3.5.1. Probabilistic-statistical method of research.

In many cases, it is necessary to investigate not only deterministic, but also random probabilistic (statistical) processes. These processes are considered on the basis of probability theory.

The totality of the random variable x is the primary mathematical material. A collection is understood as a set of homogeneous events. The set containing the most diverse variants of a mass phenomenon is called the general population, or a large sample of N. Usually only a part of the general population is studied, called sample population or small sample.

Probability R(x) developments X called the ratio of the number of cases N(x), that lead to the occurrence of the event X, to total number possible cases N:

P(x)=N(x)/N.

Probability theory considers theoretical distributions of random variables and their characteristics.

Math statistics deals with ways of processing and analyzing empirical events.

These two related sciences constitute a unified mathematical theory of mass random processes, widely used for the analysis scientific research.

Very often, the methods of probability and mathematical statistics are used in the theory of reliability, survivability and safety, which is widely used in various branches of science and technology.

3.5.2. Method of statistical modeling or statistical tests (Monte Carlo method).

This method is numerical method solving complex problems and is based on the use of random numbers simulating probabilistic processes. The results of the solution by this method make it possible to establish empirically the dependences of the processes under study.

Solving problems using the Monte Carlo method is effective only with the use of high-speed computers. To solve problems using the Monte Carlo method, it is necessary to have a statistical series, know the law of its distribution, the average value of the mathematical expectation t(x), standard deviation.

Using this method, one can obtain an arbitrarily given accuracy of the solution, i.e.

-> m(x)

3.5.3. System analysis method.

System analysis is understood as a set of techniques and methods for studying complex systems, which are a complex set of interacting elements. The interaction of the elements of the system is characterized by direct and feedback connections.

The essence of system analysis is to identify these relationships and establish their impact on the behavior of the entire system as a whole. The most complete and deep system analysis can be performed using the methods of cybernetics, which is the science of complex dynamic systems that can perceive, store and process information for the purposes of optimization and control.

System analysis consists of four stages.

The first stage consists in setting the task: they determine the object, goals and objectives of the study, as well as the criteria for studying the object and managing it.

During the second stage, the boundaries of the system under study are determined and its structure is determined. All objects and processes related to the goal are divided into two classes - the system under study and the external environment. Distinguish closed and open systems. When researching closed systems influence external environment their behavior is neglected. Then separate the individual components of the system - its elements, establish the interaction between them and the external environment.

The third stage of system analysis is the compilation of a mathematical model of the system under study. First, the system is parametrized, the main elements of the system and elementary effects on it are described using certain parameters. At the same time, there are parameters that characterize continuous and discrete, deterministic and probabilistic processes. Depending on the characteristics of the processes, one or another mathematical apparatus is used.

As a result of the third stage of system analysis, complete mathematical models of the system are formed, described in a formal, for example, algorithmic, language.

At the fourth stage, the resulting mathematical model is analyzed, its extreme conditions are found in order to optimize processes and control systems, and formulate conclusions. Optimization is evaluated according to the optimization criterion, which in this case takes extreme values ​​(minimum, maximum, minimax).

Usually, one criterion is chosen, and threshold maximum permissible values ​​are set for others. Sometimes mixed criteria are used, which are a function of the primary parameters.

Based on the selected optimization criterion, the dependence of the optimization criterion on the parameters of the model of the object (process) under study is compiled.

There are various mathematical methods for optimizing the models under study: methods of linear, non-linear or dynamic programming; probabilistic-statistical methods based on the theory of queuing; game theory, which considers the development of processes as random situations.

Questions for self-control of knowledge

Methodology of theoretical research.

The main sections of the stage of theoretical development of scientific research.

Types of models and types of modeling of the object of study.

Analytical methods of research.

Analytical research methods using experiment.

Probabilistic-analytical method of research.

Methods of static modeling (Monte Carlo method).

Method of system analysis.

Of particular interest is quantification entrepreneurial risk using methods of mathematical statistics. The main tools of this evaluation method are:

§ the probability of occurrence of a random variable ,

§ mathematical expectation or average value of the random variable under study,

§ variance,

§ standard (root mean square) deviation,

§ the coefficient of variation ,

§ probability distribution of the random variable under study.

To make a decision, you need to know the magnitude (degree) of risk, which is measured by two criteria:

1) average expected value (mathematical expectation),

2) fluctuations (variability) of a possible result.

