Probabilistic (statistical) method of risk assessment. Probabilistic-statistical research methods and the method of system analysis Questions for self-testing of knowledge
In scientific knowledge there is a complex, dynamic, holistic, subordinated system of diverse methods used at different stages and levels of knowledge. Thus, in the process of scientific research, various general scientific methods and means of cognition are used both at the empirical and theoretical levels. In turn, general scientific methods, as already noted, include a system of empirical, general logical and theoretical methods and means of knowing reality.
1. General logical methods of scientific research
General logical methods are used primarily at the theoretical level of scientific research, although some of them can also be used at the empirical level. What are these methods and what is their essence?
One of them, widely used in scientific research, is analysis method (from the Greek analysis - decomposition, dismemberment) - a method of scientific knowledge, which is the mental division of the object under study into its component elements in order to study its structure, individual characteristics, properties, internal connections, relationships.
Analysis allows the researcher to penetrate into the essence of the phenomenon being studied by dividing it into its component elements and identifying the main, essential. Analysis as a logical operation is an integral part of any scientific research and usually forms its first stage, when the researcher moves from an undifferentiated description of the object being studied to identifying its structure, composition, as well as its properties and connections. Analysis is already present at the sensory stage of cognition and is included in the process of sensation and perception. At the theoretical level of cognition, the highest form of analysis begins to function - mental, or abstract-logical analysis, which arises along with the skills of material and practical division of objects in the labor process. Gradually, man mastered the ability to transform material and practical analysis into mental analysis.
It should be emphasized that, being a necessary method of cognition, analysis is only one of the moments in the process of scientific research. It is impossible to know the essence of an object only by dividing it into the elements of which it consists. For example, a chemist, according to Hegel, places a piece of meat in his retort, subjects it to various operations, and then declares: I found that meat consists of oxygen, carbon, hydrogen, etc. But these substances - elements are no longer the essence of meat .
Each area of knowledge has, as it were, its own limit of division of an object, beyond which we move on to a different nature of properties and patterns. When particulars are studied through analysis, the next stage of cognition begins - synthesis.
Synthesis (from the Greek synthesis - connection, combination, composition) is a method of scientific knowledge, which is a mental connection of the constituent aspects, elements, properties, connections of the object under study, dissected as a result of analysis, and the study of this object as a single whole.
Synthesis is not an arbitrary, eclectic combination of parts, elements of the whole, but a dialectical whole with the highlighting of the essence. The result of the synthesis is a completely new formation, the properties of which are not only the external connection of these components, but also the result of their internal interconnection and interdependence.
Analysis mainly captures what is specific that distinguishes parts from each other. Synthesis reveals that essential commonality that connects the parts into a single whole.
The researcher mentally dissects an object into its component parts in order to first discover these parts themselves, find out what the whole consists of, and then consider it as consisting of these parts, already examined separately. Analysis and synthesis are in a dialectical unity: our thinking is as analytical as it is synthetic.
Analysis and synthesis originate in practical activities. Constantly dividing various objects into their component parts in his practical activities, man gradually learned to separate objects mentally. Practical activity consisted not only of dismembering objects, but also of reuniting parts into a single whole. On this basis, mental analysis and synthesis gradually arose.
Depending on the nature of the study of the object and the depth of penetration into its essence, various types of analysis and synthesis are used.
1. Direct or empirical analysis and synthesis - used, as a rule, at the stage of superficial familiarization with the object. This type of analysis and synthesis makes it possible to understand the phenomena of the object being studied.
2. Elementary theoretical analysis and synthesis - widely used as a powerful tool for understanding the essence of the phenomenon under study. The result of using such analysis and synthesis is the establishment of cause-and-effect relationships and the identification of various patterns.
3. Structural-genetic analysis and synthesis - allows you to penetrate most deeply into the essence of the object being studied. This type of analysis and synthesis requires isolating in a complex phenomenon those elements that represent the most important, significant and have a decisive influence on all other aspects of the object being studied.
Methods of analysis and synthesis in the process of scientific research function inextricably linked with the method of abstraction.
Abstraction (from Latin abstractio - abstraction) is a general logical method of scientific knowledge, which is a mental abstraction from the unimportant properties, connections, relationships of the objects being studied with the simultaneous mental highlighting of the essential aspects, properties, connections of these objects that are of interest to the researcher. Its essence lies in the fact that a thing, property or relationship is mentally isolated and at the same time abstracted from other things, properties, relationships and is considered as if in its “pure form”.
Abstraction in human mental activity has a universal character, because every step of thought is associated with this process, or with the use of its results. The essence of this method is that it allows one to mentally distract from unimportant, secondary properties, connections, relationships of objects and at the same time mentally highlight and record the aspects, properties, connections of these objects that are of interest to the study.
There is a distinction between the process of abstraction and the result of this process, which is called abstraction. Usually, the result of abstraction is understood as knowledge about certain aspects of the objects being studied. The abstraction process is a set of logical operations leading to obtaining such a result (abstraction). Examples of abstractions include countless concepts that people operate not only in science, but also in everyday life.
The question of what in objective reality is highlighted by the abstracting work of thinking and what thinking is distracted from is decided in each specific case depending on the nature of the object being studied, as well as on the objectives of the research. In the course of its historical development, science ascends from one level of abstraction to another, higher one. The development of science in this aspect is, in the words of W. Heisenberg, “the deployment of abstract structures.” The decisive step into the realm of abstraction was made when people mastered counting (number), thereby opening the path leading to mathematics and mathematical science. In this regard, W. Heisenberg notes: “Concepts, initially obtained by abstracting from concrete experience, take on a life of their own. They turn out to be more meaningful and productive than could be expected at first. In subsequent development, they discover their own constructive capabilities: they contribute to the construction of new forms and concepts, allow us to establish connections between them and can be, to a certain extent, applicable in our attempts to understand the world of phenomena."
A brief analysis suggests that abstraction is one of the most fundamental cognitive logical operations. Therefore, it is the most important method of scientific research. The method of generalization is also closely related to the method of abstraction.
Generalization - a logical process and the result of a mental transition from the individual to the general, from the less general to the more general.
Scientific generalization is not just a mental selection and synthesis of similar features, but penetration into the essence of a thing: the discernment of the unified in the diverse, the general in the individual, the natural in the random, as well as the unification of objects according to similar properties or connections into homogeneous groups, classes.
In the process of generalization, a transition occurs from individual concepts to general ones, from less general concepts to more general ones, from individual judgments to general ones, from judgments of lesser generality to judgments of greater generality. Examples of such a generalization can be: a mental transition from the concept of “mechanical form of motion of matter” to the concept of “form of motion of matter” and “movement” in general; from the concept of “spruce” to the concept of “coniferous plant” and “plant” in general; from the judgment “this metal is electrically conductive” to the judgment “all metals are electrically conductive.”
In scientific research, the following types of generalization are most often used: inductive, when the researcher proceeds from individual (single) facts or events to their general expression in thoughts; logical, when the researcher goes from one, less general, thought to another, more general. The limit of generalization is philosophical categories that cannot be generalized because they do not have a generic concept.
The logical transition from a more general thought to a less general one is a process of limitation. In other words, this is a logical operation, the inverse of generalization.
