Methods of making management decisions. Probabilistic and statistical models of decision making Methods for making management decisions

depending on the type of data “at the input”:

2.1. Numbers.

2.2. Finite-dimensional vectors.

2.3. Functions (time series).

2.4. Objects of non-numerical nature.

The most interesting classification is based on those controlling problems for which econometric methods are used. With this approach, blocks can be allocated:

3.1. Support forecasting and planning.

3.2. Tracking controlled parameters and anomaly detection.

3.3. Support decision making, and etc.

What factors determine the frequency of use of certain econometric controlling tools? As with other applications of econometrics, there are two main groups of factors - the tasks being solved and the qualifications of specialists.

At practical application econometric methods in the operation of the controller, it is necessary to apply appropriate software systems. General statistical systems like SPSS, Statgraphics, Statistica, ADDA, and more specialized Statcon, SPC, NADIS, REST(according to interval data statistics), Matrixer and many others. Mass introduction of easy-to-use software products, including modern econometric tools for analyzing specific economic data, can be considered as one of effective ways acceleration scientific and technological progress, dissemination of modern econometric knowledge.

Econometrics is constantly evolving. Applied research leads to the need for a more in-depth analysis of classical methods.

A good example to discuss is methods for testing the homogeneity of two samples. There are two aggregates, and we must decide whether they are different or the same. To do this, a sample is taken from each of them and one or another is used. statistical method homogeneity checks. About 100 years ago, the Student method was proposed, which is still widely used today. However, it has a whole bunch of shortcomings. Firstly, according to Student, the distributions of sample elements must be normal (Gaussian). As a rule, this is not the case. Secondly, it is aimed not at checking homogeneity in general (the so-called absolute homogeneity, i.e. the coincidence of distribution functions corresponding to two populations), but only at checking the equality of mathematical expectations. But, thirdly, it is necessarily assumed that the variances for the elements of the two samples coincide. However, checking the equality of variances, and especially normality, is much more difficult than the equality of mathematical expectations. Therefore, the Student's t test is usually used without making such checks. And then the conclusions based on the Student’s criterion hang in the air.

More theoretically advanced specialists turn to other criteria, for example, the Wilcoxon test. It is non-parametric, i.e. does not rely on the assumption of normality. But it is not without its shortcomings. It cannot be used to check absolute homogeneity (coincidence of distribution functions corresponding to two populations). This can only be done using the so-called. consistent criteria, in particular, Smirnov’s criteria and the omega-square type.

From a practical point of view, the Smirnov criterion has a disadvantage - its statistics take only a small number of values, its distribution is concentrated in a small number of points, and it is not possible to use the traditional significance levels of 0.05 and 0.01.

The term "high statistical technologies". In the term “high statistical technologies” each of the three words carries its own meaning.

“High”, as in other areas, means that the technology is based on modern achievements of theory and practice, in particular, probability theory and applied mathematical statistics. At the same time, “based on modern scientific achievements” means, firstly, that the mathematical basis of the technology within the framework of the relevant scientific discipline was obtained relatively recently, and secondly, that the calculation algorithms were developed and justified in accordance with it (and are not the so-called "heuristic"). Over time, if new approaches and results do not force one to reconsider the assessment of the applicability and capabilities of the technology or replace it with a more modern one, “high econometric technology” turns into “classical statistical technology.” Such as least square method. So, high statistical technologies are the fruits of recent serious scientific research. There are two here key concepts- the “youth” of the technology (in any case, no older than 50 years, and better, no older than 10 or 30 years) and reliance on “high science.”

The term “statistical” is familiar, but has many shades. There are more than 200 definitions of the term "statistics".

Finally, the term “technology” is relatively rarely used in relation to statistics. Data analysis typically involves a number of procedures and algorithms performed sequentially, in parallel, or in a more complex manner. In particular, the following typical stages can be distinguished:

planning a statistical study;
organizing data collection according to an optimal or at least rational program (sampling planning, creating organizational structure and selection of a team of specialists, training of personnel who will collect data, as well as data controllers, etc.);
direct collection of data and their recording on certain media (with quality control of collection and rejection of erroneous data for reasons of the subject area);
primary description of data (calculation of various sample characteristics, distribution functions, nonparametric density estimates, construction of histograms, correlation fields, various tables and diagrams, etc.),
assessment of certain numerical or non-numerical characteristics and parameters of distributions (for example, non-parametric interval estimation of the coefficient of variation or restoration of the relationship between the response and factors, i.e. function estimation),
testing statistical hypotheses (sometimes their chains - after testing the previous hypothesis, a decision is made to test one or another subsequent hypothesis),
more in-depth study, i.e. application of various algorithms for multivariate statistical analysis, diagnostic and classification algorithms, statistics of non-numerical and interval data, time series analysis, etc.;
checking the stability of the obtained estimates and conclusions regarding the permissible deviations of the initial data and the premises of the probabilistic-statistical models used, admissible transformations of measurement scales, in particular, the study of the properties of estimates by the method of multiplying samples;
application of the obtained statistical results for applied purposes (for example, for diagnosing specific materials, making forecasts, selecting investment project from the proposed options, finding the optimal mode for implementing the technological process, summing up the results of testing samples technical devices and etc.),
preparation of final reports, in particular intended for those who are not specialists in econometric and statistical methods of data analysis, including for management - “decision makers”.

Other structuring of statistical technologies is possible. It is important to emphasize that the qualified and effective use of statistical methods is by no means a test of one individual statistical hypothesis or an assessment of the parameters of one given distribution from a fixed family. This kind of operations are just the building blocks of statistical technology. Meanwhile, textbooks and monographs on statistics and econometrics usually talk about individual building blocks, but do not discuss the problems of organizing them into a technology intended for applied use. The transition from one statistical procedure to another remains in the shadows.

The problem of “joining” statistical algorithms requires special consideration, since as a result of using the previous algorithm, the conditions of applicability of the subsequent one are often violated. In particular, the results of observations may cease to be independent, their distribution may change, etc.

