Probabilistic statistical methods. Probabilistic-statistical methods of research and the method of system analysis. Probabilistic-statistical decision-making methods
When conducting psychological and pedagogical research, an important role is given to mathematical methods for modeling processes and processing 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 us carry out a comparative analysis of these methods.
Classical probabilistic-statistical method. The classical probabilistic-statistical research method is based on the theory of probability 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 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 of objects. In order to solve this problem, a certain hypothesis is adopted about the nature of the statistical regularity that manifests itself in the phenomenon under study, for example, the hypothesis that the phenomenon under study obeys the normal distribution law. Such a hypothesis is called the null hypothesis, which may turn out to be erroneous, therefore, along with the 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. The methodology for testing a statistical hypothesis is rigorously developed and outlined 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 characteristic parameters are calculated, for example, such as expected value or mean, variance, standard deviation, mode, median, skewness 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 of natural, technical and social sciences. A detailed description of the essence of this method and its application to the solution specific tasks 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
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 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 likely that he will see this event.
In accordance with the three main possibilities - decision-making under conditions of complete certainty, risk and uncertainty - decision-making methods and algorithms can be divided into three main types: analytical, statistical and based on fuzzy formalization. In each specific case, the decision-making method is selected based on the task, the available initial data, the available problem models, the decision-making environment, the decision-making process, the required solution accuracy, and the analyst's personal preferences.
In some information systems, the algorithm selection process can be automated:
The corresponding automated system has the ability to use a variety of different types of algorithms (library of algorithms);
The system interactively prompts the user to answer a number of questions about the main characteristics of the problem under consideration;
Based on the results of the user's responses, the system offers the most appropriate (according to the criteria specified in it) algorithm from the library.
2.3.1 Probabilistic-statistical methods of decision making
Probabilistic-statistical decision-making methods (MPD) are used when the effectiveness of decisions made depends on factors that are random variables for which probability distribution laws and other statistical characteristics are known. Moreover, each decision can lead to one of the many possible outcomes, and each outcome has a certain probability of occurrence, which can be calculated. The indicators characterizing the problem situation are also described with the help of probabilistic characteristics. With such DPR, the decision maker always runs the risk of getting the wrong result, which he is guided by, choosing the optimal solution based on the averaged statistical characteristics of random factors, that is, the decision is made under risk conditions.
In practice, probabilistic and statistical methods are often used when conclusions drawn from sample data are transferred to the entire population (for example, from a sample to an entire batch of products). However, in this case, in each specific situation, one should first assess the fundamental possibility of obtaining sufficiently reliable probabilistic and statistical data.
When using the ideas and results of probability theory and mathematical statistics in making decisions, the basis is a mathematical model in which objective relationships are expressed in terms of probability theory. Probabilities are primarily used to describe the randomness that must be taken into account when making decisions. This refers to both undesirable opportunities (risks) and attractive ones (“lucky chance”).
The essence of probabilistic-statistical decision-making methods is the use of probabilistic models based on estimation and testing of hypotheses using sample characteristics.
We emphasize that the logic of using sample characteristics for decision-making based on theoretical models involves the simultaneous use of two parallel series of concepts– related to theory (probabilistic model) and related to practice (sample of observational 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 characteristics.
The advantages of using these methods include the ability to take into account various scenarios for the development of events and their probabilities. The disadvantage of these methods is that the scenario probabilities used in the calculations are usually very difficult to obtain in practice.
The application of a specific probabilistic-statistical decision-making method consists of three stages:
The transition from economic, managerial, technological reality to an abstract mathematical and statistical scheme, i.e. building a probabilistic model of a control system, a technological process, a decision-making procedure, in particular based on the results of statistical control, etc.
Carrying out calculations and obtaining conclusions by purely mathematical means within the framework of a probabilistic model;
Interpretation of mathematical and statistical conclusions in relation to a real situation and making an appropriate decision (for example, on the conformity or non-compliance of product quality with established requirements, the need to adjust the technological process, etc.), in particular, conclusions (on the proportion of defective units of products in a batch, on a specific form of laws of distribution of controlled parameters of the technological process, 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. The adequacy of the probabilistic model is substantiated, in particular, using statistical methods for testing hypotheses.