Average expected value is the weighted average of a random variable, which is associated with the uncertainty of the situation:

,

where is the value of the random variable.

The mean expected value measures the outcome we expect on average.

The mean value is a generalized qualitative characteristic and does not allow making a decision in favor of any individual value of a random variable.

To make a decision, it is necessary to measure the fluctuations of indicators, that is, to determine the measure of the variability of a possible result.

The fluctuation of the possible result is the degree of deviation of the expected value from the average value.

To do this, in practice, two closely related criteria are usually used: "dispersion" and "standard deviation".

Dispersion – weighted average of squares actual results from the average expected:

standard deviation is the square root of the variance. It is a dimensional quantity and is measured in the same units in which the random variable under study is measured:

.

Dispersion and standard deviation serve as a measure of absolute fluctuation. For analysis, the coefficient of variation is usually used.

The coefficient of variation is the ratio of the standard deviation to the mean expected value , multiplied by 100%

or .

The coefficient of variation is not affected by the absolute values ​​of the studied indicator.

With the help of the coefficient of variation, even fluctuations of features expressed in different units of measurement can be compared. The coefficient of variation can vary from 0 to 100%. The larger the ratio, the greater the fluctuation.

AT economic statistics the following estimate of different values ​​of the coefficient of variation was established:

up to 10% - weak fluctuation, 10 - 25% - moderate, over 25% - high.

Accordingly, the higher the fluctuations, the greater the risk.

Example. The owner of a small store at the beginning of each day buys some perishable product for sale. A unit of this product costs 200 UAH. Selling price - 300 UAH. for a unit. From observations it is known that the demand for this product during the day can be 4, 5, 6 or 7 units with the corresponding probabilities 0.1; 0.3; 0.5; 0.1. If the product is not sold during the day, then at the end of the day it will always be bought at a price of 150 UAH. for a unit. How many units of this product should the store owner purchase at the beginning of the day?

Solution. Let's build a profit matrix for the store owner. Let's calculate the profit that the owner will receive if, for example, he buys 7 units of the product, and sells during the day 6 and at the end of the day one unit. Each unit of the product sold during the day gives a profit of 100 UAH, and at the end of the day - a loss of 200 - 150 = 50 UAH. Thus, the profit in this case will be:

Calculations are carried out similarly for other combinations of supply and demand.

The expected profit is calculated as the mathematical expectation of possible profit values ​​for each row of the constructed matrix, taking into account the corresponding probabilities. As you can see, among the expected profits, the largest is 525 UAH. It corresponds to the purchase of the product in question in the amount of 6 units.

To substantiate the final recommendation on the purchase of the required number of units of the product, we calculate the variance, standard deviation and coefficient of variation for each possible combination of supply and demand of the product (each line of the profit matrix):

With regard to the purchase of 6 units of the product by the store owner compared to 5 and 4 units, this is not obvious, since the risk in purchasing 6 units of the product (19.2%) is greater than in purchasing 5 units (9.3%), and even more so, than when purchasing 4 units (0%).

Thus, we have all the information about the expected profits and risks. And decide how many units of the product you need to buy every morning for the store owner, taking into account his experience, risk appetite.

In our opinion, the store owner should be advised to buy 5 units of the product every morning and his average expected profit will be 485 UAH. and if we compare this with the purchase of 6 units of the product, in which the average expected profit is 525 UAH, which is 40 UAH. more, but the risk in this case will be 2.06 times greater.

When conducting psychological and pedagogical research, an important role is given to mathematical methods modeling of processes and processing of experimental data. These methods include, first of all, the so-called probabilistic statistical methods research. This is due to the fact that the behavior of both an individual person in the process of his activity and a person in a team is significantly influenced by many random factors. Randomness does not allow one to describe phenomena within the framework of deterministic models, since it manifests itself as insufficient regularity in mass phenomena and, therefore, does not make it possible to reliably predict the occurrence of certain events. However, when studying such phenomena, certain regularities are revealed. The irregularity inherent in random events, with a large number of tests, as a rule, is compensated by the appearance of a statistical pattern, stabilization of the frequency of occurrence of random events. Therefore, the data random events have a certain probability. There are two fundamentally different probabilistic-statistical methods of psychological and pedagogical research: classical and non-classical. Let's spend comparative analysis these methods.