It must be emphasized that the human ability to abstract and generalize has formed and developed on the basis of social practice and mutual communication of people. It is of great importance both in the cognitive activity of people and in the general progress of the material and spiritual culture of society.
Induction (from Latin i nductio - guidance) - a method of scientific knowledge in which the general conclusion represents knowledge about the entire class of objects obtained as a result of the study of individual elements of this class. In induction, the researcher’s thought goes from the particular, the individual, through the particular to the general and universal. Induction, as a logical method of research, is associated with the generalization of the results of observations and experiments, with the movement of thought from the individual to the general. Since experience is always infinite and incomplete, inductive conclusions always have a problematic (probabilistic) nature. Inductive generalizations are usually regarded as empirical truths or empirical laws. The immediate basis of induction is the repeatability of the phenomena of reality and their signs. Finding similar features in many objects of a certain class, we come to the conclusion that these features are inherent in all objects of this class.
Based on the nature of the conclusion, the following main groups of inductive inferences are distinguished:
1. Complete induction is an inference in which a general conclusion about a class of objects is made based on the study of all objects of this class. Complete induction produces reliable conclusions, which is why it is widely used as evidence in scientific research.
2. Incomplete induction is a conclusion in which the general conclusion is obtained from premises that do not cover all objects of a given class. There are two types of incomplete induction: popular, or induction through simple enumeration. It represents an inference in which a general conclusion about a class of objects is made on the basis that among the observed facts there is not a single one that contradicts the generalization; scientific, i.e. an inference in which a general conclusion about all objects of a class is made on the basis of knowledge about the necessary characteristics or causal relationships of some objects of a given class. Scientific induction can produce not only probabilistic but also reliable conclusions. Scientific induction has its own methods of cognition. The fact is that it is very difficult to establish a causal relationship between phenomena. However, in some cases, this connection can be established using logical techniques called methods of establishing a cause-and-effect relationship, or methods of scientific induction. There are five such methods:
1. Single similarity method: if two or more cases of the phenomenon under study have only one circumstance in common, and all other circumstances are different, then this only similar circumstance is the cause of this phenomenon:
Therefore -+ A is the cause of a.
In other words, if the preceding circumstances ABC cause the phenomenon abc, and the circumstances ADE cause the phenomenon ade, then the conclusion is drawn that A is the cause of a (or that the phenomenon A and a are causally related).
2. Method of single difference: if the cases in which a phenomenon occurs or does not occur differ only in one thing: - the preceding circumstance, and all other circumstances are identical, then this one circumstance is the cause of this phenomenon:
In other words, if the preceding circumstances ABC cause the phenomenon abc, and the circumstances BC (the phenomenon A is eliminated during the experiment) cause the phenomenon bc, then the conclusion is drawn that A is the cause of a. The basis for this conclusion is the disappearance of a when A is eliminated.
3. The combined similarity and difference method is a combination of the first two methods.
4. Method of accompanying changes: if the occurrence or change of one phenomenon always necessarily causes a certain change in another phenomenon, then both of these phenomena are in a causal relationship with each other:
Change A change a
No change in B, C
Therefore A is the cause of a.
In other words, if when a previous phenomenon A changes, the observed phenomenon a also changes, and the remaining previous phenomena remain unchanged, then we can conclude that A is the cause of a.
5. Method of residuals: if it is known that the cause of the phenomenon being studied is not the circumstances necessary for it, except for one, then this one circumstance is probably the cause of this phenomenon. Using the residual method, the French astronomer Nevereux predicted the existence of the planet Neptune, which was soon discovered by the German astronomer Halle.
The considered methods of scientific induction to establish causal relationships are most often used not in isolation, but in conjunction, complementing each other. Their value depends mainly on the degree of probability of the conclusion that a particular method gives. It is believed that the most powerful method is the method of difference, and the weakest is the method of similarity. The remaining three methods occupy an intermediate position. This difference in the value of methods is based mainly on the fact that the method of similarity is associated primarily with observation, and the method of difference with experiment.
Even a brief description of the induction method allows one to verify its dignity and importance. The significance of this method lies primarily in its close connection with facts, experiment, and practice. In this regard, F. Bacon wrote: “If we mean to penetrate into the nature of things, then we everywhere turn to induction. For we believe that induction is a real form of proof, protecting the senses from all kinds of errors, closely following nature, bordering and almost merging with practice."
In modern logic, induction is considered as a theory of probabilistic inference. Attempts are being made to formalize the inductive method based on the ideas of probability theory, which will help to more clearly understand the logical problems of this method, as well as determine its heuristic value.
Deduction (from Latin deductio - deduction) - a thought process in which knowledge about an element of a class is derived from knowledge of the general properties of the entire class. In other words, the researcher’s thought in deduction goes from the general to the particular (individual). For example: “All the planets of the solar system move around the sun”; "Planet Earth"; hence: “The earth moves around the sun.” In this example, thought moves from the general (first premise) to the specific (conclusion). Thus, deductive inference allows us to better understand an individual, since with its help we obtain new knowledge (inferential) that a given object has a characteristic characteristic of the entire class.
The objective basis of deduction is that each object combines the unity of the general and the individual. This connection is inextricable, dialectical, which allows us to cognize the individual on the basis of knowledge of the general. Moreover, if the premises of a deductive inference are true and correctly connected with each other, then the conclusion - the conclusion will certainly be true. This feature distinguishes deduction from other methods of cognition. The fact is that general principles and laws do not allow the researcher to go astray in the process of deductive knowledge; they help to correctly understand individual phenomena of reality. However, it would be wrong to overestimate the scientific significance of the deductive method on this basis. Indeed, in order for the formal power of inference to come into its own, we need initial knowledge, general premises that are used in the process of deduction, and acquiring them in science is a task of great complexity.
The important cognitive significance of deduction is manifested when the general premise is not just an inductive generalization, but some kind of hypothetical assumption, for example, a new scientific idea. In this case, deduction is the starting point for the emergence of a new theoretical system. The theoretical knowledge created in this way predetermines the construction of new inductive generalizations.
All this creates real preconditions for the steady increase in the role of deduction in scientific research. Science is increasingly encountering objects that are inaccessible to sensory perception (for example, the microcosm, the Universe, the past of humanity, etc.). When learning about objects of this kind, one has to resort to the power of thought much more often than to the power of observation and experiment. Deduction is indispensable in all areas of knowledge where theoretical principles are formulated to describe formal rather than real systems, for example, in mathematics. Since formalization in modern science is used more and more widely, the role of deduction in scientific knowledge increases accordingly.
However, the role of deduction in scientific research cannot be absolutized, much less opposed to induction and other methods of scientific knowledge. Extremes of both a metaphysical and rationalistic nature are unacceptable. On the contrary, deduction and induction are closely interrelated and complement each other. Inductive research involves the use of general theories, laws, principles, i.e., it includes the element of deduction, and deduction is impossible without general provisions obtained inductively. In other words, induction and deduction are related to each other in the same necessary way as analysis and synthesis. We must try to apply each of them in its place, and this can only be achieved if we do not lose sight of their connection with each other, their mutual complement to each other. “Great discoveries,” notes L. de Broglie, “leaps of scientific thought forward are created by induction, a risky, but truly creative method... Of course, there is no need to conclude that the rigor of deductive reasoning has no value. In fact, only it prevents the imagination from falling into error, only it allows, after establishing new starting points by induction, to draw consequences and compare conclusions with facts. Only deduction can provide testing of hypotheses and serve as a valuable antidote against over-extended imagination." With such a dialectical approach, each of the mentioned and other methods of scientific knowledge will be able to fully demonstrate all their advantages.
