The essence of the first element of the statistical methodology is the collection of primary data on the object under study. For example: during the census of the country's population, data is collected about each person living in its territory, which is entered in a special form.

The second element: summary and grouping is the division of the totality of data obtained at the observation stage into homogeneous groups according to one or more characteristics. For example, as a result of grouping materials, the census is divided into groups (by sex, age, population, education, etc.).

The essence of the third element of statistical methodology lies in the calculation and socio-economic interpretation of generalizing statistical indicators:

1. Absolute

2. Relative

3. Medium

4. Indicators of variation

5. Speakers

The three main elements of statistical methodology also constitute the three stages of any statistical study.

3. Law big numbers and statistical regularity.

The law of large numbers plays an important role in statistical methodology. In the most general view it can be formulated as follows:

The law of large numbers is a general principle by virtue of which cumulative actions a large number random factors leads under certain general conditions to a result almost independent of chance.

The law of large numbers is generated by special properties of mass phenomena. The mass phenomena of the latter, in turn, on the one hand, due to their individuality, differ from each other, and on the other hand, they have something in common that determines their belonging to a certain class.

A single phenomenon is more susceptible to the influence of random and insignificant factors than a mass of phenomena as a whole. Under certain conditions, the value of a feature of an individual unit can be considered as a random variable, given that it obeys not only a general pattern, but is also formed under the influence of conditions that do not depend on this pattern. It is for this reason that statistics widely use averages, which characterize the entire population with one number. Only with a large number of observations, random deviations from the main direction of development are balanced, canceled out and the statistical regularity manifests itself more clearly. Thus, the essence of the law of large numbers lies in the fact that in numbers summarizing the result of mass statistical observation, the pattern of development of socio-economic phenomena is revealed more clearly than with a small statistical study.

4. Branches of statistics.

In progress historical development As part of statistics as a single science, the following branches have emerged and gained a certain independence:

1. General theory statistics, which develops the concept of categories and methods for measuring the quantitative patterns of social life.

2. Economic statistics studying the quantitative patterns of reproduction processes at various levels.

3. Social statistics, which studies the quantitative side of the development of the social infrastructure of society (statistics of health care, education, culture, moral, judicial, etc.).

4. Industry statistics (statistics of industry, agro-industrial complex, transport, communications, etc.).

All branches of statistics, developing and improving their methodology, contribute to the development of statistical science as a whole.

5. Basic concepts and categories of statistical science in general.

A statistical aggregate is a set of elements of the same type that are similar to each other in some respects and differ in others. For example: this is a set of sectors of the economy, a set of universities, a set of cooperation between design bureaus, etc.

Individual elements of a statistical population are called its units. In the examples discussed above, the units of the population are, respectively, the industry, the university (one) and the employee.

Units of the population usually have many characteristics.

A sign is a property of the units of the population, expressing their essence and having the ability to vary, i.e. change. Signs that take a single value in individual units of the population are called varying, and the values ​​themselves are options.

Variable signs are subdivided into attributive or qualitative ones. An attribute is called attributive or qualitative if its separate value (variants) are expressed as a state or properties inherent in the phenomenon. Variants of attributive features are expressed in verbal form. Examples of such signs can serve - economic.

An attribute is called quantitative if its individual value is expressed in the form of numbers. For example: wage, scholarship, age, size of OF.

According to the nature of variation, quantitative signs are divided into discrete and continuous.

Discrete - such quantitative signs that can only take on a well-defined, as a rule, integer value.

Continuous - are such signs that, within certain limits, can take on the value of both integer and fractional. For example: GNP of a country, etc.

There are also primary and secondary features.

The main features characterize the main content and essence of the phenomenon or process being studied.

Secondary signs give Additional information and are directly related to the inner content of the phenomenon.

Depending on the goals of a particular study, the same signs in the same cases may be primary, and in others secondary.