Classical probabilistic-statistical method. The classical probabilistic-statistical method of research is based on the theory of probability and math statistics. This method is used in the study of mass phenomena of a random nature, it includes several stages, the main of which are as follows.

1. Construction of a probabilistic model of reality based on the analysis of statistical data (determination of the law of distribution of a random variable). Naturally, the patterns of mass random phenomena are expressed the more clearly, the greater the volume of statistical material. The sample data obtained during the experiment are always limited and, strictly speaking, are of a random nature. In this regard, an important role is given to the generalization of patterns obtained in the sample and their distribution to the entire general population objects. In order to solve this problem, a certain hypothesis is adopted about the nature of the statistical pattern, which manifests itself in the phenomenon under study, for example, the hypothesis that the phenomenon under study obeys the law normal distribution. Such a hypothesis is called the null hypothesis, which may turn out to be erroneous, therefore, along with null hypothesis an alternative or competing hypothesis is also put forward. Checking how the obtained experimental data correspond to one or another statistical hypothesis is carried out using the so-called non-parametric statistical tests or goodness-of-fit tests. At present, the Kolmogorov, Smirnov, omega-square, and other goodness-of-fit criteria are widely used. The main idea of ​​these criteria is to measure the distance between the function empirical distribution and a fully known theoretical distribution function. Validation methodology statistical hypothesis rigorously developed and presented in a large number of works on mathematical statistics.

2. Carrying out the necessary calculations by mathematical means within the framework of a probabilistic model. In accordance with the established probabilistic model of the phenomenon, the calculation of characteristic parameters is carried out, for example, such as the mathematical expectation or mean value, variance, standard deviation, mode, median, asymmetry index, etc.

3. Interpretation of probabilistic-statistical conclusions in relation to a real situation.

At present, the classical probabilistic-statistical method is well developed and widely used in research in various fields natural, technical and social sciences. A detailed description of the essence of this method and its application to solving specific problems can be found in a large number of literary sources, for example, in.

Non-classical probabilistic-statistical method. The non-classical probabilistic-statistical research method differs from the classical one in that it is applied not only to mass, but also to individual events that are fundamentally random. This method can be effectively used in the analysis of the behavior of an individual in the process of performing a particular activity, for example, in the process of acquiring knowledge by students. We will consider the features of the non-classical probabilistic-statistical method of psychological and pedagogical research using the example of the behavior of students in the process of mastering knowledge.

For the first time, a probabilistic-statistical model of student behavior in the process of mastering knowledge was proposed in the work. Further development of this model was done in . Teaching as a type of activity, the purpose of which is the acquisition of knowledge, skills and abilities by a person, depends on the level of development of the student's consciousness. The structure of consciousness includes such cognitive processes as sensation, perception, memory, thinking, imagination. An analysis of these processes shows that they have elements of randomness due to the random nature of the mental and somatic states of the individual, as well as physiological, psychological and informational noises during the work of the brain. The latter led to the refusal to use the model of a deterministic dynamic system in the description of the processes of thinking in favor of the model of a random dynamic system. This means that the determinism of consciousness is realized through chance. From this we can conclude that human knowledge, which is actually a product of consciousness, also has a random character, and, therefore, a probabilistic-statistical method can be used to describe the behavior of each individual student in the process of mastering knowledge.

In accordance with this method, a student is identified by a distribution function (probability density) that determines the probability of being in a single area of ​​the information space. In the learning process, the distribution function with which the student is identified, evolving, moves in the information space. Each student has individual properties and independent localization (spatial and kinematic) of individuals relative to each other is allowed.

Based on the law of conservation of probability, the system is written differential equations, which are continuity equations that relate the change in the probability density per unit time in the phase space (the space of coordinates, velocities and accelerations of various orders) with the divergence of the probability density flow in the considered phase space. In the analysis of analytical solutions of a number of continuity equations (distribution functions) characterizing the behavior of individual students in the learning process.

When conducting experimental studies the behavior of students in the process of mastering knowledge, probabilistic-statistical scaling is used, according to which the measurement scale is an ordered system , where A is some completely ordered set of objects (individuals) that have features of interest to us (empirical system with relations); Ly - functional space (space of distribution functions) with relations; F is the operation of a homomorphic mapping of A into the subsystem Ly; G - group of admissible transformations; f is the operation of mapping distribution functions from the subsystem Ly to numerical systems with relations of the n-dimensional space M. Probabilistic-statistical scaling is used to find and process experimental distribution functions and includes three stages.