For example, when testing statistical hypotheses, significance level and power are of great importance. Methods for calculating them and using them to test a single hypothesis are usually well known. If one hypothesis is first tested, and then, taking into account the results of its testing, a second one, then the final procedure, which can also be considered as testing some (more complex) statistical hypothesis, has characteristics (level of significance and power) that, as a rule, cannot simply expressed in terms of the characteristics of the two component hypotheses, and therefore they are usually unknown. As a result, the final procedure cannot be considered as scientifically based; it refers to heuristic algorithms. Of course, after appropriate study, for example, using the Monte Carlo method, it can become one of the scientifically based procedures of applied statistics.

So, the procedure for econometric or statistical data analysis is an information technological process, in other words, one or another information technology. At present, it would be frivolous to talk about automating the entire process of econometric (statistical) data analysis, since there are too many unresolved problems that cause discussions among specialists.

The entire arsenal of currently used statistical methods can be divided into three streams:

high statistical technologies;
classical statistical technologies,
low statistical technologies.

It is necessary to ensure that only the first two types of technologies are used in specific studies. At the same time, by classical statistical technologies we mean technologies of venerable age that have retained their scientific value and significance for modern statistical practice. These are least square method, Kolmogorov, Smirnov statistics, omega square, nonparametric Spearman and Kendall correlation coefficients and many others.

We have an order of magnitude fewer econometricians than in the USA and Great Britain (the American Statistical Association has more than 20,000 members). Russia needs training of new specialists - econometricians.

Whatever new scientific results are obtained, if they remain unknown to students, then a new generation of researchers and engineers is forced to master them, acting alone, or even rediscover them. To put it somewhat roughly, we can say this: those approaches, ideas, results, facts, algorithms that were included in training courses and the corresponding teaching aids- are saved and used by descendants, those that are not included disappear into the dust of libraries.

Growth points. There are five current trends, in which modern applied statistics are developed, i.e. five “growth points”: nonparametrics, robustness, bootstrap, interval statistics, statistics of objects of non-numerical nature. Let us briefly discuss these current trends.

Nonparametrics, or nonparametric statistics, allows you to draw statistical conclusions, evaluate distribution characteristics, and test statistical hypotheses without weakly substantiated assumptions that the distribution function of sample elements is part of a particular parametric family. For example, there is a widespread belief that statistics often follow a normal distribution. However, an analysis of specific observational results, in particular, measurement errors, shows that in the vast majority of cases, real distributions differ significantly from normal ones. Uncritical use of the normality hypothesis often leads to significant errors, for example, when rejecting outliers, during statistical quality control, and in other cases. Therefore, it is advisable to use nonparametric methods in which only very weak requirements are imposed on the distribution functions of observational results. Usually only their continuity is assumed. To date, using nonparametric methods it is possible to solve almost the same range of problems that were previously solved by parametric methods.

The main idea of work on robustness (stability): conclusions should change little with small changes in the initial data and deviations from the model assumptions. There are two sets of tasks here. One is to study the robustness of common data mining algorithms. The second is the search for robust algorithms for solving certain problems.

The term “robustness” itself does not have a clear meaning. It is always necessary to specify a specific probabilistic-statistical model. However, the Tukey-Huber-Hampel “clogging” model is usually not practically useful. It is focused on “weighting the tails”, and in real situations “tails are cut off” by a priori restrictions on the results of observations associated, for example, with the measuring instruments used.

Bootstrap is a direction of nonparametric statistics based on the intensive use of information technology. The main idea is to “multiply samples”, i.e. in obtaining a set of many samples resembling that obtained in the experiment. Using this set, one can evaluate the properties of various statistical procedures. The simplest way“sample multiplication” consists of excluding one observation result from it. We exclude the first observation, we get a sample similar to the original one, but with a size reduced by 1. Then we return the excluded result of the first observation, but exclude the second observation. We get a second sample, similar to the original one. Then we return the result of the second observation, and so on. There are other ways to “reproduce samples.” For example, you can use the original sample to construct one or another estimate of the distribution function, and then use statistical tests to simulate a number of samples from elements in applied statistics it is a sample, i.e. a collection of independent identically distributed random elements. What is the nature of these elements? In classical mathematical statistics, sample elements are numbers or vectors. And in non-numerical statistics, sample elements are objects of a non-numerical nature that cannot be added and multiplied by numbers. In other words, objects of a non-numerical nature lie in spaces that do not have a vector structure.

Methods for making decisions under risk conditions are also developed and justified within the framework of the so-called theory of statistical decisions. Statistical decision theory is a theory of conducting statistical observations, processing these observations and using them. As is known, the task of economic research is to understand the nature of an economic object and to reveal the mechanism of the relationship between its most important variables. This understanding allows us to develop and implement the necessary measures to manage this object, or economic policy. To do this, we need methods adequate to the task that take into account the nature and specificity of economic data that serves as the basis for qualitative and quantitative statements about the economic object or phenomenon being studied.

Any economic data represents quantitative characteristics of any economic objects. They are formed under the influence of many factors, not all of which are accessible to external control. Uncontrollable factors can take on random values from some set of values and thereby cause the data they define to be random. The stochastic nature of economic data necessitates the use of special statistical methods adequate to them for their analysis and processing.

Quantitative assessment of business risk, regardless of the content of a specific task, is possible, as a rule, using the methods of mathematical statistics. The main tools of this assessment method are dispersion, standard deviation, and coefficient of variation.

Typical designs based on measures of variability or probability of risk conditions are widely used in applications. So, financial risks, caused by fluctuations in an outcome around an expected value, such as efficiency, are assessed using variance or expected absolute deviation from the mean. In capital management problems, a common measure of the degree of risk is the probability of losses or loss of income compared to the predicted option.

To assess the magnitude of the risk (degree of risk), we will focus on the following criteria:

1) average expected value;
2) fluctuation (variability) of the possible result.