A statistical indicator is a category that reflects the dimensions and quantitative ratios of the signs of socio-economic phenomena and their qualitative certainty in specific conditions of place and time. It is necessary to distinguish between the content of a statistical indicator and its specific numerical expression. Content, i.e. qualitative certainty lies in the fact that indicators always characterize socio-economic categories (population, economy, financial institutions, etc.). Quantitative dimensions of statistical indicators, i.e. their numerical values ​​depend primarily on the time and place of the object that is subjected to statistical research.

Socio-economic phenomena, as a rule, cannot be characterized by any one indicator, for example: the standard of living of the population. A scientifically substantiated system of statistical indicators is necessary for a comprehensive comprehensive characterization of the phenomena under study. Such a system is not permanent. It is constantly being improved based on the needs of social development.

6. Tasks of statistical science and practice in the conditions of market economy development.

The main tasks of statistics in the context of the development of market relations in Russia are the following:

1. Improving accounting and reporting and reducing the document flow on this basis.

You have to study the following main topics of the topic:

Connection of statistics with the theory and practice of market economy

Tasks of statistics

Concepts and methods of statistics

Law of large numbers, statistical regularity

Lesson 1. Introduction

1. The history of statistics

Statistics is an independent social science that has its own subject and method of research. It arose from the practical needs of social life. Already in ancient world there was a need to count the number of inhabitants of the state, to take into account people suitable for military affairs, to determine the number of livestock, the size of land and other property. Information of this kind was necessary for collecting taxes, waging wars, and so on. In the future, as social life develops, the range of phenomena taken into account gradually expands.

The volume of collected information has especially increased with the development of capitalism and world economic ties. The needs of this period compelled government bodies and capitalist enterprises to collect extensive and varied information on labor markets and the sale of goods and raw materials for practical purposes.

In the middle of the 17th century, a scientific direction arose in England, called "political arithmetic". This trend was initiated by William Petit (1623-1687) and John Graunt (1620-1674). "Political arithmetic", based on the study of information about mass social phenomena, sought to discover the patterns of social life and, thus, to point out the questions that arose in connection with the development of capitalism.

Along with the school of "political arithmetic" in England, a school of descriptive statistics or "state studies" developed in Germany. The emergence of this science dates back to 1660.

The development of political arithmetic and state science led to the emergence of the science of statistics.

The concept of "statistics" comes from the Latin word "status", which in translation means position, state, order of phenomena.

The term "statistics" was introduced into scientific circulation by Gottfried Achenwal (1719-1772), a professor at the University of Göttingen.

Depending on the object of study, statistics as a science is divided into social, demographic, economic, industrial, commercial, banking, financial, medical, etc. General properties statistical data, regardless of their nature and methods of their analysis are considered mathematical statistics and the general theory of statistics.

The subject of statistics . Statistics deals primarily with the quantitative side of the phenomena and processes of social life. One of the characteristic features of statistics is that when studying the quantitative side of social phenomena and processes, it always reflects the qualitative features of the phenomena under study, i.e. studies quantity in inseparable connection, unity with quality.

Quality in the scientific and philosophical understanding is the properties inherent in an object or phenomenon that distinguish this object or phenomenon from others. Quality is what makes objects and phenomena certain. Using philosophical terminology, we can say that statistics studies social phenomena as the unity of their qualitative and quantitative certainty, i.e. studies the measure of social phenomena.

Statistical methodology . The most important constituent elements of the statistical methodology are:

mass surveillance

grouping, application of generalizing (summary) characteristics;

analysis and generalization of statistical facts and detection of regularities in the studied phenomena.

Let's take a closer look at these elements.

In order to characterize any mass phenomenon from a quantitative point of view, one must first collect information about its constituent elements. This is achieved with the help of mass observation, carried out on the basis of the rules and methods developed by statistical science.

The information collected in the process of statistical observation is subject to further summary (primary scientific processing), during which characteristic parts (groups) are distinguished from the entire set of surveyed units. The selection of groups and subgroups of units from the entire surveyed mass is called in statistics grouping . Grouping in statistics is the basis for processing and analyzing the collected information. It is carried out on the basis of certain principles and rules.