Mathematical statistics is usually divided into three sections according to the type of problems to be solved: 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 result of an observation 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 a non-numerical nature, in which the result of an observation is of a non-numerical nature, for example, it is a set (a geometric figure), an ordering, or obtained as a result of a measurement by a qualitative attribute.
An example of when it is advisable to use probabilistic-statistical models.
When controlling the quality of any product, a sample is taken from it to decide whether the batch of products produced meets the established requirements. Based on the results of the sample control, a conclusion is made about the entire batch. In this case, it is very important to avoid subjectivity in the formation of the sample, i.e. it is necessary that each unit of product in the controlled lot has the same probability of being selected in the sample. The choice based on the lot in such a situation is not sufficiently objective. Therefore, in production conditions, the selection of units of production in the sample is usually carried out not by lot, but by special tables of random numbers or by means of computer random number generators.
In statistical regulation of technological processes, based on the methods of mathematical statistics, rules and plans for statistical control of processes are developed, aimed at timely detection of the disorder of technological processes and taking measures to adjust them and prevent the release of products that do not meet the established requirements. These measures are aimed at reducing production costs and losses from the supply of low-quality products. With statistical acceptance control, based on the methods of mathematical statistics, quality control plans are developed by analyzing samples from product batches. The difficulty lies in being able to correctly build probabilistic-statistical decision-making models, on the basis of which it is possible to answer the questions posed above. In mathematical statistics, probabilistic models and methods for testing hypotheses have been developed for this purpose3.
In addition, in a number of managerial, industrial, economic, national economic situations, problems of a different type arise - problems of estimating the characteristics and parameters of probability distributions.
Or, in a statistical analysis of the accuracy and stability of technological processes, it is necessary to evaluate such quality indicators as the average value of the controlled parameter and the degree of its spread in the process under consideration. According to the theory of probability, it is advisable to use its mathematical expectation as the mean value of a random variable, and the variance, standard deviation, or coefficient of variation as a statistical characteristic of the spread. This raises the question: how to estimate these statistical characteristics from sample data and with what accuracy can this be done? There are many similar examples in the literature. All of them show how probability theory and mathematical statistics can be used in production management when making decisions in the field of statistical product quality management.
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 control, 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 and a statistical assessment of quality are carried out. Specific methods include methods of statistical acceptance control of product quality, statistical regulation of technological processes, assessment and control of reliability, etc.
In production management, in particular, when optimizing product quality and ensuring compliance with standards, it is especially important to apply statistical methods at the initial stage of the product life cycle, i.e. at the stage of research preparation of experimental design developments (development of promising requirements for products, preliminary design, terms of reference for experimental design development). This is due to the limited information available at the initial stage of the product life cycle and the need to predict the technical possibilities and economic situation for the future.
The most common probabilistic-statistical methods are regression analysis, factor analysis, analysis of variance, statistical methods for risk assessment, scenario method, etc. The field of statistical methods, devoted to the analysis of statistical data of a non-numeric nature, is gaining more and more importance. measurement results on qualitative and heterogeneous features. One of the main applications of statistics of objects of non-numerical nature is the theory and practice of expert assessments related to the theory of statistical decisions and voting problems.
The role of a person in solving problems using the methods of the theory of statistical decisions is to formulate the problem, i.e., to bring the real problem to the corresponding model one, to determine the probabilities of events based on statistical data, and also to approve the resulting optimal solution.
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Introduction
1. Chi-square distribution
Conclusion
Appendix
Introduction
How are the approaches, ideas and results of probability theory used in our lives? mathematical square theory
The base 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 the situation, for example, when drawing lots, random selection of units for control, conducting lotteries or consumer surveys.
Probability theory allows one to calculate other probabilities that are 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 (a probabilistic model) and those related to practice (a sample of observational 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, the quantities related to the theoretical series "are in the minds of researchers", refer to the world of ideas (according to the ancient Greek philosopher Plato), and are not available for direct measurement. Researchers have only selective data, with the help of which they try to establish the properties of a theoretical probabilistic model that are of interest to them.