Analogy. When studying the properties, signs, connections of objects and phenomena of real reality, we cannot cognize them immediately, entirely, in their entirety, but we study them gradually, revealing step by step more and more new properties. Having studied some of the properties of an object, we may find that they coincide with the properties of another, already well-studied object. Having established such similarity and discovered many matching features, we can assume that other properties of these objects also coincide. The course of such reasoning forms the basis of the analogy.
Analogy is a method of scientific research with the help of which, from the similarity of objects of a given class in some characteristics, a conclusion is drawn about their similarity in other characteristics. The essence of the analogy can be expressed using the formula:
A has signs of aecd
B has signs of ABC
Therefore, B appears to have attribute d.
In other words, in an analogy the researcher’s thought goes from knowledge of a known generality to knowledge of the same generality, or, in other words, from particular to particular.
Regarding specific objects, conclusions obtained by analogy are, as a rule, only plausible in nature: they are one of the sources of scientific hypotheses, inductive reasoning and play an important role in scientific discoveries. For example, the chemical composition of the Sun is similar to the chemical composition of the Earth in many ways. Therefore, when the element helium, not yet known on Earth, was discovered on the Sun, they concluded by analogy that a similar element should be on Earth. The correctness of this conclusion was established and confirmed later. In a similar way, L. de Broglie, having assumed a certain similarity between particles of matter and the field, came to the conclusion about the wave nature of particles of matter.
To increase the likelihood of drawing conclusions by analogy, it is necessary to strive to:
not only the external properties of the compared objects were identified, but mainly the internal ones;
these objects were similar in the most important and essential characteristics, and not in random and secondary ones;
the range of matching features was as wide as possible;
Not only similarities, but also differences were taken into account so that the latter would not be transferred to another object.
The analogy method gives the most valuable results when an organic relationship is established not only between similar features, but also with the feature that is transferred to the object under study.
The truth of conclusions by analogy can be compared with the truth of conclusions by the method of incomplete induction. In both cases, reliable conclusions can be obtained, but only when each of these methods is not applied in isolation from other methods of scientific knowledge, but in an inextricable dialectical connection with them.
The method of analogy, understood extremely broadly as the transfer of information about one object to another, constitutes the epistemological basis of modeling.
Modeling - a method of scientific knowledge, with the help of which the study of an object (original) is carried out by creating a copy (model) of it, replacing the original, which is then learned from certain aspects that interest the researcher.
The essence of the modeling method is to reproduce the properties of the object of knowledge on a specially created analogue, model. What is a model?
A model (from the Latin modulus - measure, image, norm) is a conventional image of an object (original), a certain way of expressing the properties, connections of objects and phenomena of reality on the basis of analogy, establishing similarities between them and on this basis reproducing them on a material or ideal object-likeness. In other words, a model is an analogue, a “substitute” of the original object, which in cognition and practice serves to acquire and expand knowledge (information) about the original for the purpose of constructing, transforming or managing the original.
There must be a certain similarity (relationship of similarity) between the model and the original: physical characteristics, functions, behavior of the object being studied, its structure, etc. It is this similarity that allows the information obtained as a result of studying the model to be transferred to the original.
Since modeling is very similar to the method of analogy, the logical structure of inference by analogy is, as it were, an organizing factor that unites all aspects of modeling into a single, purposeful process. One might even say that in a certain sense, modeling is a type of analogy. The analogy method serves as a logical basis for the conclusions that are drawn during modeling. For example, based on the property abcd belonging to model A and the properties abc belonging to the original A, it is concluded that the property d discovered in model A also belongs to the original A.
The use of modeling is dictated by the need to reveal aspects of objects that either cannot be comprehended through direct study, or are unprofitable to study for purely economic reasons. A person, for example, cannot directly observe the process of natural formation of diamonds, the origin and development of life on Earth, a number of phenomena of the micro- and mega-world. Therefore, we have to resort to artificial reproduction of such phenomena in a form convenient for observation and study. In some cases, it is much more profitable and economical to build and study its model instead of directly experimenting with an object.
Modeling is widely used to calculate the trajectories of ballistic missiles, to study the operating modes of machines and even entire enterprises, as well as in the management of enterprises, in the distribution of material resources, in the study of life processes in the body and in society.
Models used in everyday and scientific knowledge are divided into two large classes: real, or material, and logical (mental), or ideal. The former are natural objects that obey natural laws in their functioning. They materially reproduce the subject of research in a more or less visual form. Logical models are ideal formations, fixed in the appropriate symbolic form and functioning according to the laws of logic and mathematics. The importance of iconic models is that, with the help of symbols, they make it possible to reveal such connections and relationships of reality that are almost impossible to detect by other means.
At the present stage of scientific and technological progress, computer modeling has become widespread in science and in various fields of practice. A computer running a special program is capable of simulating a wide variety of processes, for example, fluctuations in market prices, population growth, takeoff and entry into orbit of an artificial Earth satellite, chemical reactions, etc. The study of each such process is carried out using an appropriate computer model.
System method . The modern stage of scientific knowledge is characterized by the increasing importance of theoretical thinking and theoretical sciences. Systems theory, which analyzes systemic research methods, occupies an important place among the sciences. In the systemic method of cognition, the dialectic of the development of objects and phenomena of reality finds the most adequate expression.
A systematic method is a set of general scientific methodological principles and methods of research, which are based on an orientation toward revealing the integrity of an object as a system.
The basis of the systems method is the system and structure, which can be defined as follows.
A system (from the Greek systema - a whole made up of parts; a connection) is a general scientific position that expresses a set of elements that are interconnected both with each other and with the environment and form a certain integrity, the unity of the object being studied. The types of systems are very diverse: material and spiritual, inorganic and living, mechanical and organic, biological and social, static and dynamic, etc. Moreover, any system is a collection of various elements that make up its specific structure. What is structure?
Structure ( from lat. structura - structure, arrangement, order) is a relatively stable way (law) of connecting the elements of an object, which ensures the integrity of a particular complex system.
The specificity of the systems approach is determined by the fact that it focuses research on revealing the integrity of the object and the mechanisms that provide it, identifying the diverse types of connections of a complex object and bringing them together into a single theoretical picture.
The main principle of the general theory of systems is the principle of system integrity, which means viewing nature, including society, as a large and complex system, breaking up into subsystems that, under certain conditions, act as relatively independent systems.
The whole variety of concepts and approaches in the general theory of systems can, with a certain degree of abstraction, be divided into two large classes of theories: empirical-intuitive and abstract-deductive.