In the process of processing statistical information, the totality of the surveyed units and its selected parts based on the use of the grouping method are characterized by a system of digital indicators: absolute and average values, relative values, dynamics indicators, etc.

3. Tasks of statistics

Complete and reliable statistical information is the necessary basis on which the process of economic management is based. Making managerial decisions at all levels, from a national or regional level to the level of an individual corporation or private firm, is impossible without official statistical support.

It is statistical data that make it possible to determine the volume of gross domestic product and national income, to identify the main trends in the development of economic sectors, to assess the level of inflation, to analyze the state of financial and commodity markets, to study the standard of living of the population and other socio-economic phenomena and processes.

Statistics is a science that studies the quantitative side of mass phenomena and processes in close connection with their qualitative side, a quantitative expression of the laws of social development in specific conditions of place and time.

For getting statistical information bodies of state and departmental statistics, as well as commercial structures conduct various kinds of statistical research. As already noted, the process of statistical research includes three main stages: data collection, their summary and grouping, analysis and calculation of generalizing indicators.

The results and quality of all subsequent work largely depend on how the primary statistical material is collected, how it is processed and grouped. Insufficient study of the program-methodological and organizational aspects of statistical observation, the lack of logical and arithmetic control of the collected data, non-compliance with the principles of group formation can ultimately lead to absolutely erroneous conclusions.

No less complex, time-consuming and responsible is the final, analytical stage of the study. At this stage, average indicators and distribution indicators are calculated, the structure of the population is analyzed, the dynamics and relationships between the studied phenomena and processes are studied.

The techniques and methods of data collection, processing and analysis used at all stages of the study are the subject of study of the general theory of statistics, which is the basic branch of statistical science. The developed methodology is used in macroeconomic statistics, sectoral statistics (industry, agriculture, other trade), population statistics, social statistics, and other statistical fields. The great importance of statistics in society is explained by the fact that it is one of the most basic, one of the most important means by which an economic entity keeps records in the economy.

Accounting is a way to systematically measure and study generalized phenomena using quantitative methods.

For every study of quantitative relationships there is an account. Various quantitative relations between phenomena can be represented in the form of certain mathematical formulas, and this, in itself, will not yet be an account. One of the characteristic features of accounting is the calculation of INDIVIDUAL elements, INDIVIDUAL units that make up this or that phenomenon. Various mathematical formulas are used in accounting, but their application is necessarily associated with counting elements.

Accounting is a means of control and generalization of the results obtained in the process of generalized development.

Thus, statistics is the most important tool for understanding and using economic and other laws of social development.

The economic reform poses qualitatively new tasks for statistical science and practice. In accordance with the state program for Russia's transition to the internationally accepted system of accounting and statistics, the system for collecting statistical information is being reorganized and the methodology for analyzing market processes and phenomena is being improved.

The System of National Accounts (SNA), which is widely used in world practice, corresponds to the peculiarities and requirements of market relations. Therefore, the transition to a market economy made it possible to introduce the SNA into statistical and accounting records, reflecting the functioning of the sectors of the market economy.

This is necessary for a comprehensive analysis of the economy at the macro level and for providing information to international economic organizations with which Russia cooperates.

Statistics play a big role in information and analytical support for development economic reform. The single purpose of this process is the assessment, analysis and forecasting of the state and development of the economy at the present stage.

The law of large numbers plays an important role in statistical methodology. In its most general form, it can be formulated as follows:

The law of large numbers is a general principle by virtue of which the cumulative action of a large number of random factors leads, under certain general conditions, to a result almost independent of chance.

The law of large numbers is generated by special properties of mass phenomena. The mass phenomena of the latter, in turn, on the one hand, due to their individuality, differ from each other, and on the other hand, they have something in common that determines their belonging to a certain class.

A single phenomenon is more susceptible to the influence of random and insignificant factors than a mass of phenomena as a whole. Under certain conditions, the value of a feature of an individual unit can be considered as a random variable, given that it obeys not only a general pattern, but is also formed under the influence of conditions that do not depend on this pattern. It is for this reason that statistics widely use averages, which characterize the entire population with one number. Only with a large number of observations, random deviations from the main direction of development are balanced, canceled out and the statistical regularity manifests itself more clearly. Thus, essence of the law of large numbers lies in the fact that in the numbers summarizing the result of mass statistical observation, the pattern of development of socio-economic phenomena is revealed more clearly than with a small statistical study.