Why do we need a probabilistic model? The fact is that only with its help it is possible to transfer the properties established by the results of the analysis of a particular sample to other samples, as well as to the entire so-called general population. The term "population" is used to refer to a large but finite population 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 purpose of marketing or sociological surveys is to transfer statements received from a sample of hundreds or thousands of people to general populations of several million people. In quality control, a batch of products acts as a general population.
To transfer inferences from a sample to a larger population, some assumptions are needed about the relationship of 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 the sample arithmetic mean, calculate the frequency of fulfillment of certain conditions, etc. However, the results of the calculations will apply only to a specific sample; transferring the conclusions obtained with their help to any other set is incorrect. This activity is sometimes referred to as "data analysis". Compared to probabilistic-statistical methods, data analysis has limited cognitive value.
So, the use of probabilistic models based on estimation and testing of hypotheses with the help of sample characteristics is the essence of probabilistic-statistical decision-making methods.
1. Chi-square distribution
The normal distribution defines three distributions that are now commonly used in statistical data processing. These are the distributions of Pearson ("chi - square"), Student and Fisher.
We will focus on the distribution ("chi - square"). This distribution was first studied by the astronomer F. Helmert in 1876. In connection with the Gaussian theory of errors, he studied the sums of squares of n independent standard normally distributed random variables. Later, Karl Pearson named this distribution function "chi-square". 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 defined by the h2 distribution (for example, the Student's distribution), describe sample distributions of various functions from normally distributed observations and are used to construct confidence intervals and statistical tests.
Pearson distribution (chi - squared) - 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-squared distribution. As the number of degrees of freedom increases, the distribution slowly approaches normal.
The density of this distribution
So, the distribution of 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 the χ2 distribution function for different degrees of freedom.
Figure 1 Dependence of the probability density q (x) in the distribution of h2 (chi - squared) for a different number of degrees of freedom
Moments of the "chi-square" distribution:
The chi-squared distribution is used in variance estimation (using a confidence interval), in 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 substantiate 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 20th century passed under the sign of parametric statistics. Methods based on the analysis of data from parametric families of distributions described by Pearson family curves were studied. The most popular was the normal distribution. The Pearson, Student, and Fisher criteria were used to test the hypotheses. The maximum likelihood method, analysis of variance were proposed, and the main ideas for planning the experiment were formulated.
The chi-square distribution is one of the most widely used in statistics for testing statistical hypotheses. On the basis of the "chi-square" distribution, one of the most powerful goodness-of-fit tests, Pearson's "chi-square" test, is constructed.
The goodness-of-fit test is a criterion for testing the hypothesis about the proposed law of the unknown distribution.
The p2 ("chi-square") test is used to test the hypothesis of different distributions. This is his merit.
The calculation formula of the criterion is equal to
where m and m" are empirical and theoretical frequencies, respectively
distribution under consideration;
n is the number of degrees of freedom.
For verification, 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 ch2 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 the criterion p2, which theoretically can vary from zero to infinity. This is done by comparing the actually obtained value of ch2f with its critical value (ch2st). (a) and 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 p2 criterion to the estimation of discrete distributions is associated with some errors that affect its value, especially for small samples. To obtain more accurate estimates, the sample distributed in the variation series should have at least 50 options. The correct application of the p2 criterion also requires that the frequencies of variants in the 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 their total amount is greater than or equal to 5. According to the combination of frequencies, the number of classes (N) also decreases. The number of degrees of freedom is set according to the secondary number of classes, taking into account the number of restrictions on the freedom of variation.
Since the accuracy of determining the criterion p2 largely depends on the accuracy of calculating the theoretical frequencies (T), unrounded theoretical frequencies should be used to obtain the difference between the empirical and calculated frequencies.
As an example, take a study published on a website dedicated to the application of statistical methods in the humanities.
The Chi-square test allows comparison of frequency distributions, whether they are normally distributed or not.
Frequency refers to the number of occurrences of an event. Usually, the frequency of occurrence of an event is dealt with when the variables are measured in the scale of names and their other characteristics, except for the frequency, are impossible or problematic to select. In other words, when the variable has qualitative characteristics. Also, many researchers tend to translate test scores into levels (high, medium, 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 more (less), the Chi-square coefficient is also used.
Let's take a look at the simplest example.
