The sampling

Selection from the final totality and its theory are the basis of the statist. methods of quality control and frequently adapt in the sociological studies (see selective observation). According to the probability theory, the sample will correctly reflect the properties of entire totality, if selection is produced randomly, i.e. so that any of the possible samples of the assigned volume p from the totality of volume N [ the number of such samples is equal N!/.n!(.N - n)! ] it has identical probability to be actually selected.

The sampling without replacement (nonrepetitive sample) most frequently is used in practice, when each selected object before the selection track, object to the totality being investigated does not return (this selection it adapts with statist. quality control). Sampling with replacement (sample with the repetition) is examined usually only in the theoretical studies (example of sampling with replacement is the registration of the number of particles, which were touched in this period of the walls of the vessel, inside which is accomplished the Brownian motion). If p < <..N, then repeated and nonrepetitive selections give practically equivalent results.

The properties of totality, investigated by v. m., can be qualitative and quantitative. In the first case the task of sample survey consists in the determination of a quantity M of objects of totality, which possess some sign (for example, with the statist. control they are frequently interested in a quantity M of defective articles in the party of volume N). As estimation for M serves the relation of m.N/n, where Q. - the number of objects with the data by sign in the sample of volume p. In the case of measurable characteristic they deal concerning the determination of sr. of the value of totality X = (x₁ + X_g +. . . + Xn)/.N. Estimation for X is selective average E = (E₁ + E₂ +. . . to +E_nof)/.n. where E₁., E_n - those values from the totality x₁, x_{2 being investigated}.. ., x_N, which belong to sample. From matematich. of the point of view the 1st case - particular variety 2- GO, which occurs, when M of values x_i are equal to 1, and rest (N - M)ravny 0; in this situation X = M/N and E = of m/.n.

In matematich. of theory v. m. the estimation sr. of values occupies center, place because the study of the changeability of sign inside the totality is reduced to it to the known degree, since for the characteristic of changeability usually tyuey ppinimayut the dispersion

o² = y/.N[(.kh₁-.x)² +. . . + (x_N - x)²],

being sr. the value of the standard deviations Of the xi from their sr. value X. In the case of studying the quality characteristic o² = M (N - M)/.N².

The accuracy of the estimations of ts/ya and g they judge by their dispersions

which in the terms of the dispersion of the final totality o² are expressed in the form of relations about²/.n (in the case of samples with the repetition) and o²(N - n)/.n(.N - 1) (in the case of nonrepetitive samples). Since in many practically interesting tasks the random variables of ts/k and E with p > = 30 are approximately subordinated to normal distribution, then the deviations of m/.n from M/N and E from X, exceeding in absolute value т_m/_n and т_E , correspondingly, can with p > = 30 be carried out on the average approximately in one case of twenty. More complete information about the distribution of measurable characteristic in this totality can be obtained with the aid of empirich. the distribution of this sign in the sample. Selection from the unlimited aggregate. In matematich. statistician the results of any uniform observations (most frequently independent variables) it is accepted NAZ by sample even when these results do not correspond to the concept of sample with the repetitions or without the repetitions from the final totality. For example, the results of measurements is angular to the localities, subjected to the independent continuous random errors, frequently NAZ by sample from the unlimited aggregate. It is assumed that principally it is possible to carry out any number of such observations. The obtained actually results are considered sample from the infinite set of possible results, NAZ by general population.

The concept of general population is not logically irreproachable and necessary. For the solution of practical problems it is necessary not infinite universe itself, but only these or other characteristics, which are placed with it in the correspondence. These characteristics from the point of view of the probability theory are the numerical or functional characteristics of the some probability distribution, and selection elements - by random variables, which are subordinated to this distribution. This interpretation makes it possible to extend the general theory of statistical evaluations to the selective estimations.

For this reason, for example, in the probabilistic theory of working observations the concept of infinite universe is substituted by the concept of the probability distribution, which contains the unknown ' parameters. The results of observations interpret as the experimentally observed values of the random variables, which are subordinated to this distribution. Purpose of working - calculation according to observations in one or other sense or another of optimum statist. estimations for the unknown parameters of distribution.

:-Dunin- BarkovskyI is cast. V., Smirnov N. V., the probability theory and mathematical statistics in technology (general part), M., 1955, main shch;Kendall M., St'yuart a., the distribution theory, translated from English, M., 1966. L. n. bol'sheye.