LAW OF GREAT NUMBERS

Economy. Dictionary. - M.: "INFRA-M", Publishing house "Ves Mir". J. Black. General editorial staff: Doctor of Economics Osadchaya I.M. . 2000 .

Raizberg B.A., Lozovsky L.Sh., Starodubtseva E.B. . Modern economic dictionary. - 2nd ed., corrected. Moscow: INFRA-M. 479 p. . 1999

Economic dictionary. 2000 .

See what the "LAW OF GREAT NUMBERS" is in other dictionaries:

LAW OF GREAT NUMBERS- see LARGE NUMBERS LAW. Antinazi. Encyclopedia of Sociology, 2009 ... Encyclopedia of Sociology

Law of Large Numbers- the principle according to which the quantitative patterns inherent in mass social phenomena are most clearly manifested with a sufficiently large number of observations. Single phenomena are more susceptible to random and ... ... Glossary of business terms

LAW OF GREAT NUMBERS- states that with a probability close to one, the arithmetic mean of a large number random variables approximately one order of magnitude will differ little from a constant equal to the arithmetic mean of the mathematical expectations of these quantities. Diff. ... ... Geological Encyclopedia

law of large numbers- - [Ya.N. Luginsky, M.S. Fezi Zhilinskaya, Yu.S. Kabirov. English Russian Dictionary of Electrical Engineering and Power Industry, Moscow, 1999] Topics in electrical engineering, basic concepts EN law of averageslaw of large numbers ... Technical translator's guide

Law of Large Numbers- in probability theory asserts that the empirical mean (arithmetic mean) of a sufficiently large finite sample from a fixed distribution is close to the theoretical mean (expectation) of this distribution. Depending ... Wikipedia

law of large numbers- didžiųjų skaičių dėsnis statusas T sritis fizika atitikmenys: engl. law of large numbers vok. Gesetz der großen Zahlen, n rus. law of large numbers, m pranc. loi des grands nombres, f … Fizikos terminų žodynas

LAW OF GREAT NUMBERS- a general principle, due to which the combined action of random factors leads, under certain very general conditions, to a result that is almost independent of chance. The convergence of the frequency of occurrence of a random event with its probability with an increase in the number ... ... Russian sociological encyclopedia

Law of Large Numbers- a law stating that the cumulative action of a large number of random factors leads, under some very general conditions, to a result that is almost independent of chance ... Sociology: a dictionary

LAW OF GREAT NUMBERS- statistical law expressing the relationship of statistical indicators (parameters) of the sample and the general population. The actual values ​​of statistical indicators obtained from a certain sample always differ from the so-called. theoretical ... ... Sociology: Encyclopedia

LAW OF GREAT NUMBERS- the principle that the frequency of financial losses of a certain type can be predicted with high accuracy when there are a large number of losses of similar types ... encyclopedic Dictionary economics and law

Law of Large Numbers

Interacting daily in work or study with numbers and numbers, many of us do not even suspect that there is a very interesting law of large numbers, used, for example, in statistics, economics, and even psychological and pedagogical research. It refers to probability theory and says that the arithmetic mean of any large sample from a fixed distribution is close to the mathematical expectation of this distribution.

You probably noticed that it is not easy to understand the essence of this law, especially for those who are not particularly friendly with mathematics. Based on this, we would like to talk about it plain language(as far as possible, of course), so that everyone can at least roughly understand for themselves what it is. This knowledge will help you better understand some mathematical patterns, become more erudite and positively influence the development of thinking.

Concepts of the law of large numbers and its interpretation

In addition to the above definition of the law of large numbers in probability theory, we can give its economic interpretation. In this case, it represents the principle that the frequency of a particular type of financial loss can be predicted from a high degree reliability when observed high level losses of such types in general.