A self-esteem test was conducted among younger adolescents. Test scores were translated into three levels: high, medium, low. The frequencies were distributed as follows:
High (H) 27 pers.
Medium (C) 12 people
Low (H) 11 pers.
It is obvious that the majority of children with high self-esteem, however, this needs to be proved statistically. To do this, we use the Chi-square test.
Our task is to check whether the obtained empirical data differ from the theoretically equally probable ones. To do this, it is necessary to find the theoretical frequencies. In our case, theoretical frequencies are equiprobable frequencies that are found by adding all frequencies and dividing by the number of categories.
In our case:
(B + C + H) / 3 \u003d (27 + 12 + 11) / 3 \u003d 16.6
The formula for calculating the chi-square test is:
h2 \u003d? (E - T) I / T
We build a table:
|
Empirical (Uh) |
Theoretical (T) |
(E - T)І / T |
||
Find the sum of the last column:
Now you need to find the critical value of the criterion according to 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 empirical value obtained is greater than the critical value - the frequency differences are significant (n2= 9.64; p≤0.05).
As you can see, the calculation of 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 in the analysis of responses to questionnaires.
Let's take a more complex example.
For example, a psychologist wants to know if 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, regarding the frequency of occurrence of three words: "active", "diligent", "disciplined", synonyms of words were also counted.
Data on the frequency of occurrence of words were entered in the table:
To process the obtained data, we use the chi-square test.
To do this, we construct a table of distribution of empirical frequencies, i.e. the frequencies that we observe:
Theoretically, we expect the frequencies to be distributed equally, 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 resulting table for calculations will look like this:
|
Empirical (Uh) |
Theoretical (T) |
(E - T)І / T |
|||
|
boys |
"Active" |
||||
|
"Diligent" |
|||||
|
"Disciplined" |
|||||
|
"Active" |
|||||
|
"Diligent" |
|||||
|
"Disciplined" |
|||||
|
Amount: 4.21 |
h2 \u003d? (E - T) I / T
where R is the number of rows in the table.
In our case, chi-square = 4.21; n = 2.
According to 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 that the null hypothesis is accepted.
Conclusion: teachers do not attach importance to the gender of the child when writing his characteristics.
Conclusion
Students of almost all specialties study the section "probability theory and mathematical statistics" at the end of the course of higher mathematics; in reality, they get acquainted only with some basic concepts and results, which are clearly not enough for practical work. Students meet some mathematical methods of research in special courses (for example, such as "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 prescription in nature. As a result, the knowledge of applied statisticians is insufficient.
Therefore, the course "Applied Statistics" in technical universities is of great importance, and in economic universities - the course "Econometrics", since econometrics, as you know, is a 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 considered a continuous probabilistic model and tried to show its useability with examples.
And at the end of my work, I came to the conclusion that the competent implementation of the basic procedures of mathematical and static data analysis, 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. - 479s.
3. Ayvozyan S.A. Probability Theory and Applied Statistics, v.1. M.: Unity, 2001. - 656s.
4. Khamitov G.P., Vedernikova T.I. Probabilities and statistics. Irkutsk: BSUEP, 2006 - 272p.
5. Ezhova L.N. Econometrics. Irkutsk: BSUEP, 2002. - 314p.
6. Mosteller F. Fifty entertaining probabilistic problems with solutions. M.: Nauka, 1975. - 111p.
7. Mosteller F. Probability. M.: Mir, 1969. - 428s.
8. Yaglom A.M. Probability and information. M.: Nauka, 1973. - 511s.
9. Chistyakov V.P. Probability course. M.: Nauka, 1982. - 256s.
10. Kremer N.Sh. Theory of Probability and Mathematical Statistics. M.: UNITI, 2000. - 543s.
11. Mathematical encyclopedia, v.1. M.: Soviet Encyclopedia, 1976. - 655s.
12. http://psystat.at.ua/ - Statistics in psychology and pedagogy. Article Chi-square test.
Appendix
Critical distribution points p2
Table 1
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Probability theory and mathematical statistics are sciences about methods of quantitative analysis of mass random phenomena. A set of values of a random variable is called a sample, and the elements of the set are called sample values of a random variable.