For statistical sampling

Where Xj - expected value for each observation case (/" = 1, 2,...), l, - number of observation cases (frequency) value l:, x=E - average expected value, st - variance,

V - coefficient of variation, we have:

Let's consider the problem of assessing risk under business contracts. Interproduct LLC decides to enter into an agreement for the supply of food products from one of three bases. Having collected data on the terms of payment for goods by these bases (Table 6.7), it is necessary, after assessing the risk, to select the base that pays for the goods in the shortest possible time when concluding a contract for the supply of products.

Table 6.7

Payment terms in days	Number of cases observed P	HP	(x-x)	(x-x ) 2	(x-x) 2 p

For the first base, based on formulas (6.4.1):

For second base

For third base

The coefficient of variation for the first base is the smallest, which indicates the advisability of concluding a product supply agreement with this base.

The considered examples show that risk has a mathematically expressed probability of loss, which is based on statistical data and can be calculated with a fairly high degree of accuracy. When choosing the most acceptable solution, the rule of optimal probability of the result was used, which consists in choosing from among the possible solutions the one at which the probability of the result is acceptable for the entrepreneur.

In practice, the application of the rule of optimal probability of a result is usually combined with the rule of optimal variability of the result.

As is known, the variability of indicators is expressed by their dispersion, standard deviation and coefficient of variation. The essence of the rule of optimal fluctuation of the result is that from the possible solutions, the one is selected in which the probabilities of winning and losing for the same risky investment of capital have a small gap, i.e. the smallest amount of variance, the standard deviation of the variation. In the problems under consideration, the choice of optimal solutions was made using these two rules.

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Introduction

1. Probability theory and mathematical statistics in decision making

1.1 How probability theory and mathematical statistics are used

1.2 Examples of the application of probability theory and mathematical statistics

1.3 Assessment objectives

1.4 What is “mathematical statistics”

1.5 Briefly about the history of mathematical statistics

1.6 Probabilistic-statistical methods and optimization

2. Typical practical problems of probabilistic-statistical decision making and methods for solving them

2.1 Statistics and applied statistics

2.2 Tasks of statistical analysis of the accuracy and stability of technological processes and product quality

2.3 Problems of one-dimensional statistics (statistics of random variables)

2.4 Multivariate statistical analysis

2.5 Statistics of random processes and time series

2.6 Statistics of objects of non-numeric nature

3. Application of probabilistic and statistical methods of decision making in solving economic problems

Conclusion

References

Introduction

Probabilistic-statistical methods of decision-making are used in the case when the effectiveness of the decisions made depends on factors that are random variables for which the laws of probability distribution and other statistical characteristics are known. Moreover, each decision can lead to one of many possible outcomes, and each outcome has a certain probability of occurrence, which can be calculated. Indicators characterizing problematic situation, are also described using probabilistic characteristics. In such decision-making tasks, the decision maker always runs the risk of getting a result that is not the one he orients himself towards when 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 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 when making decisions, the basis is a mathematical model in which objective relationships are expressed in terms of probability theory. Probabilities are used primarily 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.

The logic of using sample characteristics to make decisions based on theoretical models involves the simultaneous use of two parallel series of concepts - 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). Typically, 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 probability values for the scenarios 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. construction of a probabilistic model of a control system, technological process, decision-making procedure, in particular based on the results of statistical control, etc.;

A probabilistic model of a real phenomenon should be considered constructed if the quantities under consideration and the connections 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.

Based on the type of problem being solved, mathematical statistics is usually divided into three sections: data description, estimation, and hypothesis testing. Based on the type of statistical data processed, mathematical statistics is divided into four areas:

An example when it is advisable to use probabilistic-statistical models.

When controlling the quality of any product, a sample is selected from it to decide whether the batch of products being 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 when forming a sample, that is, it is necessary that each unit of product in the controlled batch has the same probability of being selected for the sample. Selection based on lot in such a situation is not sufficiently objective. Therefore, in production conditions, the selection of product units for the sample is usually carried out not by lot, but by special tables of random numbers or using computer random number sensors.

In the statistical regulation of technological processes, based on the methods of mathematical statistics, rules and plans for statistical process control are developed, aimed at timely detection of problems in technological processes and taking measures to adjust them and prevent the release of products that do not meet established requirements. These measures are aimed at reducing production costs and losses from the supply of low-quality units. During 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 of decision-making, on the basis of which the questions posed above can be answered. In mathematical statistics, probabilistic models and methods for testing hypotheses have been developed for this purpose.

In addition, in a number of managerial, production, economic, and national economic situations, problems of a different type arise - problems of assessing the characteristics and parameters of probability distributions.

Or, when statistically analyzing 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 scatter in the process under consideration. According to probability theory, it is advisable to use its mathematical expectation as the average value of a random variable, and dispersion, 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. They all 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 and statistical methods of general application and specific ones are used. For example, in the section of production management devoted to statistical methods of product quality management, applied mathematical statistics (including design of experiments) are used. Using its methods, statistical analysis of the accuracy and stability of technological processes and statistical quality assessment are carried out. Specific methods include methods of statistical acceptance control of product quality, statistical regulation of technological processes, reliability assessment and control
and etc.

In production management, in particular, when optimizing product quality and ensuring compliance with standard requirements, 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 product requirements, preliminary design, technical specifications 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 capabilities and economic situation for the future.

The most common probabilistic statistical methods are regression analysis, factor analysis, variance analysis, statistical methods for risk assessment, scenario method, etc. The area of statistical methods devoted to the analysis of statistical data of a non-numerical nature, i.e., is becoming increasingly important. measurement results based on qualitative and different types of characteristics. One of the main applications of statistics of objects of a 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 when solving problems using the methods of the theory of statistical solutions is to state the problem, i.e., to reduce a real problem to the corresponding standard one, to determine the probabilities of events based on statistical data, and also to approve the resulting optimal solution.