1. In empirical-intuitive concepts, specific, really existing objects are considered as the primary object of research. In the process of ascending from the specific individual to the general, the concepts of the system and systemic principles of research at different levels are formulated. This method has an external resemblance to the transition from the individual to the general in empirical knowledge, but behind the external similarity lies a certain difference. It consists in the fact that if the empirical method proceeds from the recognition of the primacy of elements, then the systems approach proceeds from the recognition of the primacy of systems. In the systems approach, systems are taken as a starting point for research as an integral formation consisting of many elements along with their connections and relationships, subject to certain laws; the empirical method is limited to the formulation of laws expressing the relationships between the elements of a given object or a given level of phenomena. And although there is a moment of generality in these laws, this generality, however, refers to a narrow class of mostly identical objects.
2. In abstract-deductive concepts, abstract objects - systems characterized by extremely general properties and relationships - are taken as the starting point for research. A further descent from extremely general systems to more and more specific ones is accompanied by the simultaneous formulation of such system principles that apply to specifically defined classes of systems.
Empirical-intuitive and abstract-deductive approaches are equally legitimate; they are not opposed to each other, but on the contrary - their joint use opens up extremely large cognitive possibilities.
The systems method allows you to scientifically interpret the principles of organization of systems. The objectively existing world appears as a world of certain systems. Such a system is characterized not only by the presence of interconnected components and elements, but also by their certain orderliness, organization on the basis of a certain set of laws. Therefore, systems are not chaotic, but ordered and organized in a certain way.
In the process of research, one can, of course, “ascend” from elements to integral systems, as well as vice versa - from integral systems to elements. But under all circumstances, research cannot be isolated from systemic connections and relationships. Ignoring such connections inevitably leads to one-sided or erroneous conclusions. It is no coincidence that in the history of knowledge, straightforward and one-sided mechanism in explaining biological and social phenomena slid into the position of recognition of the first impulse and spiritual substance.
Based on the above, the following basic requirements of the system method can be identified:
Identification of the dependence of each element on its place and functions in the system, taking into account the fact that the properties of the whole are not reducible to the sum of the properties of its elements;
Analysis of the extent to which the behavior of a system is determined both by the characteristics of its individual elements and the properties of its structure;
Study of the mechanism of interdependence, interaction between the system and the environment;
Studying the nature of hierarchy inherent in a given system;
Ensuring a plurality of descriptions for the purpose of multidimensional coverage of the system;
Consideration of the dynamism of the system, its presentation as a developing integrity.
An important concept of the systems approach is the concept of “self-organization”. It characterizes the process of creating, reproducing or improving the organization of a complex, open, dynamic, self-developing system, the connections between the elements of which are not rigid, but probabilistic. The properties of self-organization are inherent in objects of a very different nature: a living cell, an organism, a biological population, human groups.
The class of systems capable of self-organization are open and nonlinear systems. The openness of a system means the presence of sources and sinks, exchange of matter and energy with the environment. However, not every open system self-organizes and builds structures, because everything depends on the relationship between two principles - on the basis that creates the structure, and on the basis that scatters and erodes this principle.
In modern science, self-organizing systems are a special subject of study of synergetics - a general scientific theory of self-organization, focused on the search for the laws of evolution of open nonequilibrium systems of any basic basis - natural, social, cognitive (cognitive).
Currently, the systemic method is gaining increasing methodological importance in solving natural science, socio-historical, psychological and other problems. It is widely used by almost all sciences, which is due to the urgent epistemological and practical needs of the development of science at the present stage.
Probabilistic (statistical) methods - these are methods with the help of which the action of many random factors characterized by a stable frequency is studied, which makes it possible to detect the need that “breaks through” the cumulative effect of many randomnesses.
Probabilistic methods are formed on the basis of probability theory, which is often called the science of randomness, and in the minds of many scientists, probability and randomness are practically inseparable. The categories of necessity and chance are by no means outdated; on the contrary, their role in modern science has increased immeasurably. As the history of knowledge has shown, “we are only now beginning to appreciate the significance of the entire range of problems associated with necessity and chance.”
To understand the essence of probabilistic methods, it is necessary to consider their basic concepts: “dynamic patterns”, “statistical patterns” and “probability”. These two types of patterns differ in the nature of the predictions that follow from them.
In dynamic type laws, predictions are unambiguous. Dynamic laws characterize the behavior of relatively isolated objects, consisting of a small number of elements, in which it is possible to abstract from a number of random factors, which makes it possible to more accurately predict, for example, in classical mechanics.
In statistical laws, predictions are not reliable, but only probabilistic. This nature of predictions is due to the action of many random factors that occur in statistical phenomena or mass events, for example, a large number of molecules in a gas, the number of individuals in populations, the number of people in large groups, etc.
A statistical pattern arises as a result of the interaction of a large number of elements that make up an object - a system, and therefore characterizes not so much the behavior of an individual element, but rather the behavior of the object as a whole. The necessity manifested in statistical laws arises as a result of mutual compensation and balancing of many random factors. “Although statistical patterns can lead to statements whose degree of probability is so high that it borders on certainty, nevertheless, in principle, exceptions are always possible.”
Statistical laws, although they do not give unambiguous and reliable predictions, are nevertheless the only possible ones in the study of mass phenomena of a random nature. Behind the combined action of various factors of a random nature, which are practically impossible to cover, statistical laws reveal something stable, necessary, and repeating. They serve as confirmation of the dialectic of the transition of the accidental into the necessary. Dynamic laws turn out to be a limiting case of statistical ones, when probability becomes practically certainty.
Probability is a concept that characterizes the quantitative measure (degree) of the possibility of the occurrence of some random event under certain conditions that can be repeated many times. One of the main tasks of probability theory is to clarify the patterns that arise from the interaction of a large number of random factors.
Probabilistic-statistical methods are widely used in the study of mass phenomena, especially in such scientific disciplines as mathematical statistics, statistical physics, quantum mechanics, cybernetics, and synergetics.
When conducting psychological and pedagogical research, an important role is given to mathematical methods of modeling processes and processing experimental data. These methods include, first of all, the so-called probabilistic-statistical research methods. This is due to the fact that the behavior of both an individual in the process of his activity and a person in a team is significantly influenced by many random factors. Randomness does not allow us 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 patterns are discovered. The irregularity inherent in random events, with a large number of tests, is usually compensated by the emergence of a statistical pattern, stabilization of the frequency of occurrence of random events. Therefore, these random events have a certain probability. There are two fundamentally different probabilistic-statistical methods of psychological and pedagogical research: classical and non-classical. Let us conduct a comparative analysis of these methods.
Classic probabilistic-statistical method. The classical probabilistic-statistical research method is based on probability theory and mathematical statistics. This method is used in the study of mass phenomena of a random nature; it includes several stages, the main of which are the following.
1. Construction of a probabilistic model of reality based on the analysis of statistical data (determination of the distribution law of a random variable). Naturally, the patterns of mass random phenomena are expressed more clearly the larger the volume of statistical material. Sample data obtained during an experiment are always limited and, strictly speaking, of a random nature. In this regard, an important role is given to generalizing the patterns obtained from the sample and extending them to the entire population of objects. In order to solve this problem, a certain hypothesis is accepted about the nature of the statistical pattern that manifests itself in the phenomenon under study, for example, the hypothesis that the phenomenon under study obeys the law of normal distribution. This hypothesis is called the null hypothesis, which may turn out to be false, therefore, along with the null hypothesis, an alternative or competing hypothesis is also put forward. Checking how well the obtained experimental data corresponds to a particular statistical hypothesis is carried out using so-called nonparametric statistical tests or goodness-of-fit tests. Currently, Kolmogorov, Smirnov, omega-square, etc. goodness-of-fit criteria are widely used. The basic idea of these tests is to measure the distance between the empirical distribution function and the fully known theoretical distribution function. The methodology for testing a statistical hypothesis has been rigorously developed and outlined in a large number of works on mathematical statistics.