Of particular interest is the quantitative assessment of entrepreneurial risk using the methods of mathematical statistics. The main tools of this assessment 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 particular 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 the squares of the 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.
In economic statistics, such an assessment of different values of the coefficient of variation is 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?
Decision. 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):
| 400 | 0,1 | 40 | 16000 |
| 400 | 0,3 | 120 | 48000 |
| 400 | 0,5 | 200 | 80000 |
| 400 | 0,1 | 40 | 16000 |
| 1,0 | 400 | 160000 |
| 350 | 0,1 | 35 | 12250 |
| 500 | 0,3 | 150 | 75000 |
| 500 | 0,5 | 250 | 125000 |
| 500 | 0,1 | 50 | 25000 |
| 1,0 | 485 | 2372500 |
| 300 | 0,1 | 30 | 9000 |
| 450 | 0,3 | 135 | 60750 |
| 600 | 0,5 | 300 | 180000 |
| 600 | 0,1 | 60 | 36000 |
| 1,0 | 525 | 285750 |
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.
How are probability and mathematical statistics used? These disciplines are the basis of probabilistic-statistical methods decision making. To use their mathematical apparatus, you need tasks decision making express in terms of probabilistic-statistical models. Application of a specific probabilistic-statistical method decision making consists of three stages:
- transition from economic, managerial, technological reality to an abstract mathematical and statistical scheme, i.e. building a probabilistic model of a control system, a technological process, decision-making procedures, in particular according to the results of statistical control, etc.;
- carrying out calculations and obtaining conclusions by purely mathematical means within the framework of a probabilistic model;
- interpretation of mathematical and statistical conclusions in relation to a real situation and making an appropriate decision (for example, on the conformity or non-compliance of product quality with established requirements, the need to adjust the technological process, etc.), in particular, conclusions (on the proportion of defective units of products in a batch, on specific form of distribution laws controlled parameters technological process, etc.).
Mathematical statistics uses the concepts, methods and results of probability theory. Consider the main issues of building probabilistic models decision making in economic, managerial, technological and other situations. For the active and correct use of normative-technical and instructive-methodical documents on probabilistic-statistical methods decision making prior knowledge is required. So, it is necessary to know under what conditions one or another document should be applied, what initial information is necessary to have for its selection and application, what decisions should be made based on the results of data processing, etc.
Examples of application of probability theory and mathematical statistics. Let's consider several examples when probabilistic-statistical models are a good tool for solving managerial, industrial, economic, and national economic problems. So, for example, in the novel by A.N. Tolstoy's "Walking through the torments" (vol. 1) says: "the workshop gives twenty-three percent of the marriage, you hold on to this figure," Strukov said to Ivan Ilyich.
The question arises how to understand these words in the conversation of factory managers, since one unit of production cannot be defective by 23%. It can be either good or defective. Perhaps Strukov meant that a large batch contains approximately 23% of defective units. Then the question arises, what does "about" mean? Let 30 out of 100 tested units of products turn out to be defective, or out of 1000-300, or out of 100000-30000, etc., should Strukov be accused of lying?
Or another example. The coin that is used as a lot must be "symmetrical", i.e. when it is thrown, on average, in half the cases, the coat of arms should fall out, and in half the cases - the lattice (tails, number). But what does "average" mean? If you spend many series of 10 throws in each series, then there will often be series in which a coin drops out 4 times with a coat of arms. For a symmetrical coin, this will happen in 20.5% of the series. And if there are 40,000 coats of arms for 100,000 tosses, can the coin be considered symmetrical? Procedure decision making is based on the theory of probability and mathematical statistics.
The example under consideration may not seem serious enough. However, it is not. Drawing lots is widely used in the organization of industrial feasibility experiments, for example, when processing the results of measuring the quality index (friction moment) of bearings depending on various technological factors (the influence of a conservation environment, methods of preparing bearings before measurement, the effect of bearing load in the measurement process, etc.). P.). Suppose it is necessary to compare the quality of bearings depending on the results of their storage in different conservation oils, i.e. in composition oils and . When planning such an experiment, the question arises of which bearings should be placed in the composition oil, and which - in the composition oil, but in such a way as to avoid subjectivity and ensure the objectivity of the decision.