1. Probability theory and mathematical statistics in decision making

1.1 How probability theory is usedand mathematical statistics

These disciplines are the basis of probabilistic and statistical methods of decision making. To use their mathematical apparatus, it is necessary to express decision-making problems in terms of probabilistic-statistical models. The application of a specific probabilistic-statistical decision-making method consists of three stages:

Carrying out calculations and drawing conclusions using 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 compliance 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 product in a batch, on specific form of laws of distribution of controlled parameters of the technological process, etc.).

Mathematical statistics uses the concepts, methods and results of probability theory. Let's consider the main issues of constructing probabilistic models of decision-making in economic, managerial, technological and other situations. For the active and correct use of regulatory, technical and instructional documents on probabilistic and statistical methods of decision-making, preliminary knowledge is required. Thus, it is necessary to know under what conditions a particular document should be used, 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.

1.2 Examples of application of probability theoryand mathematical statistics

Let's consider several examples where probabilistic-statistical models are a good tool for solving management, production, economic, and national economic problems. So, for example, in A.N. Tolstoy’s novel “Walking through Torment” (vol. 1) it is said: “the workshop produces twenty-three percent of rejects, you stick to this figure,” Strukov told Ivan Ilyich.”

The question arises of how to understand these words in the conversation of factory managers, since one unit of production cannot be 23% defective. It can be either good or defective. Strukov probably meant that a large-volume batch contains approximately 23% defective units of production. The question then arises, what does “approximately” mean? Let 30 out of 100 tested units of production turn out to be defective, or out of 1000 - 300, or out of 100,000 - 30,000, etc., is it necessary to accuse Strukov of lying?

Or another example. The coin used as a lot must be “symmetrical”, i.e. when throwing it, on average, in half the cases the coat of arms should fall out, and in half the cases - a hash (tails, number). But what does "on average" mean? If you conduct many series of 10 tosses in each series, then you will often encounter series in which the coin lands as a coat of arms 4 times. For a symmetrical coin, this will happen in 20.5% of runs. And if after 100,000 tosses there are 40,000 coats of arms, can the coin be considered symmetrical? The decision-making procedure is based on probability theory and mathematical statistics.

The example in question may not seem serious enough. However, it is not. Drawing lots is widely used in organizing industrial technical and economic experiments, for example, when processing the results of measuring the quality indicator (friction torque) of bearings depending on various technological factors (the influence of the conservation environment, methods of preparing bearings before measurement, the influence of bearing loads during the measurement process, etc.). P.). Let's say it is necessary to compare the quality of bearings depending on the results of their storage in different preservation oils, i.e. in oils of composition A and B. When planning such an experiment, the question arises of which bearings should be placed in oil of composition A, and which ones should be placed in oil of composition B, but in such a way as to avoid subjectivity and ensure the objectivity of the decision made.

The answer to this question can be obtained by drawing lots. A similar example can be given with quality control of any product. To decide whether the controlled batch of products meets or does not meet the established requirements, a sample is selected from it. 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 when forming a sample, that is, it is necessary that each unit of product in the controlled batch has the same probability of being selected for the sample. In production conditions, the selection of product units for the sample is usually carried out not by lot, but by special tables of random numbers or using computer random number sensors.

Similar problems of ensuring objectivity of comparison arise when comparing various schemes for organizing production, remuneration, during tenders and competitions, and selecting candidates for vacant positions and so on. Everywhere we need a draw or similar procedures. Let us explain with 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 defeat 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, the second strongest team will not make it to the final. The one who plans the tournament can either “knock out” the second-strongest team from the tournament ahead of schedule, pitting it against the leader in the first meeting, or provide it with second place by ensuring meetings with weaker teams right up to the final. To avoid subjectivity, a draw is carried out. For an 8-team tournament, the probability that the top two teams will meet in the final is 4/7. Accordingly, with a probability of 3/7, the second strongest team will leave the tournament early.

Any measurement of product units (using a caliper, micrometer, ammeter, etc.) contains errors. To find out whether there are systematic errors, it is necessary to make repeated measurements of a unit of product whose characteristics are known (for example, a standard sample). It should be remembered that in addition to systematic error, there is also random error.

Therefore, the question arises of how to find out from the measurement results whether there is a systematic error. If we only note whether the error obtained during the next measurement is positive or negative, then this task can be reduced to the previous one. Indeed, let’s compare a measurement to throwing a coin, a positive error to the loss of a coat of arms, and a negative error to a grid (a zero error with a sufficient number of scale divisions almost never occurs). Then checking for the absence of 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 “sign criterion” in mathematical statistics.

1.3 Assessment objectives

In a number of managerial, production, economic, and national economic situations, problems of a different type arise - problems of assessing the characteristics and parameters of probability distributions.

Let's look at an example. Let a batch of N electric lamps arrive for inspection. A sample of n electric lamps was randomly selected from this batch. A number of natural questions arise. How to determine the average service life of electric lamps based on the test results of sample elements and with what accuracy can this characteristic be assessed? How will the accuracy change if we take a larger sample? At what number of hours T can it be guaranteed that at least 90% of electric lamps will last T or more hours?

Let us assume that when testing a sample of n electric lamps, X electric lamps turned out to be defective. Then the following questions arise. What limits can be specified for the number D of defective electric lamps in a batch, for the level of defectiveness D/N, etc.?

Or, when statistically analyzing 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 scatter in the process under consideration. According to probability theory, it is advisable to use its mathematical expectation as the average value of a random variable, and dispersion, 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 that can be given. 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 management of product quality.

1.4 What is “mathematical statistics”

Mathematical statistics is understood as “a branch of mathematics devoted to 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 allows us to evaluate the accuracy and reliability of the conclusions obtained in each problem based on the available statistical material.” In this case, statistical data refers to information about the number of objects in any more or less extensive collection that have certain characteristics.