In addition, depending on the level of convergence of features, we can distinguish the weak and strengthened laws of large numbers. We are talking about weak when convergence exists in probability, and about strong when convergence exists in almost everything.

If we interpret it a little differently, then we should say this: it is always possible to find such a finite number of trials, where, with any preprogrammed probability less than one, the relative frequency of occurrence of some event will differ very little from its probability.

Thus, the general essence of the law of large numbers can be expressed as follows: the result of the complex action of a large number of identical and independent random factors will be such a result that does not depend on chance. And speaking in even simpler language, then in the law of large numbers, the quantitative laws of mass phenomena will clearly manifest themselves only when there are a large number of them (that is why the law of large numbers is called the law).

From this we can conclude that the essence of the law is that in the numbers that are obtained by mass observation, there are some correctness, which is impossible to detect in a small number of facts.

The essence of the law of large numbers and its examples

The law of large numbers expresses the most general patterns of the accidental and the necessary. When random deviations "extinguish" each other, the averages determined for the same structure take on the form of typical ones. They reflect the operation of essential and permanent facts under the specific conditions of time and place.

Regularities defined by the law of large numbers are strong only when they represent mass tendencies, and they cannot be laws for individual cases. Thus, the principle mathematical statistics, which says that the complex action of a number of random factors can cause a non-random result. And the most striking example of the operation of this principle is the convergence of the frequency of occurrence of a random event and its probability when the number of trials increases.

Let's remember the usual coin toss. Theoretically, heads and tails can fall out with the same probability. This means that if, for example, a coin is tossed 10 times, 5 of them should come up heads and 5 should come up heads. But everyone knows that this almost never happens, because the ratio of the frequency of heads and tails can be 4 to 6, and 9 to 1, and 2 to 8, etc. However, with an increase in the number of coin tosses, for example, up to 100, the probability that heads or tails will fall out reaches 50%. If, theoretically, an infinite number of such experiments are carried out, the probability of a coin falling out on both sides will always tend to 50%.

How exactly the coin will fall is influenced by a huge number of random factors. This is the position of the coin in the palm of your hand, and the force with which the throw is made, and the height of the fall, and its speed, etc. But if there are many experiments, regardless of how the factors act, it can always be argued that the practical probability is close to the theoretical probability.

And here is another example that will help to understand the essence of the law of large numbers: suppose we need to estimate the level of earnings of people in a certain region. If we consider 10 observations, where 9 people receive 20 thousand rubles, and 1 person - 500 thousand rubles, the arithmetic mean will be 68 thousand rubles, which, of course, is unlikely. But if we take into account 100 observations, where 99 people receive 20 thousand rubles, and 1 person - 500 thousand rubles, then when calculating the arithmetic mean, we get 24.8 thousand rubles, which is already closer to the real state of affairs. By increasing the number of observations, we will force the average value to tend to the true value.

It is for this reason that in order to apply the law of large numbers, it is first necessary to collect statistical material in order to obtain truthful results by studying a large number of observations. That is why it is convenient to use this law, again, in statistics or social economics.

Summing up

The importance of the fact that the law of large numbers works is difficult to overestimate for any field of scientific knowledge, and especially for scientific developments in the field of the theory of statistics and methods of statistical knowledge. The action of the law is also of great importance for the objects under study themselves with their mass regularities. Almost all methods of statistical observation are based on the law of large numbers and the principle of mathematical statistics.

But, even without taking into account science and statistics as such, we can safely conclude that the law of large numbers is not just a phenomenon from the field of probability theory, but a phenomenon that we encounter almost every day in our lives.

We hope that now the essence of the law of large numbers has become more clear to you, and you can easily and simply explain it to someone else. And if the topic of mathematics and probability theory is interesting to you in principle, then we recommend reading about Fibonacci numbers and the Monty Hall paradox. See also approximate calculations in life situations and the most popular numbers. And, of course, pay attention to our cognitive science course, because after passing it, you will not only master new thinking techniques, but also improve your cognitive abilities in general, including mathematical ones.