1. Finding experimental distribution functions based on the results of a control event, for example, an exam. A typical view of individual distribution functions found using a twenty-point scale is shown in fig. 1. The technique for finding such functions is described in.

2. Mapping of distribution functions to a number space. For this purpose, the moments of individual distribution functions are calculated. In practice, as a rule, it suffices to confine ourselves to determining the moments of the first order (mathematical expectation), the second order (dispersion) and the third order, characterizing the asymmetry of the distribution function.

3. Ranking of students according to the level of knowledge based on a comparison of the moments of different orders of their individual distribution functions.

Rice. 1. A typical view of the individual distribution functions of students who received different grades in the exam in general physics: 1 - traditional grade "2"; 2 - traditional rating "3"; 3 - traditional rating "4"; 4 - traditional rating "5"

On the basis of the additivity of individual distribution functions in the experimental distribution functions for the flow of students are found (Fig. 2).

Rice. Fig. 2. Evolution of the complete distribution function of the flow of students, approximated by smooth lines: 1 - after the first year; 2 - after the second course; 3 - after the third course; 4 - after the fourth course; 5 - after the fifth course

Analysis of the data presented in fig. 2 shows that as you move through the information space, the distribution functions blur. This is due to the fact that the mathematical expectations of the distribution functions of individuals move at different speeds, and the functions themselves are blurred due to dispersion. Further analysis of these distribution functions can be carried out within the framework of the classical probabilistic-statistical method.

The discussion of the results. An analysis of the classical and non-classical probabilistic-statistical methods of psychological and pedagogical research has shown that there is a significant difference between them. It, as can be understood from the above, lies in the fact that the classical method is applicable only to the analysis of mass events, while the non-classical method is applicable to both the analysis of mass and single events. In this regard, the classical method can be conventionally called the mass probabilistic-statistical method (MBSM), and the non-classical method - the individual probabilistic-statistical method (IMSM). In 4] it is shown that none of the classical methods of assessing students' knowledge in the framework of a probabilistic-statistical model of an individual can be applied for these purposes.

We will consider the distinctive features of the IMSM and IVSM methods using the example of measuring the completeness of students' knowledge. To this end, we will conduct a thought experiment. Suppose that there are a large number of students who are absolutely identical in mental and physical characteristics and have the same background, and let them, without interacting with each other, simultaneously participate in the same cognitive process, experiencing absolutely the same strictly determined influence. Then, in accordance with the classical ideas about the objects of measurement, all students should receive the same assessments of the completeness of knowledge with any given measurement accuracy. However, in reality, with a sufficiently high accuracy of measurements, assessments of the completeness of students' knowledge will differ. It is not possible to explain such a result of measurements within the framework of the IMSM, since it is initially assumed that the impact on absolutely identical students who do not interact with each other is of a strictly deterministic nature. The classical probabilistic-statistical method does not take into account the fact that the determinism of the process of cognition is realized through randomness, inherent in each individual who cognizes the surrounding world.

The random nature of the student's behavior in the process of mastering knowledge is taken into account by the IVSM. The use of an individual probabilistic-statistical method for analyzing the behavior of the idealized group of students under consideration would show that it is impossible to indicate exactly the position of each student in the information space, one can only say the probabilities of being in one or another area of ​​the information space. In fact, each student is identified by an individual distribution function, and its parameters, such as mathematical expectation, variance, etc., are individual for each student. This means that the individual distribution functions will be in different areas of the information space. The reason for this behavior of students lies in the random nature of the process of cognition.

However, in a number of cases, the results of studies obtained within the framework of the MVSM can also be interpreted within the framework of the IVSM. Let's assume that the teacher uses a five-point measurement scale when evaluating a student's knowledge. In this case, the error in the assessment of knowledge is ±0.5 points. Therefore, when a student is given a score of, say, 4 points, this means that his knowledge is in the range from 3.5 points to 4.5 points. In fact, the position of an individual in the information space in this case is determined by a rectangular distribution function, the width of which is equal to the measurement error of ±0.5 points, and the estimate is the mathematical expectation. This error is so large that it does not allow us to observe the true form of the distribution function. However, despite such a rough approximation of the distribution function, the study of its evolution makes it possible to obtain important information both about the behavior of an individual and a group of students as a whole.