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

Based on the type of statistical data processed, mathematical statistics is divided into four areas:

Univariate 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 observing an object is described by several numbers (vector);

Statistics of random processes and time series, where the result of an 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 based on a qualitative criterion.

Historically, some areas of statistics of objects of a non-numerical nature (in particular, problems of estimating the proportion of defects and testing hypotheses about it) and one-dimensional statistics were the first to appear. The mathematical apparatus is simpler for them, so their example is usually used to demonstrate the basic ideas of mathematical statistics.

Only those data processing methods, i.e. 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, risk occurrence, functioning technological equipment, obtaining the results of the experiment, the course of the disease, etc. A probabilistic model of a real phenomenon should be considered constructed if the quantities under consideration and the connections 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.

Non-probabilistic methods of data processing 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 conclusions obtained on the basis of limited statistical material.

Probabilistic and statistical methods are applicable wherever it is possible to construct and justify 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).

Applied probabilistic and statistical disciplines such as reliability theory and queuing theory are widely used. The content of the first of them is clear from the name, 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 telephone sets. The duration of servicing 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.

1.5 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 probability theory, investigated and justified the least squares method, created by him in 1795 and used for processing astronomical data (in order to clarify the orbit of a small 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.

IN late XIX V. - early 20th century Major contributions to mathematical statistics were made by English researchers, primarily K. Pearson (1857-1936) and R. A. Fisher (1890-1962). In particular, Pearson developed the chi-square test for testing statistical hypotheses, and Fisher developed analysis of variance, the theory of experimental design, and the maximum likelihood method for estimating parameters.

In the 30s of the twentieth century. Pole Jerzy Neyman (1894-1977) and Englishman E. Pearson developed general theory 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 nonparametric statistics. In the forties of the twentieth century. Romanian A. Wald (1902-1950) built the theory of sequential statistical analysis.

Mathematical statistics is developing rapidly at the present time. Thus, over the past 40 years, four fundamentally new areas of research can be distinguished:

Development and implementation of mathematical methods for planning experiments;

Development of statistics of objects of non-numerical nature as independent direction in applied mathematical statistics;

Development of statistical methods that are resistant to small deviations from the probabilistic model used;

Widespread development of work on the creation of computer software packages designed for statistical data analysis.

1.6 Probabilistic-statistical methods and optimization

The idea of optimization permeates modern applied mathematical statistics and other statistical methods. Namely, methods of planning experiments, statistical acceptance control, statistical regulation of technological processes, etc. On the other hand, optimization formulations in decision-making theory, for example, the applied theory of optimization of product quality and standard requirements, 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 of the product life cycle, i.e. at the stage of research preparation of experimental design developments (development of promising product requirements, preliminary design, technical specifications 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 capabilities and economic situation for the future. Statistical methods should be used at all stages of solving an optimization problem - when scaling variables, developing mathematical models of the 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, statistics of random variables, multivariate statistical analysis, statistics of random processes and time series, statistics of objects of non-numerical nature. It is advisable to select a statistical method for analyzing specific data according to the recommendations.

2. Typical practical problems of probability-statistic decision makingand methods for solving them

2.1 Statistics and applied statistics

Applied statistics is understood as the part of mathematical statistics devoted to methods of processing real statistical data, as well as the corresponding mathematical and software. Thus, purely mathematical problems are not included in applied statistics.

Statistical data is understood as numerical or non-numerical values of controlled parameters (signs) of the objects under study, which are obtained as a result of observations (measurements, analyses, tests, experiments, etc.) of a certain number of signs for each unit included in the study. Methods for obtaining statistical data and sample sizes are established based on the formulation of a specific applied problem based on the methods of the mathematical theory of experiment planning.

The result of observation xi of the studied characteristic X (or a set of studied characteristics X) of the yi-th sampling unit reflects the quantitative and/or qualitative properties of the surveyed unit with number i (here i = 1, 2, ..., n, where n is the sample size).

The results of observations x1, x2,…, xn, where xi is the result of observation of the i-th sampling unit, or the results of observations for several samples, are processed using methods of applied statistics corresponding to the task. Typically used analytical methods, i.e. methods based on numerical calculations (objects of a non-numerical nature are described using numbers). In some cases, it is permissible to use graphic methods(visual analysis).

2.2 Tasks of statistical analysis of the accuracy and stability of technological processes and product quality

Statistical methods are used, in particular, to analyze the accuracy and stability of technological processes and product quality. The goal is to prepare solutions that ensure the effective functioning of technological units and improve the quality and competitiveness of manufactured products. Statistical methods should be used in all cases where, based on the results of a limited number of observations, it is necessary to establish the reasons for the improvement or deterioration of the accuracy and stability of technological equipment. The accuracy of a technological process is understood as a property of a technological process that determines the proximity of the actual and nominal values of the parameters of the manufactured product. The stability of a technological process is understood as a property of a technological process that determines the constancy of the probability distributions for its parameters over a certain period of time without outside intervention.

The goals of applying statistical methods for analyzing the accuracy and stability of technological processes and product quality at the stages of development, production and operation (consumption) of products are, in particular:

* determination of actual indicators of accuracy and stability of the technological process, equipment or product quality;

* establishing compliance of product quality with the requirements of regulatory and technical documentation;

* checking compliance with technological discipline;

* study of random and systematic factors that can lead to defects;

* identification of production and technology reserves;

* justification technical standards and product approvals;

* assessment of test results of prototypes when justifying product requirements and standards for them;

* justification for the choice of technological equipment and measuring and testing instruments;

* comparison of various product samples;

* justification for replacing continuous control with statistical control;

* identifying the possibility of introducing statistical methods for product quality management, etc.

To achieve the above goals, use various methods describing data, evaluating and testing hypotheses. Let us give examples of problem statements.