1.1.4. Statistics Method

Statistics Method involves the following sequence of actions:

development of a statistical hypothesis,

summary and grouping of statistical data,

The passage of each stage is associated with the use of special methods, explained by the content of the work performed.

1.1.5. Tasks of statistics

Development of a system of hypotheses characterizing the development, dynamics, state of socio-economic phenomena.

Organization of statistical activity.

Development of analysis methodology.

Development of a system of indicators for managing the economy at the macro and micro levels.

To popularize the data of statistical observation.

1.1.6. The law of large numbers and its role in the study of statistical regularities

The mass nature of social laws and the originality of their actions predetermine the need for the study of aggregate data.

The law of large numbers is generated by special properties of mass phenomena. The latter, by virtue of their individuality, on the one hand, differ from each other, and on the other hand, they have something in common, due to their belonging to a certain class, species. Moreover, single phenomena are more susceptible to the influence of random factors than their totality.

The law of large numbers in its simplest form states that the quantitative regularities of mass phenomena are clearly manifested only in a sufficiently large number of them.

Thus, its essence lies in the fact that in the numbers obtained as a result of mass observation, certain regularities appear that cannot be detected in a small number of facts.

The law of large numbers expresses the dialectic of the accidental and the necessary. As a result of the mutual cancellation of random deviations, the average values ​​calculated for a value of the same type become typical, reflecting the actions of constant and significant facts under given conditions of place and time.

The tendencies and regularities revealed by the law of large numbers are valid only as mass tendencies, but not as laws for each individual case.

The manifestation of the operation of the law of large numbers can be seen in many areas of the phenomena of social life studied by statistics. For example, the average output per worker, the average unit cost of a product, the average wage, and other statistical characteristics express patterns common to a given mass phenomenon. Thus, the law of large numbers contributes to the disclosure of the patterns of mass phenomena as an objective necessity for their development.

1.1.7. The main categories and concepts of statistics: statistical population, population unit, sign, variation, statistical indicator, system of indicators

Since statistics deals with mass phenomena, the main concept is the statistical totality.

Population - this is a set of objects or phenomena studied by statistics that have one or more common features and differ from each other in other ways. So, for example, when determining the volume of retail trade turnover, all trade enterprises selling goods to the population are considered as a single statistical aggregate - “retail trade”.

E population unit – this is the primary element of the statistical population, which is the carrier of the signs to be registered, and the basis of the account maintained during the survey.

For example, in a census of commercial equipment, the observation unit is the trade enterprise, and the population unit is their equipment (counters, refrigeration units, etc.).

sign — This characteristic property phenomenon under study that distinguishes it from other phenomena. Signs can be characterized by a number of statistical values.

In different branches of statistics, different signs are studied. So, for example, the object of study is an enterprise, and its features are the type of product, output volume, number of employees, etc. Or the object is a separate person, and the signs are gender, age, nationality, height, weight, etc.

Thus, statistical features, i.e. there are a lot of properties, qualities of objects of observation. All their diversity is usually divided into two large groups: signs of quality and signs of quantity.

Qualitative sign (attributive) - a sign, the individual meanings of which are expressed in the form of concepts, names.

Profession - turner, locksmith, technologist, teacher, doctor, etc.

Quantitative sign - a sign, certain values ​​of which have quantitative expressions.

Height - 185, 172, 164, 158.

Weight - 105, 72, 54, 48.

Each object of study may have a number of statistical features, but from object to object, some features change, others remain unchanged. Changing features from one object to another are called variable. It is these features that are studied in statistics, since it is not interesting to study an unchanging feature. Suppose that there are only men in your group, everyone has one attribute (gender - male) and there is nothing more to say on this basis. And if there are women, then you can already calculate their percentage in the group, the dynamics of changes in the number of women by months school year and etc.

Variation sign - this is the diversity, the variability of the value of the attribute in individual units of the observation population.

Variation of the trait - gender - male, female.

Salary variation - 10000, 100000, 1000000.

The individual characteristic values ​​are called options this sign.