The result of measuring the completeness of a student's knowledge is directly or indirectly influenced by the consciousness of the teacher (meter), who is also characterized by randomness. In the process of pedagogical measurements, in fact, there is an interaction of two random dynamic systems that identify the behavior of the student and teacher in this process. The interaction of the student subsystem with the faculty subsystem is considered and it is shown that the speed of movement of the mathematical expectation of the individual distribution functions of students in the information space is proportional to the impact function of the teaching staff and inversely proportional to the inertia function characterizing the resistance to changing the position of the mathematical expectation in space (analogous to Aristotle's law in mechanics).

At present, despite significant achievements in the development of the theoretical and practical foundations of measurements in the conduct of psychological and pedagogical research, the problem of measurements as a whole is still far from being solved. This is primarily due to the fact that there is still not enough information about the influence of consciousness on the measurement process. A similar situation has developed in solving the measurement problem in quantum mechanics. So, in the paper, when considering the conceptual problems of quantum measurement theory, it is said that it is hardly possible to resolve some paradoxes of measurements in quantum mechanics without directly including the consciousness of the observer in the theoretical description of quantum measurement. It goes on to say that “... it is consistent with the assumption that consciousness can make some event possible, even if, according to the laws of physics (quantum mechanics), the probability of this event is small. Let us make an important clarification of the formulation: the consciousness of a given observer can make it likely that he will see this event.

Statistical Methods

Statistical methods- methods of analysis of statistical data. There are methods of applied statistics, which can be applied in all areas of scientific research and any sectors of the national economy, and other statistical methods, the applicability of which is limited to a particular area. This refers to methods such as statistical acceptance control, statistical control of technological processes, reliability and testing, and design of experiments.

Classification of statistical methods

Statistical methods of data analysis are used in almost all areas of human activity. They are used whenever it is necessary to obtain and substantiate any judgments about a group (objects or subjects) with some internal heterogeneity.

It is advisable to distinguish three types of scientific and applied activities in the field of statistical methods of data analysis (according to the degree of specificity of methods associated with immersion in specific problems):

a) development and research of general purpose methods, without taking into account the specifics of the field of application;

b) development and research of statistical models of real phenomena and processes in accordance with the needs of a particular field of activity;

c) application of statistical methods and models for statistical analysis of specific data.

Applied Statistics

Description of the type of data and the mechanism of their generation is the beginning of any statistical research. Both deterministic and probabilistic methods are used to describe data. With the help of deterministic methods, it is possible to analyze only those data that are at the disposal of the researcher. For example, they were used to obtain tables calculated by official state statistics bodies on the basis of statistical reports submitted by enterprises and organizations. It is possible to transfer the obtained results to a wider set, to use them for prediction and control only on the basis of probabilistic-statistical modeling. Therefore, only methods based on probability theory are often included in mathematical statistics.

We do not consider it possible to oppose deterministic and probabilistic-statistical methods. We consider them as successive stages of statistical analysis. At the first stage, it is necessary to analyze the available data, present them in a form convenient for perception using tables and charts. Then it is advisable to analyze the statistical data on the basis of certain probabilistic-statistical models. Note that the possibility of a deeper insight into the essence of a real phenomenon or process is provided by the development of an adequate mathematical model.

In the simplest situation, statistical data are the values ​​of some feature characteristic of the objects under study. Values ​​can be quantitative or represent an indication of the category to which the object can be assigned. In the second case, we talk about a qualitative sign.

When measuring by several quantitative or qualitative characteristics, we obtain a vector as statistical data about the object. It can be considered as a new kind of data. In this case, the sample consists of a set of vectors. If part of the coordinates is numbers, and part is qualitative (categorized) data, then we are talking about a vector of heterogeneous data.

One element of the sample, that is, one dimension, can be a function as a whole. For example, describing the dynamics of the indicator, that is, its change over time, is the patient's electrocardiogram or the amplitude of the beats of the motor shaft. Or a time series that describes the dynamics of the performance of a particular firm. Then the sample consists of a set of functions.

The elements of the sample can also be other mathematical objects. For example, binary relations. Thus, when polling experts, they often use ordering (ranking) of objects of expertise - product samples, investment projects, options for management decisions. Depending on the regulations of the expert study, the elements of the sample can be various types of binary relations (ordering, partitioning, tolerance), sets, fuzzy sets etc.