The answer to this question can be obtained by drawing lots. A similar example can be given with the quality control of any product. Sampling is done to decide whether or not an inspected lot of products meets the specified requirements. Based on the results of the sample control, a conclusion is made about the entire batch. In this case, it is very important to avoid subjectivity in the formation of the sample, i.e. it is necessary that each unit of product in the controlled lot has the same probability of being selected in the sample. Under production conditions, the selection of units of production in the sample is usually carried out not by lot, but by special tables of random numbers or with the help of computer random number generators.
Similar problems of ensuring the objectivity of comparison arise when comparing different schemes. production organization, remuneration, during tenders and competitions, selection of candidates for vacant positions, etc. Everywhere you need a lottery or similar procedures. Let us explain by the example of identifying the strongest and second strongest teams when organizing a tournament according to the Olympic system (the loser is eliminated). Let the stronger team always win over the weaker one. It is clear that the strongest team will definitely become the champion. The second strongest team will reach the final if and only if it has no games with the future champion before the final. If such a game is planned, then the second strongest team will not reach the final. The one who plans the tournament can either "knock out" the second strongest team from the tournament ahead of time, bringing it together in the first meeting with the leader, or ensure it the second place, ensuring meetings with weaker teams until the final. To avoid subjectivity, draw lots. For an 8-team tournament, the probability that the two strongest teams will meet in the final is 4/7. Accordingly, with a probability of 3/7, the second strongest team will leave the tournament ahead of schedule.
In any measurement of product units (using a caliper, micrometer, ammeter, etc.), there are errors. To find out if there are systematic errors, it is necessary to make repeated measurements of a unit of production, the characteristics of which are known (for example, a standard sample). It should be remembered that in addition to the systematic error, there is also a random error.
Therefore, the question arises of how to find out from the measurement results whether there is a systematic error. If we note only whether the error obtained during the next measurement is positive or negative, then this problem can be reduced to the previous one. Indeed, let's compare the measurement with the throwing of a coin, the positive error - with the loss of the coat of arms, the negative - with the lattice (zero error with a sufficient number of divisions of the scale almost never occurs). Then checking the absence of a systematic error is equivalent to checking the symmetry of the coin.
The purpose of these considerations is to reduce the problem of checking the absence of a systematic error to the problem of checking the symmetry of a coin. The above reasoning leads to the so-called "criterion of signs" in mathematical statistics.
In the statistical regulation of technological processes, based on the methods of mathematical statistics, rules and plans for statistical control of processes are developed, aimed at timely detection of the disorder of technological processes, taking measures to adjust them and prevent the release of products that do not meet the established requirements. These measures are aimed at reducing production costs and losses from the supply of low-quality products. With statistical acceptance control, based on the methods of mathematical statistics, quality control plans are developed by analyzing samples from product batches. The difficulty lies in being able to correctly build probabilistic-statistical models decision making on the basis of which the above questions can be answered. In mathematical statistics, probabilistic models and methods for testing hypotheses have been developed for this, in particular, hypotheses that the proportion of defective units of production is equal to a certain number, for example, (remember the words of Strukov from the novel by A.N. Tolstoy).
Assessment tasks. In a number of managerial, industrial, economic, national economic situations, problems of a different type arise - problems of estimating the characteristics and parameters of probability distributions.
Consider an example. Let a batch of N electric lamps come to the control. A sample of n electric lamps was randomly selected from this batch. A number of natural questions arise. How can the average service life of electric lamps be determined from the results of testing the sample elements and with what accuracy can this characteristic be estimated? How will the accuracy change if a larger sample is taken? At what number of hours can it be guaranteed that at least 90% of the electric lamps will last more than hours?
Suppose that when testing a sample with a volume of electric lamps, electric lamps turned out to be defective. Then the following questions arise. What limits can be specified for the number of defective electric lamps in a batch, for the level of defectiveness, etc.?