Phenomena and processes in the life of society are studied by statistics through statistical indicators.

statistic - this is a generalizing characteristic of some property of the statistical population or its part. In this it differs from a sign (property inherent in a unit of the population). For example, GPA per semester for a group of students is a statistical indicator. A score in some subject of a particular student is a sign.

Statistical indicator system is a set of interconnected statistical indicators that comprehensively reflect the processes of social life in certain conditions of place and time.

The law of large numbers. statistical regularity

The concept of statistics and its main provisions

Statistics as a Population Parameter

The law of large numbers. statistical regularity

Boy or girl

Research methods used in population statistics

Bibliography

Word statistics V mid-eighteenth V. began to designate a set of various kinds of factual information about states (from the Latin “status” - state). Such information included data on the size and movement of the population of states, their territorial division and administrative structure, economy, etc.

Currently, the term "statistics" has several related meanings. One of them closely corresponds to the above. Statistics is often referred to as a set of facts about a particular country. The main ones are systematically published in special editions in the prescribed form.

However, modern statistics in the considered sense of the word is distinguished from the “state of reference” of past centuries not only by the vastly increased completeness and versatility of the information contained in it. With regard to the nature of the information, it now includes only what is received quantitative expression. So, statistics do not include information about whether a given state is a monarchy or a republic. What language is adopted in it as the state language, etc.

But it includes quantitative data on the number of people who use this or that language as their spoken language. Statistics do not include the list and location on the map of individual territorial units states, but include quantitative data on the distribution of population, industry, etc.

A common feature of the information that makes up statistics is that they always refer not to one single (individual) phenomenon, but cover summary characteristics whole line such phenomena, or, as they say, their totality. An individual phenomenon differs from the totality by its indecomposability into independently existing and similar constituent elements. The totality consists of just such elements. The disappearance of one of the elements of the aggregate does not destroy it as such.

Thus, the population of a city remains its population even after one of its members has died or moved to another.

Different aggregates and their units in reality are combined and intertwined with each other, sometimes in very complex complexes. A specific feature of statistics is that in all cases its data refer to the population. The characteristics of individual individual phenomena fall into its field of vision only as a basis for obtaining summary characteristics of the totality.

For example, the registration of a marriage has a certain meaning for a given individual couple entering into it; certain rights and obligations follow from it for each spouse. Statistics include only summary data on the number of marriages, on the composition of those who entered into them - by age, by source of livelihood, etc. Individual cases of marriage are of interest to statistics only in so far as, based on information about them, it is possible to obtain summary data.

Statistics as a Population Parameter

Recently, the term "statistics" has often been understood in a somewhat narrower, but more precisely defined sense, associated with the processing of the results of a series of individual observations.

Let's imagine that as a result of observations we got the numbers x 1 , x 2 . x n. These numbers are considered as one of the possible realizations of the set n quantities in their combination.

A statistic is a parameter f depending on x 1 , x 2 . x n. Since these quantities are, as noted, one of their possible realizations, the value of this parameter also turns out to be one of a number of possible ones. Therefore, each statistic in this sense has its own probability distribution (that is, for any given number a there is a possibility that the f will be no more than a).

Compared with the content invested in the term “statistics” in the sense discussed above, here, firstly, we mean its narrowing each time to one value - a parameter, which does not exclude the joint consideration of several parameters (several statistics) in one complex problem . Secondly, it emphasizes the presence of a mathematical rule (algorithm) for obtaining the parameter value from the totality of observation results: calculate their arithmetic mean, take the maximum of the delivered values, calculate the ratio of the number of some of their special group to total number etc.

Finally, in the indicated sense, the term "statistics" is applied to a parameter obtained from the results of observations in any field of phenomena - social and others. It could be the average yield, or the average span of the pine trees in the forest, or the average result of repeated measurements of the parallax of some star, and so on. in this sense, the term "statistics" is used mainly in mathematical statistics, which, like any branch of mathematics, cannot be limited to one or another area of ​​phenomena.