2. Carrying out the necessary calculations using mathematical means within the framework of a probabilistic model. In accordance with the established probabilistic model of the phenomenon, calculations of characteristic parameters are carried out, for example, such as the mathematical expectation or mean value, dispersion, standard deviation, mode, median, asymmetry index, etc.
3. Interpretation of probabilistic and statistical conclusions in relation to the real situation.
Currently, the classical probabilistic-statistical method is well developed and widely used in research in various fields of 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 events, but also to individual events that are fundamentally random in nature. This method can be effectively used in analyzing the behavior of an individual in the process of performing a particular activity, for example, in the process of assimilation of knowledge by a student. We will consider the features of the non-classical probabilistic-statistical method of psychological and pedagogical research using the example of student behavior in the process of acquiring knowledge.
For the first time, a probabilistic-statistical model of student behavior in the process of acquiring knowledge was proposed in the work. Further development of this model was done in the work. 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. Analysis of these processes shows that they are characterized by elements of randomness, due to the random nature of the mental and somatic states of the individual, as well as physiological, psychological and informational noise during the work of the brain. The latter led, when describing thinking processes, to abandoning the use of a deterministic dynamic system model in favor of a random dynamic system model. 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 nature, and, therefore, a probabilistic-statistical method can be used to describe the behavior of each individual student in the process of acquiring knowledge.
In accordance with this method, a student is identified by a distribution function (probability density), which determines the probability of finding him in a single region of the information space. During the learning process, the distribution function with which the student is identified moves in the information space as it evolves. 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, a system of differential equations is written, which are continuity equations that relate the change in probability density per unit time in phase space (the space of coordinates, velocities and accelerations of various orders) with the divergence of the probability density flow in the phase space under consideration. The analysis of analytical solutions of a number of continuity equations (distribution functions) characterizing the behavior of individual students in the learning process was carried out.
When conducting experimental studies of student behavior in the process of acquiring knowledge, probabilistic-statistical scaling is used, according to which the measurement scale is an ordered system
1. Finding experimental distribution functions based on the results of a control event, for example, an exam. A typical form of individual distribution functions found using a twenty-point scale is presented in Fig. 1. The method for finding such functions is described in.
2. Mapping distribution functions onto number space. For this purpose, the moments of individual distribution functions are calculated. In practice, as a rule, it is enough to limit ourselves to determining the moments of the first order (mathematical expectation), second order (variance) and third order, characterizing the asymmetry of the distribution function.
3. Ranking of students by level of knowledge based on comparison of moments of different orders of their individual distribution functions.
Rice. 1. Typical form of individual distribution functions of students who received different grades in the general physics exam: 1 - traditional grade “2”; 2 - traditional rating “3”; 3 - traditional rating “4”; 4 - traditional rating “5”
Based on the additivity of individual distribution functions, experimental distribution functions for the flow of students were found (Fig. 2).
Rice. 2. Evolution of the complete distribution function of the student flow, approximated by smooth lines: 1 - after the first year; 2 - after the second year; 3 - after the third year; 4 - after the fourth year; 5 - after the fifth year
Analysis of the data presented in Fig. 2 shows that as we move through the information space, the distribution functions become blurred. This occurs due to the fact that the mathematical expectations of the distribution functions of individuals move at different speeds, and the functions themselves blur 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, is that the classical method is applicable only to the analysis of mass events, and the non-classical method is applicable to both the analysis of mass and single events. In this regard, the classical method can be conditionally called the mass probabilistic-statistical method (MPSM), and the non-classical method - the individual probabilistic-statistical method (IPSM). In 4] it is shown that none of the classical methods for assessing students’ knowledge within the framework of a probabilistic-statistical model of an individual can be applied for these purposes.
Let's consider the distinctive features of the MVSM and IVSM methods using the example of measuring the completeness of students' knowledge. To this end, let's conduct a thought experiment. Let us assume 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 classical ideas about objects of measurement, all students should receive the same assessments of the completeness of knowledge with any given measurement accuracy. However, in reality, with sufficiently high measurement accuracy, assessments of the completeness of students' knowledge will differ. It is not possible to explain this measurement result within the framework of MVSM, 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 cognition process is realized through randomness, which is inherent in every individual cognizing the world around him.
The random nature of student behavior in the process of acquiring knowledge takes into account IVSM. The use of an individual probabilistic-statistical method to analyze the behavior of the idealized group of students under consideration would show that it is impossible to indicate the exact position of each student in the information space; one can only say the probability of finding him 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 individual distribution functions will be located in different areas of the information space. The reason for this behavior of students lies in the random nature of the learning process.
However, in a number of cases, research results obtained within the framework of the IVSM can be interpreted within the framework of the IVSM. Let's assume that a teacher uses a five-point measurement scale when assessing a student's knowledge. In this case, the error in assessing knowledge is ±0.5 points. Therefore, when a student is given a grade of, for example, 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 allows us 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 (measurer), who is also characterized by randomness. In the process of pedagogical measurements, there actually is an interaction between two random dynamic systems that identify the behavior of the student and the teacher in this process. The interaction of the student subsystem with the teaching subsystem is considered and it is shown that the speed of movement of the mathematical expectation of individual distribution functions of students in the information space is proportional to the function of the influence of the teaching staff and is inversely proportional to the inertia function, which characterizes the intractability of changing the position of the mathematical expectation in space (an analogue of Aristotle’s law in mechanics).
At present, despite significant achievements in the development of theoretical and practical foundations for measurements when conducting psychological and pedagogical research, the problem of measurement as a whole is still far from being solved. This is due, first of all, to the fact that there is still not enough information about the influence of consciousness on the measurement process. A similar situation arose when solving the measurement problem in quantum mechanics. Thus, in the work, when considering the conceptual problems of the quantum theory of measurements, it is said that resolving some paradoxes of measurements in quantum mechanics “... is hardly possible without directly including the consciousness of the observer in the theoretical description of quantum measurement.” It goes on to say that “... it is consistent to assume that consciousness can make some event probable, 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 probable that he will see this event.”
Statistical methods
Statistical methods- methods of statistical data analysis. There are methods of applied statistics, which can be used in all areas of scientific research and any sectors of the national economy, and other statistical methods, the applicability of which is limited to one or another area. This refers to methods such as statistical acceptance control, statistical control of technological processes, reliability and testing, and planning 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 justify 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 the 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 area of activity;
c) application of statistical methods and models for statistical analysis of specific data.
Applied Statistics
A description of the type of data and the mechanism for its generation is the beginning of any statistical study. Both deterministic and probabilistic methods are used to describe data. Using deterministic methods, it is possible to analyze only the data that is available to the researcher. For example, with their help, tables were obtained that were calculated by official state statistics bodies based on statistical reports submitted by enterprises and organizations. The obtained results can be transferred to a wider population and used 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 contrast deterministic and probabilistic-statistical methods. We consider them as sequential steps of statistical analysis. At the first stage, it is necessary to analyze the available data and present it in an easy-to-read form using tables and charts. Then it is advisable to analyze the statistical data on the basis of certain probabilistic and statistical models. Note that the possibility of deeper insight into the essence of a real phenomenon or process is ensured by the development of an adequate mathematical model.