2.3 Problems of one-dimensional statistics (statistics of random variables)

Comparison of mathematical expectations is carried out in cases where it is necessary to establish the correspondence of the quality indicators of the manufactured product and the reference sample. This is the task of testing the hypothesis:

H0: M(X) = m0,

where m0 is the value corresponding to the reference sample; X is a random variable that models the results of observations. Depending on the formulation of the probabilistic model of the situation and the alternative hypothesis, the comparison of mathematical expectations is carried out either by parametric or non-parametric methods.

Comparison of dispersions is carried out when it is necessary to establish the difference between the dispersion of a quality indicator and the nominal one. To do this, we test the hypothesis:

No less important than the problems of testing hypotheses are the problems of parameter estimation. They, like hypothesis testing problems, are divided into parametric and nonparametric depending on the probabilistic model of the situation used.

In parametric estimation problems, a probabilistic model is adopted, according to which the results of observations x1, x2,..., xn are considered as realizations of n independent random variables with a distribution function F(x;u). Here and is an unknown parameter lying in the parameter space specified by the probabilistic model used. The estimation task is to determine point estimates and confidence limits (or confidence region) for the parameter and.

The parameter and is either a number or a vector of fixed finite dimension. So, for a normal distribution and = (m, y2) is a two-dimensional vector, for a binomial distribution and = p is a number, for a gamma distribution
and = (a, b, c) is a three-dimensional vector, etc.

In modern mathematical statistics, a number of common methods determining estimates and confidence limits - method of moments, maximum likelihood method, one-step estimation method, stable (robust) estimation method, unbiased estimation method, etc.

Let's briefly look at the first three of them.

The method of moments is based on the use of expressions for the moments of the random variables under consideration through the parameters of their distribution functions. Estimates of the method of moments are obtained by substituting sample moments instead of theoretical ones into functions expressing parameters in terms of moments.

In the maximum likelihood method, developed mainly by R.A. Fisher, the value u* for which the so-called likelihood function is maximum is taken as an estimate of the parameter u

f(x1, u) f(x2, u) … f(xn, u),

where x1, x2,…, xn are the results of observations; f(x, u) is their distribution density, depending on the parameter u, which needs to be estimated.

Maximum likelihood estimators tend to be efficient (or asymptotically efficient) and have less variance than method of moments estimators. In some cases, formulas for them are written out explicitly (normal distribution, exponential distribution without shift). However, more often, to find them, it is necessary to numerically solve a system of transcendental equations (Weibull-Gnedenko distributions, gamma). In such cases, it is advisable to use not maximum likelihood estimates, but other types of estimates, primarily one-step estimates.

In nonparametric estimation problems, a probabilistic model is adopted, in which the results of observations x1, x2,..., xn are considered as realizations of n independent random variables with a distribution function F(x) general view. F(x) is only required to fulfill certain conditions such as continuity, the existence of mathematical expectation and dispersion, etc. Such conditions are not as stringent as the condition of belonging to a certain parametric family.

In the nonparametric setting, either the characteristics of a random variable (mathematical expectation, dispersion, coefficient of variation) or its distribution function, density, etc. are estimated. Thus, by virtue of the law of large numbers, the sample arithmetic mean is a consistent estimate of the mathematical expectation M(X) (for any distribution function F(x) of observation results for which the mathematical expectation exists). Using the central limit theorem, asymptotic confidence limits are determined

(M(X))H = , (M(X))B = .

where g - confidence probability, - quantile of the order of the standard normal distribution N(0;1) with zero mathematical expectation and unit variance, - sample arithmetic mean, s - sample standard deviation. The term "asymptotic confidence limits" means that the probabilities

P((M(X))H< M(X)}, P{(M(X))B >M(X)),

P((M(X))H< M(X) < (M(X))B}

tend to, and r, respectively, for n > ?, but, generally speaking, are not equal to these values for finite n. In practice, asymptotic confidence limits provide sufficient accuracy for n of order 10.

The second example of nonparametric estimation is estimating the distribution function. According to Glivenko's theorem, the empirical distribution function Fn(x) is a consistent estimate of the distribution function F(x). If F(x) is a continuous function, then, based on Kolmogorov’s theorem, the confidence limits for the distribution function F(x) are specified in the form

(F(x))Н = max, (F(x))B = min,

where k(r,n) is the order r quantile of the distribution of the Kolmogorov statistic for a sample size n (recall that the distribution of this statistic does not depend on F(x)).

The rules for determining estimates and confidence limits in the parametric case are based on the parametric family of distributions F(x;u). When processing real data, the question arises: do these data correspond to the accepted probabilistic model? Those. statistical hypothesis that the observation results have a distribution function from the family (F(x;u), and U) for some u = u0? Such hypotheses are called agreement hypotheses, and the criteria for testing them are called agreement criteria.

If the true value of the parameter u = u0 is known, the distribution function F(x; u0) is continuous, then the Kolmogorov test, based on statistics, is often used to test the goodness-of-fit hypothesis

where Fn(x) is the empirical distribution function.

If the true value of the parameter u0 is unknown, for example, when testing the hypothesis about the normality of the distribution of observation results (i.e., when testing whether this distribution belongs to the family of normal distributions), then statistics are sometimes used

It differs from the Kolmogorov statistics Dn in that instead of the true value of the parameter u0 its estimate u* is substituted.

The distribution of the Dn(u*) statistic is very different from the distribution of the Dn statistic. As an example, consider a normality test when u = (m, y2), and u* = (, s2). For this case, the quantiles of the distributions of statistics Dn and Dn(u*) are given in Table 1. Thus, the quantiles differ by approximately 1.5 times.

Table 1 - Quantiles of statistics Dn and Dn(and*) when checking normality

During the initial processing of statistical data, an important task is to exclude observational results obtained as a result of gross errors and misses. For example, when viewing data on the weight (in kilograms) of newborn children, along with the numbers 3,500, 2,750, 4,200, the number 35.00 may appear. It is clear that this is a mistake, and an erroneous number was obtained due to an erroneous recording - the decimal point was shifted by one sign, as a result, the observation result was mistakenly increased by 10 times.

