the distribution of the latter (called 'frequencies') will follow the coefficients of a binomial raised to the power n-1, where n denotes the number of members in the series. Binomial distribution is also obtained when measuring a large number of different objects belonging to a certain homogeneous group, as, for example, when measuring the height of individuals of a certain age-sex group, homogeneous in racial, domestic, economic, etc. respects. An example of binomial distribution can be the following table of binomial distribution of body height measurements in male workers (according to F. F. Erisman): Height in cm Men Number of cases Per thousand According to binomial formula 134-136.5 137-139.5 - - 140-142.5 - - 143-145.5 146-148.5 149-151.5 152-154.5 155-157.5 158-160.5 161-163.5 164-166.5 167-169.5 170-172.5 173-175.5 176-178.5 179-181.5 182-184.5 185-187.5 The credit for discovering this distribution law belongs to the Belgian mathematician and statistician Quetelet. Quetelet's law closely approaches the so-called law of errors of Gauss and can be considered as a consequence of two theorems of probability theory - the theorem of J. Bernoulli and the theorem of Poisson,- also called the 'law of large numbers' (see Law of large numbers). This law provides the basis for studying the structure of empirical distribution series obtained as a result of statistical research (see Variational statistics).