Or, in a statistical analysis of the accuracy and stability of technological processes, it is necessary to evaluate such quality indicators, as an average controlled parameter and the degree of its spread in the process under consideration. According to the theory of probability, it is advisable to use its mathematical expectation as the average value of a random variable, and the variance, standard deviation, or the coefficient of variation. This raises the question: how to estimate these statistical characteristics from sample data and with what accuracy can this be done? There are many similar examples. Here it was important to show how probability theory and mathematical statistics can be used in production management when making decisions in the field of statistical product quality management.
What is "mathematical statistics"? Mathematical statistics is understood as "a branch 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 probability theory, which makes it possible to evaluate the accuracy and reliability of the conclusions obtained in each task based on the available statistical material" [ [ 2.2], p. 326]. At the same time, statistical data refers to information about the number of objects in a 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 random variables), in which the observation result is described by a real number;
- multidimensional 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 a non-numerical nature, in which the result of the observation is of a non-numerical nature, for example, is a set ( geometric figure), ordering or obtained as a result of measurement on a qualitative basis.
Historically, some areas of statistics of non-numerical objects (in particular, problems of estimating the percentage of marriage 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. We are talking 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, using 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 applications, they are used as probabilistic statistical methods wide application, as well as specific ones. For example, in the section of production management devoted to statistical methods of product quality control, applied mathematical statistics (including the design of experiments) are used. With the help of its methods, statistical analysis accuracy and stability of technological processes and statistical quality assessment. 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. the duration of conversations is also modeled by 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.
Briefly about the history of mathematical statistics. Mathematical statistics as a science begins with the works of the famous German mathematician Carl Friedrich Gauss (1777-1855), who, based on the theory of probability, investigated and substantiated method least squares , created by him in 1795 and used to process astronomical data (in order to refine the orbit of the minor planet Ceres). One of the most popular probability distributions, the normal one, is often named after him, and in the theory of random processes, the main object of study is Gaussian processes.
AT late XIX in. - the beginning of the twentieth century. a major contribution to mathematical statistics was made by English researchers, primarily K. Pearson (1857-1936) and R.A. Fisher (1890-1962). In particular, Pearson developed the "chi-square" criterion for testing statistical hypotheses, and Fisher - analysis of variance, the theory of experiment planning, the maximum likelihood method of parameter estimation.
In the 30s of the twentieth century. Pole Jerzy Neumann (1894-1977) and Englishman E. Pearson developed a general theory of testing statistical hypotheses, and Soviet mathematicians Academician A.N. Kolmogorov (1903-1987) and Corresponding Member of the USSR Academy of Sciences N.V. Smirnov (1900-1966) laid the foundations of non-parametric statistics. In the forties of the twentieth century. Romanian A. Wald (1902-1950) built the theory of consistent statistical analysis.
Mathematical statistics is rapidly developing at the present time. So, over the past 40 years, four fundamentally new areas of research can be distinguished [ [ 2.16 ] ]:
- development and implementation of mathematical methods for planning experiments;
- development of statistics of objects of non-numerical nature as an independent direction in applied mathematical statistics;
- development of statistical methods resistant to small deviations from the used probabilistic model;
- wide deployment of work on the creation of computer software packages designed for statistical data analysis.
Probabilistic-statistical methods and optimization. The idea of optimization permeates modern applied mathematical statistics and other statistical methods. Namely, methods for planning experiments, statistical acceptance control, statistical regulation of technological processes, etc. On the other hand, optimization formulations in theory decision making, For example, applied theory optimization of product quality and the requirements of standards, provide for the widespread use of probabilistic-statistical methods, primarily applied mathematical statistics.
In production management, in particular, when optimizing product quality and standard requirements, it is especially important to apply statistical methods at the initial stage life cycle products, i.e. at the stage of research preparation of experimental design developments (development of promising requirements for products, preliminary design, terms of reference for experimental design development). This is due to the limited information available at the initial stage of the product life cycle and the need to predict the technical possibilities and economic situation for the future. Statistical Methods should be applied at all stages of solving the optimization problem - when scaling variables, developing mathematical models functioning of products and systems, conducting technical and economic experiments, etc.
In optimization problems, including optimization of product quality and standard requirements, all areas of statistics are used. Namely - the statistics of random variables, multivariate statistical analysis, statistics of random processes and time series, statistics of objects of non-numerical nature. The choice of a statistical method for the analysis of specific data should be carried out according to the recommendations [