Statistical methods for excluding outliers are based on the assumption that such observations have distributions that differ sharply from those being studied and therefore should be excluded from the sample.

The simplest probabilistic model that's how it is. Under the null hypothesis, the observation results are considered as realizations of independent identically distributed random variables X1, X2, Xn with the distribution function F(x). Under the alternative hypothesis, X1, X2, Xn-1 are the same as under the null hypothesis, and Xn corresponds to the gross error and has a distribution function G(x) = F(x - c), where c is large. Then with a probability close to 1 (more precisely, tending to 1 as the sample size increases),

Xn = max (X1, X2, Xn) = Xmax,

those. When describing data, Xmax should be considered as a possible blunder. The critical region has the form

Ш = (x: x > d).

The critical value d = d(b,n) is chosen depending on the significance level b and sample size n from the condition

P(Xmax > d | H0) = b (1)

Condition (1) is equivalent to the following for large n and small b:

If the distribution function of observation results F(x) is known, then the critical value d is found from relation (2). If F(x) is known up to parameters, for example, it is known that F(x) - normal function distribution, then rules for testing the hypothesis under consideration have also been developed.

However, often the form of the distribution function of observational results is not known absolutely accurately and not to an accuracy of parameters, but only with some error. Then relation (2) becomes practically useless, since a small error in determining F(x), as can be shown, leads to a large error in determining the critical value d from condition (2), and for a fixed d the level of significance of the criterion can differ significantly from the nominal .

Therefore, in a situation where there is no complete information about F(x), but the mathematical expectation M(X) and the variance y2 = D(X) of the observation results X1, X2, Xn are known, nonparametric rejection rules based on Chebyshev’s inequality can be used. Using this inequality, we find the critical value d = d(b,n) such that

then relation (3) will be satisfied if

By Chebyshev's inequality

therefore, in order for (4) to be satisfied, it is sufficient to equate the right-hand sides of formulas (4) and (5), i.e. determine d from the condition

The rejection rule, based on the critical value d calculated using formula (6), uses minimal information about the distribution function F(x) and therefore excludes only observational results that are very far removed from the bulk. In other words, the value of d1 given by relation (1) is usually much less than the value of d2 given by relation (6).

2.4 Multivariate statistical analysis

Multivariate statistical analysis is used to solve the following problems:

* study of the dependence between signs;

* classification of objects or features specified by vectors;

* reducing the dimension of the feature space.

In this case, the result of observations is a vector of values of a fixed number of quantitative and sometimes qualitative characteristics measured in an object. A quantitative characteristic is a characteristic of an observable unit that can be directly expressed by a number and a unit of measurement. A quantitative characteristic is contrasted with a qualitative characteristic - a characteristic of an observed unit, determined by assignment to one of two or more conditional categories (if there are exactly two categories, then the characteristic is called alternative). Statistical analysis of qualitative characteristics is part of the statistics of objects of non-numerical nature. Quantitative characteristics are divided into characteristics measured on the scales of intervals, ratios, differences, and absolute.

And qualitative ones - for characteristics measured in a scale of names and an ordinal scale. Data processing methods must be consistent with the scales in which the characteristics in question are measured.

The goals of studying the dependence between characteristics are to prove the existence of a connection between characteristics and to study this connection. To prove the existence of a connection between two random variables X and Y, correlation analysis is used. If the joint distribution of X and Y is normal, then statistical conclusions are based on the sample linear correlation coefficient; in other cases, the Kendall and Spearman rank correlation coefficients are used, and for qualitative characteristics, the chi-square test is used.

Regression analysis is used to study the functional dependence of the quantitative trait Y on the quantitative traits x(1), x(2), ..., x(k). This dependence is called regression or, for short, regression. The simplest probabilistic model of regression analysis (in the case of k = 1) uses as initial information a set of pairs of observation results (xi, yi), i = 1, 2, … , n, and has the form

yi = axi + b + ei, i = 1, 2, … , n,

where ei are observation errors. It is sometimes assumed that ei are independent random variables with the same normal distribution N(0, y2). Since the distribution of observation errors is usually different from normal, it is advisable to consider the regression model in a nonparametric formulation, i.e. with an arbitrary distribution of ei.

The main task of regression analysis is to estimate the unknown parameters a and b, which define the linear dependence of y on x. To solve this problem, the least squares method, developed by K. Gauss in 1794, is used, i.e. find estimates of the unknown model parameters a and b from the condition of minimizing the sum of squares

by variables a and b.

Analysis of variance is used to study the influence of qualitative characteristics on a quantitative variable. For example, let there be k samples of measurement results quantitative indicator quality of product units produced on k machines, i.e. a set of numbers (x1(j), x2(j), … , xn(j)), where j is the machine number, j = 1, 2, …, k, and n is the sample size. In a common formulation of variance analysis, it is assumed that the measurement results are independent and in each sample they have a normal distribution N(m(j), y2) with the same variance.

Checking the uniformity of product quality, i.e. absence of influence of the machine number on product quality, comes down to testing the hypothesis

H0: m(1) = m(2) = … = m(k).

Variance analysis has developed methods for testing such hypotheses.

Hypothesis H0 is tested against the alternative hypothesis H1, according to which at least one of the specified equalities is not satisfied. The test of this hypothesis is based on the following "variance decomposition" specified by R. A. Fisher:

where s2 is the sample variance in the pooled sample, i.e.

Thus, the first term on the right side of formula (7) reflects the intragroup dispersion. Finally, there is intergroup variance,

The area of applied statistics associated with variance expansions like formula (7) is called variance analysis. As an example of an analysis of variance problem, consider testing the above hypothesis H0 under the assumption that the measurement results are independent and in each sample they have a normal distribution N(m(j), y2) with the same variance. If H0 is true, the first term on the right side of formula (7), divided by y2, has a chi-square distribution with k(n-1) degrees of freedom, and the second term, divided by y2, also has a chi-square distribution, but with ( k-1) degrees of freedom, with the first and second terms being independent as random variables. Therefore the random variable

has a Fisher distribution with (k-1) numerator degrees of freedom and k(n-1) denominator degrees of freedom. Hypothesis H0 is accepted if F< F1-б, и отвергается в противном случае, где F1-б - квантиль порядка 1-б распределения Фишера с указанными числами степеней свободы. Такой выбор критической области определяется тем, что при Н1 величина F безгранично увеличивается при росте объема выборок n. Значения F1-б берут из соответствующих таблиц.