In the simplest situation, statistical data are the values of some characteristic characteristic of the objects being studied. Values can be quantitative or provide an indication of the category to which the object can be classified. In the second case, they talk about a qualitative sign.
When measuring by several quantitative or qualitative characteristics, we obtain a vector as statistical data about an object. It can be thought of as a new kind of data. In this case, the sample consists of a set of vectors. There are part of the coordinates - numbers, and part - qualitative (categorized) data, then we are talking about a vector of different types of data.
One element of the sample, that is, one dimension, can be the 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 beat of the motor shaft. Or a time series describing the dynamics of a particular company’s performance. Then the sample consists of a set of features.
Sample elements can also be other mathematical objects. For example, binary relationships. Thus, when surveying experts, they often use ordering (ranking) of objects of examination - product samples, investment projects, options for management decisions. Depending on the regulations of the expert study, the sampling elements can be various types of binary relations (ordering, partitioning, tolerance), sets, fuzzy sets, etc.
So, the mathematical nature of sample elements in various problems of applied statistics can be very different. However, two classes of statistical data can be distinguished - numerical and non-numerical. Accordingly, applied statistics is divided into two parts - numerical statistics and non-numerical statistics.
Numerical statistics are numbers, vectors, functions. They can be added and multiplied by coefficients. Therefore, in numerical statistics, various sums are of great importance. The mathematical apparatus for analyzing the sums of random elements of a sample is the (classical) laws of large numbers and central limit theorems.
Non-numerical statistical data are categorized data, vectors of different types of features, binary relations, sets, fuzzy sets, etc. They cannot be added and multiplied by coefficients. Therefore, it makes no sense to talk about sums of non-numeric statistics. They are elements of non-numerical mathematical spaces (sets). The mathematical apparatus for analyzing non-numerical statistical data is based on the use of distances between elements (as well as measures of proximity, indicators of difference) 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, diagnostic problems and cluster analysis are solved, etc. (see).
Applied research uses various types of statistical data. 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 operation of a number of devices before failure, and information that the remaining devices continued to operate at the end of the test. Censored data is often used in assessing and monitoring the reliability of technical devices.
Typically, statistical methods for analyzing data of the first three types are considered separately. This limitation is caused by the fact noted above that the mathematical apparatus for analyzing data of a non-numerical nature is significantly different than for data in the form of numbers, vectors and functions.
Probabilistic-statistical modeling
When applying statistical methods in specific fields 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 field of application. It is very instructive to compare probabilistic-statistical models used in various fields, to discover their similarities and at the same time to note some differences. Thus, one can see the similarity of problem statements and 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 "sample studies".
The difference between sample studies and expert studies is manifested, first of all, in the number of objects or subjects surveyed - in sample studies we are usually talking 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 logistic models, when processing narrative (text, chronicle) information or when studying the mutual influence of factors.
Issues of reliability and safety of technical devices and technologies, queuing theory are discussed in detail in a large number of scientific works.
Statistical analysis of specific data
The application of statistical methods and models for statistical analysis of specific data is closely tied to the problems of the relevant 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 are no less reasons 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 giving lectures on marketing research). The study of the dynamics of price growth using inflation indices calculated from independently collected information is of interest primarily from the point of view of economics and management of the 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 for the analysis of statistical data constantly arise in it, and new methods are developed and justified. Justification is often carried out by mathematical means, that is, by proving theorems. A major role is played by the methodological component - how exactly to set problems, what assumptions to accept for the purpose of further mathematical study. The role of modern information technologies, in particular, computer experiments, is great.
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 of 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. Course in probability theory 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. - P. 912. - ISBN 0-471-17082-8
See also
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- Amalgam (disambiguation)
See what “Statistical methods” are in other dictionaries:
STATISTICAL METHODS- STATISTICAL METHODS scientific methods for describing and studying mass phenomena that allow quantitative (numerical) expression. The word “statistics” (from Igal. stato state) has a common root with the word “state”. Initially it... ... Philosophical Encyclopedia
STATISTICAL METHODS –- scientific methods of describing and studying mass phenomena that allow quantitative (numerical) expression. The word “statistics” (from Italian stato – state) has a common root with the word “state”. Initially it related to the science of management and... Philosophical Encyclopedia
Statistical methods- (in ecology and biocenology) methods of variation statistics, which make it possible to study the whole (for example, phytocenosis, population, productivity) according to its partial aggregates (for example, according to data obtained at registration sites) and assess the degree of accuracy... ... Ecological dictionary
statistical methods- (in psychology) (from the Latin status state) certain 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
Statistical methods- 20.2. Statistical methods Specific statistical methods used to organize, regulate and test activities include, but are not limited to the following: a) design of experiments and factor analysis; b) analysis of variance and... Dictionary-reference book of terms of normative and technical documentation
STATISTICAL METHODS- methods for studying quantities. aspects of mass societies. phenomena and processes. S. m. make it possible to characterize in digital terms the ongoing changes in societies. processes, study various. forms of socio-economic. patterns, change... ... Agricultural Encyclopedic Dictionary
STATISTICAL METHODS- some methods of applied mathematical statistics used to process experimental results. A number of statistical methods have been developed specifically for testing the quality of psychological tests, for use in professional... ... Professional education. Dictionary
STATISTICAL METHODS- (in engineering psychology) (from the Latin status state) some methods of applied statistics used in engineering psychology to process experimental results. The main purpose of using S. m. is to increase the validity of conclusions in ... ... Encyclopedic Dictionary of Psychology and Pedagogy
3.5.1. Probabilistic-statistical research method.
In many cases, it is necessary to study not only deterministic, but also random probabilistic (statistical) processes. These processes are considered on the basis of probability theory.
The set of random variable x constitutes the primary mathematical material. A set is understood as a set of homogeneous events. A set containing the most diverse variants of a mass phenomenon is called a general population, or large sample N. Usually only a part of the population is studied, called elective population or small sample.
Probability P(x) events X called the ratio of the number of cases N(x), which lead to the occurrence of an event X, to the total number of possible cases N:
P(x)=N(x)/N.
Probability theory examines 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 single mathematical theory of mass random processes, widely used to analyze scientific research.
Methods of probability and mathematical statistics are very often 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 testing (Monte Carlo method).
This method is a numerical method for solving complex problems and is based on the use of random numbers that simulate probabilistic processes. The results of solving this method make it possible to establish empirically the dependencies 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, you must have a statistical series, know the law of its distribution, the mean value and the mathematical expectation t(x), standard deviation.
Using this method, you can obtain an arbitrarily specified accuracy of the solution, i.e.
-> t(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 system elements is characterized by direct and feedback connections.
The essence of system analysis is to identify these connections and establish their influence on the behavior of the entire system as a whole. The most complete and in-depth system analysis can be performed using the methods of cybernetics, which is the science of complex dynamic systems capable of perceiving, storing and processing information for optimization and control purposes.