So, the mathematical nature of the sample elements in various problems of applied statistics can be very different. However, two classes of statistics can be distinguished - numeric and non-numeric. Accordingly, applied statistics is divided into two parts - numerical statistics and non-numerical statistics.

Numeric statistics are numbers, vectors, functions. They can be added, multiplied by coefficients. Therefore, in numerical statistics great importance have different amounts. Mathematical apparatus analysis of the sums of random elements of the sample are the (classical) laws big numbers and central limit theorems.

Non-numeric statistical data are categorized data, vectors of heterogeneous features, binary relations, sets, fuzzy sets, etc. They cannot be added and multiplied by coefficients. So it doesn't make sense to talk about sums of non-numeric statistics. They are elements of non-numerical mathematical spaces (sets). The mathematical apparatus for the analysis of non-numerical statistical data is based on the use of distances between elements (as well as proximity measures, difference indicators) in such spaces. With the help of distances, empirical and theoretical averages are determined, the laws of large numbers are proved, nonparametric estimates of the probability distribution density are constructed, problems of diagnostics and cluster analysis are solved, etc. (see).

Applied research uses statistical data various kinds. This is due, in particular, to the methods of obtaining them. For example, if testing of some technical devices continues until a certain point in time, then we get the so-called. censored data consisting of a set of numbers - the duration of the operation of a number of devices before failure, and information that the remaining devices continued to work at the end of the test. Censored data is often used in the assessment and control of the reliability of technical devices.

Usually, statistical methods of data analysis of the first three types are considered separately. This limitation is caused by the circumstance noted above that the mathematical apparatus for analyzing data of a non-numerical nature is essentially different from that for data in the form of numbers, vectors, and functions.

Probabilistic-statistical modeling

When applying statistical methods in specific areas of knowledge and sectors of the national economy, we obtain scientific and practical disciplines such as “statistical methods in industry”, “statistical methods in medicine”, etc. From this point of view, econometrics is “statistical methods in economics”. These disciplines of group b) are usually based on probabilistic-statistical models built in accordance with the characteristics of the application area. It is very instructive to compare the probabilistic-statistical models used in various fields, to discover their closeness and, at the same time, to state some differences. Thus, one can see the closeness of the problem statements and the statistical methods used to solve them in such areas as scientific medical research, specific sociological research and marketing research, or, in short, in medicine, sociology, and marketing. These are often grouped together under the name "sampling studies".

The difference between selective studies and expert studies is manifested, first of all, in the number of objects or subjects examined - in selective studies, we usually talk about hundreds, and in expert studies, about tens. But the technology of expert research is much more sophisticated. The specificity is even more pronounced in demographic or logistical models, in the processing of narrative (textual, chronicle) information, or in the study of the mutual influence of factors.

Issues of reliability and safety of technical devices and technologies, queuing theory are considered in detail in a large number of scientific papers.

Statistical analysis of specific data

The application of statistical methods and models for the statistical analysis of specific data is closely tied to the problems of the respective field. The results of the third of the identified types of scientific and applied activities are at the intersection of disciplines. They can be considered as examples of the practical application of statistical methods. But there is no less reason to attribute them to the corresponding field of human activity.

For example, the results of a survey of instant coffee consumers are naturally attributed to marketing (which is what they do when lecturing on marketing research). The study of price growth dynamics using inflation indices calculated from independently collected information is of interest primarily from the point of view of economics and management. national economy(both at the macro level and at the level of individual organizations).

Development prospects

The theory of statistical methods is aimed at solving real problems. Therefore, new formulations of mathematical problems of statistical data analysis constantly appear in it, new methods are developed and substantiated. Justification is often carried out by mathematical means, that is, by proving theorems. An important role is played by the methodological component - how exactly to set tasks, what assumptions to accept for the purpose of further mathematical study. The role of modern information technologies in particular, a computer experiment.

An urgent task is to analyze the history of statistical methods in order to identify development trends and apply them for forecasting.

Literature

2. Naylor T. Machine simulation experiments with models economic systems. - M.: Mir, 1975. - 500 p.

3. Kramer G. Mathematical methods of statistics. - M.: Mir, 1948 (1st ed.), 1975 (2nd ed.). - 648 p.

4. Bolshev L. N., Smirnov N. V. Tables of mathematical statistics. - M.: Nauka, 1965 (1st ed.), 1968 (2nd ed.), 1983 (3rd ed.).