Nonparametric methods have been developed for solving classical problems of variance analysis, in particular, testing the hypothesis H0.

The next type of multivariate statistical analysis problems is classification problems. They are divided into three fundamentally various types- discriminant analysis, cluster analysis, grouping problems.

The task of discriminant analysis is to find a rule for classifying an observed object into one of the previously described classes. In this case, objects are described in a mathematical model using vectors, the coordinates of which are the results of observing a number of features in each object. Classes are described either directly in mathematical terms or using training samples. A training set is a sample for each element of which it is indicated which class it belongs to.

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18 Activating methods of making management decisions

Brainstorming is a method of group discussion of a problem based on non-analytical thinking.

1) The idea generation stage is separated from the criticism stage;

2) At the stage of generating ideas, any criticism is prohibited; absurd ideas are accepted.

3) All ideas are recorded in writing;

4) At this stage, critics select 3-4 ideas that can be considered as alternative options.

The “Questions and Answers” method is based on the preliminary compilation of a set of questions, the answers to which can form new approach to solving the problem.

"5 Whys" Method

Five "whys" is an effective tool that uses questions to explore the cause-and-effect relationships underlying a particular problem, identify causal factors, and identify the root cause. By examining logic in the direction of “Why?”, we gradually reveal the entire chain of sequentially interconnected causal factors that influence the problem.

Action plan

Identify the specific problem that needs to be solved.

Come to an agreement on the formulation of the problem under consideration.

When looking for a solution to a problem, you should start with the end result (the problem) and work backwards (towards the root cause), asking why the problem occurs.

Write the answer under the problem.

If the answer does not reveal the root cause of the problem, ask the question "Why?" again. and write down the new answer below.

The question "Why?" must be repeated until the root cause of the problem becomes apparent.

If the answer solves the problem and the group agrees with it, a solution is made that uses the answer.

The “game-theoretic method” is based on the creation of a human-machine system for developing solutions. The predecessor was traditional meetings. Typically, such meetings included economics, social. And specialized solutions. The interests of the participants are often different, and the range of issues is wide-ranging. The qualitative development of the meeting methodology was the introduction of the development process of SD, artificial intelligence in the form of a computer model.

The computer model of the organization includes:

1) Reference data (about suppliers, consumers);

2) Company simulation models

3) Methods of economic calculation and forecasting

4) Information about solutions in similar situations.

As a result, meetings are more productive. Such a meeting can take place in several game sessions: where in 1 session all participants enter their requirements, after processing the comp. Gives a specific solution that can be discussed and adjusted again. This can last until a general decision is reached or until the decision is refused.

And comes after “mere lies” and “blatant lies”, attributed to Benjamin Disraeli, who was the fortieth and forty-second (the periods fall in the 2nd half of the 19th century) Prime Minister of Great Britain. However, in our time, the authorship of Disraeli, advertised by Mark Twain, is denied. But, be that as it may, many experts continue to repeat this phrase in their works or the main content of which is methods of statistical analysis. As a rule, it sounds like a joke, in which there is only a grain of joke...

Statistics is a branch of specific knowledge that describes the procedure for collecting, analyzing and interpreting large amounts of data, both qualitative and quantitative. It concerns various scientific or practical areas of life. For example, applied statistics helps to choose the right statistical method for processing all kinds of data for analysis. Legal works in the field of offenses and control over them. Mathematical develops mathematical methods that make it possible to systematize and use the information obtained for practical or scientific purposes. Demographics describe patterns. Query statistics are more relevant to linguists and the Internet.

The use of statistical methods dates back to at least the 5th century BC. One of the earliest records is contained in a book written in the 9th century AD. e. Arab philosopher, doctor, mathematician and musician Al-Kindi. He gave detailed description how to use frequency analysis (histogram). The New Chronicle, dating back to the 14th century and describing the history of Florence, is considered one of the first positive works of statistics in history. They were compiled by the Florentine banker Giovanni Villani and include much information about the population, administration, commerce and trade, education and religious sites.

The early use of statistics was determined by the state’s desire to build a demographically and economically sound policy. Its scope was expanded in the early 19th century to include data collection and analysis in general. Today this area of knowledge is widely used by government agencies, business, natural and social sciences. Its mathematical foundations, the need for which arose from the study of gambling, were laid back in the 17th century with the development of probability theory by French mathematicians and Pierre de Fermat. Statistical was first described by Carl Friedrich Gauss around 1794.

Fast and steady growth computing power, starting from the second half of the 20th century, had a significant impact on the development of applied statistics. The computer revolution has placed new emphasis on its experimental and empirical components. Now available a large number of both general and special programs with which you can easily put into practice any statistical method, be it control charts, histograms, checklist, stratification method, Ishikawa scheme or Pareto analysis.

Today statistics is one of the key tools for running an effective business and organizing production. It allows you to understand and measure variability trends, resulting in improved process control, as well as improved quality of products and services. For example, managers who use statistical qualities usually make informed decisions, thereby management works effectively and brings the expected results. Therefore, statistics in this case is the key and, perhaps, the only reliable tool.

The ability to select and correctly apply a statistical method allows you to obtain reliable conclusions and not mislead those to whom the analysis data is provided. Therefore, the frequent mention by specialists of the old saying about 3 degrees of lies should be considered as a warning against mistakes that can mislead and form the basis decisions made with devastating consequences.

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18 Activating methods of making management decisions