System analysis consists of four stages.
The first stage is to state the problem: the object, goals and objectives of the study are determined, 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 itself being studied and the external environment. Distinguish closed And open systems. When studying closed systems, the influence of the external environment on their behavior is neglected. Then the individual components of the system - its elements - are identified, and the interaction between them and the external environment is established.
The third stage of system analysis is to compile a mathematical model of the system under study. First, the system is parameterized, the main elements of the system and the elementary impacts on it are described using certain parameters. At the same time, parameters characterizing continuous and discrete, deterministic and probabilistic processes are distinguished. 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. The optimization is assessed according to the optimization criterion, which in this case takes extreme values (minimum, maximum, minimax).
Usually, one criterion is selected, 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 drawn up.
Various mathematical methods for optimizing the models under study are known: methods of linear, nonlinear or dynamic programming; probabilistic-statistical methods based on queuing theory; 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 theoretical development stage of scientific research.
Types of models and types of modeling of the research object.
Analytical research methods.
Analytical methods of research using experiment.
Probabilistic-analytical research method.
Static modeling methods (Monte Carlo method).
System analysis method.
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Introduction
1. Chi-square distribution
Conclusion
Application
Introduction
How are the approaches, ideas and results of probability theory used in our lives? mathematical square theory
The basis is a probabilistic model of a real phenomenon or process, i.e. a mathematical model in which objective relationships are expressed in terms of probability theory. Probabilities are used primarily to describe the uncertainties that must be taken into account when making decisions. This refers to both undesirable opportunities (risks) and attractive ones (“lucky chance”). Sometimes randomness is deliberately introduced into a situation, for example, when drawing lots, randomly selecting units for control, conducting lotteries or conducting consumer surveys.
Probability theory allows one probabilities to be used to calculate others of interest to the researcher.
A probabilistic model of a phenomenon or process is the foundation of mathematical statistics. Two parallel series of concepts are used - those related to theory (probabilistic model) and those related to practice (sampling of observation results). For example, the theoretical probability corresponds to the frequency found from the sample. The mathematical expectation (theoretical series) corresponds to the sample arithmetic mean (practical series). As a rule, sample characteristics are estimates of theoretical ones. At the same time, quantities related to the theoretical series “are in the heads of researchers”, relate to the world of ideas (according to the ancient Greek philosopher Plato), and are not available for direct measurement. Researchers have only sample data with which they try to establish the properties of a theoretical probabilistic model that interest them.
Why do we need a probabilistic model? The fact is that only with its help can the properties established from the analysis of a specific sample be transferred to other samples, as well as to the entire so-called general population. The term "population" is used when referring to a large but finite collection of units being studied. For example, about the totality of all residents of Russia or the totality of all consumers of instant coffee in Moscow. The goal of marketing or sociological surveys is to transfer statements obtained from a sample of hundreds or thousands of people to populations of several million people. In quality control, a batch of products acts as a general population.
To transfer conclusions from a sample to a larger population requires some assumptions about the relationship of the sample characteristics with the characteristics of this larger population. These assumptions are based on an appropriate probabilistic model.
Of course, it is possible to process sample data without using one or another probabilistic model. For example, you can calculate a sample arithmetic mean, count the frequency of fulfillment of certain conditions, etc. However, the calculation results will relate only to a specific sample; transferring the conclusions obtained with their help to any other population is incorrect. This activity is sometimes called “data analysis.” Compared to probabilistic-statistical methods, data analysis has limited educational value.
So, the use of probabilistic models based on estimation and testing of hypotheses using sample characteristics is the essence of probabilistic-statistical methods of decision making.
1. Chi-square distribution
Using the normal distribution, three distributions are defined that are now often used in statistical data processing. These are the Pearson (“chi-square”), Student and Fisher distributions.
We will focus on the distribution (“chi-square”). This distribution was first studied by astronomer F. Helmert in 1876. In connection with Gaussian error theory, he studied the sums of squares of n independent standardly normally distributed random variables. Later, Karl Pearson gave the name “chi-square” to this distribution function. And now the distribution bears his name.
Due to its close connection with the normal distribution, the h2 distribution plays an important role in probability theory and mathematical statistics. The h2 distribution, and many other distributions that are determined by the h2 distribution (for example, the Student distribution), describe sample distributions of various functions from normally distributed observation results and are used to construct confidence intervals and statistical tests.
Pearson distribution (chi - square) - distribution of a random variable, where X1, X2,..., Xn are normal independent random variables, and the mathematical expectation of each of them is zero, and the standard deviation is one.
Sum of squares
distributed according to the law (“chi - square”).
In this case, the number of terms, i.e. n is called the "number of degrees of freedom" of the chi-square distribution. As the number of degrees of freedom increases, the distribution slowly approaches normal.
The density of this distribution
So, the distribution h2 depends on one parameter n - the number of degrees of freedom.
The distribution function h2 has the form:
if h2?0. (2.7.)
Figure 1 shows a graph of the probability density and h2 distribution functions for different degrees of freedom.
Figure 1 Dependence of the probability density q (x) in the distribution h2 (chi - square) for different numbers of degrees of freedom
Moments of the chi-square distribution:
The chi-square distribution is used in estimating variance (using a confidence interval), testing hypotheses of agreement, homogeneity, independence, primarily for qualitative (categorized) variables that take a finite number of values, and in many other tasks of statistical data analysis.
2. "Chi-square" in problems of statistical data analysis
Statistical methods of data analysis are used in almost all areas of human activity. They are used whenever it is necessary to obtain and justify any judgments about a group (objects or subjects) with some internal heterogeneity.
The modern stage of development of statistical methods can be counted from 1900, when the Englishman K. Pearson founded the journal "Biometrika". First third of the twentieth century. passed under the sign of parametric statistics. Methods were studied based on the analysis of data from parametric families of distributions described by Pearson family curves. The most popular was the normal distribution. To test the hypotheses, Pearson, Student, and Fisher tests were used. The maximum likelihood method and analysis of variance were proposed, and the basic ideas of experiment planning were formulated.
The chi-square distribution is one of the most widely used in statistics for testing statistical hypotheses. Based on the chi-square distribution, one of the most powerful goodness-of-fit tests is constructed - the Pearson chi-square test.
The criterion of agreement is the criterion for testing the hypothesis about the assumed law of an unknown distribution.
The h2 test ("chi-square") is used to test the hypothesis of various distributions. This is his dignity.
The calculation formula of the criterion is equal to
where m and m" are empirical and theoretical frequencies, respectively
the distribution in question;
n is the number of degrees of freedom.
To check, we need to compare empirical (observed) and theoretical (calculated under the assumption of a normal distribution) frequencies.
If the empirical frequencies completely coincide with the frequencies calculated or expected, S (E - T) = 0 and the criterion h2 will also be equal to zero. If S (E - T) is not equal to zero, this will indicate a discrepancy between the calculated frequencies and the empirical frequencies of the series. In such cases, it is necessary to evaluate the significance of criterion h2, which theoretically can vary from zero to infinity. This is done by comparing the actual value of h2f with its critical value (h2st). The null hypothesis, i.e. the assumption that the discrepancy between the empirical and theoretical or expected frequencies is random, is refuted if h2f is greater than or equal to h2st for the accepted significance level (a) and the number of degrees of freedom (n).