5. Smirnov N. V., Dunin-Barkovsky I. V. A course in the theory of probability and mathematical statistics for technical applications. Ed. 3rd, stereotypical. - M.: Nauka, 1969. - 512 p.

6. Norman Draper, Harry Smith Applied regression analysis. Multiple Regression = Applied Regression Analysis. - 3rd ed. - M .: "Dialectics", 2007. - S. 912. - ISBN 0-471-17082-8

See also

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• Amalgam (disambiguation)

See what "Statistical Methods" is in other dictionaries:

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STATISTICAL METHODS -- scientific methods of description and study of mass phenomena that allow quantitative (numerical) expression. The word "statistics" (from Italian stato - state) has a common root with the word "state". Initially, it referred to the science of management and ... Philosophical Encyclopedia

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statistical methods- (in psychology) (from Latin status status) some methods of applied mathematical statistics used in psychology mainly for processing experimental results. The main purpose of using S. m is to increase the validity of conclusions in ... ... Great Psychological Encyclopedia

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What is "mathematical statistics"

Under mathematical statistics understand “a section of mathematics devoted to the mathematical methods of collecting, systematizing, processing and interpreting statistical data, as well as using them for scientific or practical conclusions. The rules and procedures of mathematical statistics are based on the theory of probability, which makes it possible to evaluate the accuracy and reliability of the conclusions obtained in each problem on the basis of the available statistical material. At the same time, statistical data refers to information about the number of objects in any more or less extensive collection that have certain characteristics.

According to the type of problems being solved, mathematical statistics is usually divided into three sections: data description, estimation, and hypothesis testing.

According to the type of statistical data being processed, mathematical statistics is divided into four areas:

• - one-dimensional statistics (statistics of random variables), in which the observation result is described by a real number;
• - multivariate statistical analysis, where the result of observation of an object is described by several numbers (vector);
• - statistics of random processes and time series, where the result of observation is a function;
• - statistics of objects of non-numerical nature, in which the result of observation has a non-numerical nature, for example, it is a set (geometric figure), ordering, or obtained as a result of measurement by a qualitative attribute.

Historically, some areas of statistics of objects of non-numerical nature (in particular, problems of estimating the percentage of defective products and testing hypotheses about it) and one-dimensional statistics were the first to appear. The mathematical apparatus is simpler for them, therefore, by their example, they usually demonstrate the main ideas of mathematical statistics.

Only those methods of data processing, ie. mathematical statistics are evidence-based, which are based on probabilistic models of relevant real phenomena and processes. It's about about models of consumer behavior, the occurrence of risks, the functioning of technological equipment, obtaining the results of an experiment, the course of a disease, etc. A probabilistic model of a real phenomenon should be considered built if the quantities under consideration and the relationships between them are expressed in terms of probability theory. Correspondence to the probabilistic model of reality, i.e. its adequacy is substantiated, in particular, with the help of statistical methods for testing hypotheses.

Incredible data processing methods are exploratory, they can only be used in preliminary data analysis, since they do not make it possible to assess the accuracy and reliability of the conclusions obtained on the basis of limited statistical material.

Probabilistic and statistical methods are applicable wherever it is possible to construct and substantiate a probabilistic model of a phenomenon or process. Their use is mandatory when conclusions drawn from sample data are transferred to the entire population (for example, from a sample to an entire batch of products).

In specific areas of application, both probabilistic-statistical methods of wide application and specific ones are used. For example, in the section of production management devoted to statistical methods of product quality management, applied mathematical statistics (including the design of experiments) are used. With the help of its methods, a statistical analysis of the accuracy and stability of technological processes is carried out and statistical evaluation quality. Specific methods include methods of statistical acceptance control of product quality, statistical regulation of technological processes, assessment and control of reliability, etc.

Such applied probabilistic-statistical disciplines as reliability theory and queuing theory are widely used. The content of the first of them is clear from the title, the second deals with the study of systems such as a telephone exchange, which receives calls at random times - the requirements of subscribers dialing numbers on their telephones. The duration of the service of these requirements, i.e. duration of calls, also modeled random variables. A great contribution to the development of these disciplines was made by Corresponding Member of the USSR Academy of Sciences A.Ya. Khinchin (1894-1959), academician of the Academy of Sciences of the Ukrainian SSR B.V. Gnedenko (1912-1995) and other domestic scientists.