The distribution of probable values of the random variable h2 is continuous and asymmetric. It depends on the number of degrees of freedom (n) and approaches a normal distribution as the number of observations increases. Therefore, the application of the h2 criterion to the assessment of discrete distributions is associated with some errors that affect its value, especially on small samples. To obtain more accurate estimates, the sample distributed into the variation series must have at least 50 options. Correct application of criterion h2 also requires that the frequencies of variants in extreme classes should not be less than 5; if there are less than 5 of them, then they are combined with the frequencies of neighboring classes so that the total amount is greater than or equal to 5. According to the combination of frequencies, the number of classes (N) decreases. The number of degrees of freedom is established by the secondary number of classes, taking into account the number of restrictions on the freedom of variation.
Since the accuracy of determining the h2 criterion largely depends on the accuracy of calculating theoretical frequencies (T), unrounded theoretical frequencies should be used to obtain the difference between the empirical and calculated frequencies.
As an example, let's take a study published on a website dedicated to the application of statistical methods in the humanities.
The Chi-square test allows you to compare frequency distributions regardless of whether they are normally distributed or not.
Frequency refers to the number of occurrences of an event. Usually, the frequency of occurrence of events is dealt with when variables are measured on a scale of names and their other characteristics, besides frequency, are impossible or problematic to select. In other words, when a variable has qualitative characteristics. Also, many researchers tend to convert test scores into levels (high, average, low) and build tables of score distributions to find out the number of people at these levels. To prove that in one of the levels (in one of the categories) the number of people is really greater (less) the Chi-square coefficient is also used.
Let's look at the simplest example.
A test was conducted among younger adolescents to identify self-esteem. The test scores were converted into three levels: high, medium, low. The frequencies were distributed as follows:
High (B) 27 people.
Average (C) 12 people.
Low (L) 11 people
It is obvious that the majority of children have high self-esteem, but this needs to be proven statistically. To do this, we use the Chi-square test.
Our task is to check whether the obtained empirical data differ from theoretically equally probable ones. To do this, you need to find the theoretical frequencies. In our case, theoretical frequencies are equally probable frequencies, which are found by adding all frequencies and dividing by the number of categories.
In our case:
(B + C + H)/3 = (27+12+11)/3 = 16.6
Formula for calculating the chi-square test:
h2 = ?(E - T)I / T
We build the table:
|
Empirical (E) |
Theoretical (T) |
(E - T)I / T |
||
Find the sum of the last column:
Now you need to find the critical value of the criterion using the table of critical values (Table 1 in the Appendix). To do this we need the number of degrees of freedom (n).
n = (R - 1) * (C - 1)
where R is the number of rows in the table, C is the number of columns.
In our case, there is only one column (meaning the original empirical frequencies) and three rows (categories), so the formula changes - we exclude the columns.
n = (R - 1) = 3-1 = 2
For the error probability p?0.05 and n = 2, the critical value is h2 = 5.99.
The obtained empirical value is greater than the critical value - the differences in frequencies are significant (h2 = 9.64; p? 0.05).
As you can see, calculating the criterion is very simple and does not take much time. The practical value of the chi-square test is enormous. This method is most valuable when analyzing responses to questionnaires.
Let's look at a more complex example.
For example, a psychologist wants to know whether it is true that teachers are more biased towards boys than towards girls. Those. more likely to praise girls. To do this, the psychologist analyzed the characteristics of students written by teachers for the frequency of occurrence of three words: “active,” “diligent,” “disciplined,” and synonyms of the words were also counted.
Data on the frequency of occurrence of words were entered into the table:
To process the obtained data we use the chi-square test.
To do this, we will build a table of the distribution of empirical frequencies, i.e. those frequencies that we observe:
Theoretically, we expect that the frequencies will be equally distributed, i.e. the frequency will be distributed proportionally between boys and girls. Let's build a table of theoretical frequencies. To do this, multiply the row sum by the column sum and divide the resulting number by the total sum (s).
The final table for calculations will look like this:
|
Empirical (E) |
Theoretical (T) |
(E - T)I / T |
|||
|
Boys |
"Active" |
||||
|
"Diligent" |
|||||
|
"Disciplined" |
|||||
|
"Active" |
|||||
|
"Diligent" |
|||||
|
"Disciplined" |
|||||
|
Amount: 4.21 |
h2 = ?(E - T)I / T
where R is the number of rows in the table.
In our case, chi-square = 4.21; n = 2.
Using the table of critical values of the criterion, we find: with n = 2 and an error level of 0.05, the critical value h2 = 5.99.
The resulting value is less than the critical value, which means the null hypothesis is accepted.
Conclusion: teachers do not attach importance to the gender of the child when writing characteristics for him.
Conclusion
Students of almost all specialties study the section “probability theory and mathematical statistics” at the end of the higher mathematics course; in reality, they become acquainted only with some basic concepts and results, which are clearly not enough for practical work. Students are introduced to some mathematical research methods in special courses (for example, “Forecasting and technical and economic planning”, “Technical and economic analysis”, “Product quality control”, “Marketing”, “Controlling”, “Mathematical methods of forecasting”) ", "Statistics", etc. - in the case of students of economic specialties), however, the presentation in most cases is very abbreviated and formulaic in nature. As a result, the knowledge of applied statistics specialists is insufficient.
Therefore, the “Applied Statistics” course in technical universities is of great importance, and the “Econometrics” course in economic universities, since econometrics is, as is known, the statistical analysis of specific economic data.
Probability theory and mathematical statistics provide fundamental knowledge for applied statistics and econometrics.
They are necessary for specialists for practical work.
I looked at the continuous probabilistic model and tried to show its use with examples.
And at the end of my work, I came to the conclusion that competent implementation of the basic procedures of mathematical-static data analysis and static testing of hypotheses is impossible without knowledge of the chi-square model, as well as the ability to use its table.
Bibliography
1. Orlov A.I. Applied statistics. M.: Publishing house "Exam", 2004.
2. Gmurman V.E. Theory of Probability and Mathematical Statistics. M.: Higher School, 1999. - 479 p.
3. Ayvozyan S.A. Probability theory and applied statistics, vol. 1. M.: Unity, 2001. - 656 p.
4. Khamitov G.P., Vedernikova T.I. Probabilities and statistics. Irkutsk: BGUEP, 2006 - 272 p.
5. Ezhova L.N. Econometrics. Irkutsk: BGUEP, 2002. - 314 p.
6. Mosteller F. Fifty entertaining probabilistic problems with solutions. M.: Nauka, 1975. - 111 p.
7. Mosteller F. Probability. M.: Mir, 1969. - 428 p.
8. Yaglom A.M. Probability and information. M.: Nauka, 1973. - 511 p.
9. Chistyakov V.P. Probability theory course. M.: Nauka, 1982. - 256 p.
10. Kremer N.Sh. Theory of Probability and Mathematical Statistics. M.: UNITY, 2000. - 543 p.
11. Mathematical Encyclopedia, vol.1. M.: Soviet Encyclopedia, 1976. - 655 p.
12. http://psystat.at.ua/ - Statistics in psychology and pedagogy. Article Chi-square test.
Application
Critical distribution points h2
Table 1
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