Statistics is also understood as the process of its “keeping”, i.e. the process of collecting and processing information about the facts necessary to obtain statistics in both considered senses.

At the same time, the information necessary for statistics can be collected for the sole purpose of obtaining generalized characteristics for the mass of cases of a given kind, i.e. It is natural for the purposes of statistics. Such, for example, is the information collected during population censuses.

The law of large numbers. Statistical regularity.

The main generalization of the experience of studying any mass phenomena is the law of large numbers. A separate individual phenomenon, considered as one of the phenomena of this kind, contains an element of chance: it could be or not be, be this or that. When a large number of such phenomena are combined into general characteristics in their entire mass, chance disappears to the greater extent, the more individual phenomena are connected.

Mathematics, in particular the theory of probability, considered in a purely quantitative aspect, the law of large numbers, expresses it with a whole chain of mathematical theorems. They show under what conditions and to what extent one can count on the absence of randomness in the characteristics covering the mass, how this is connected with the number of individual phenomena included in them. Statistics is based on these theorems in the study of each specific mass phenomenon.

regularity, which manifested itself only in a large mass of phenomena through overcoming the randomness inherent in its individual elements, is called statistical regularity .

In some cases, statistics is faced with the task of measuring its manifestations, while its very existence is theoretically clear in advance.

In other cases, a regularity can be found empirically by statistics. In this way, for example, it was found that with an increase in family income in its budget, the percentage of expenses for food decreases.

Thus, whenever statistics in the study of a phenomenon reaches generalizations and finds a regularity operating in it, this latter immediately becomes the property of that particular science, to the circle of interests of which this phenomenon belongs. Therefore, for each statistic acts as a method.

Considering the results of mass observation, statistics finds similarities and differences in them, combines elements into groups, revealing different types, differentiating the entire observed mass according to these types. The results of observation of individual elements of the mass are used, further, to obtain the characteristics of the entire population and the special parts distinguished in it, i.e. in order to obtain general indicators.

Mass observation, grouping and summary of its results, calculation and analysis of generalizing indicators - these are the main features of the statistical method.

Statistics as a science takes care of and is reduced to mathematical statistics. In mathematics, the tasks of characterizing mass phenomena are considered only in a purely quantitative aspect, divorced from the qualitative content (which is mandatory for mathematics, as a science in general). Statistics, even in the study of the general laws of mass phenomena, proceeds not only from quantitative generalizations of these phenomena, but above all from the mechanism of the emergence of the mass phenomenon itself.

At the same time, from what has been said about the role of quantitative measurement for statistics, it follows great importance for it, mathematical methods in general, specially adapted for solving problems that arise in the study of mass phenomena (probability theory and mathematical statistics). Moreover, the role of mathematical methods here is so great that an attempt to exclude them from the course of statistics (due to the presence in the plans of a separate subject - mathematical statistics) significantly impoverishes statistics.

Refusal of this attempt, however, should not mean the opposite extreme, namely, the absorption by statistics of the entire theory of probability and mathematical statistics. If, for example, in mathematics, the average value for a series of distributions (probabilities or empirical frequencies) is considered, then statistics also cannot bypass the appropriate techniques, but here this is one of the aspects, along with which a number of others arise (general and group averages, the occurrence and the role of averages in the information system, the material content of the system of weights, chronological averages, average and relative values, etc.).

Or another example: the mathematical theory of sampling focuses all attention on the error of representativeness - for different systems of selection, different characteristics, etc. System error, i.e. the error not absorbed in the average value, it eliminates in advance by constructing the so-called unbiased estimates free of it. In statistics, perhaps the main question in this matter is the question of how to avoid this systemic error.

In the study of the quantitative side of mass phenomena, a number of problems of a mathematical nature arise. To solve them, mathematics develops appropriate techniques, but for this it must consider them in a general form, for which the qualitative content of the mass phenomenon is indifferent. So the manifestation of the law of large numbers was first noticed precisely in the socio-economic field and almost simultaneously in gambling (the very distribution of which was explained by the fact that they were a cast from the economy, in particular, developing commodity-money relations). From the moment, however, when the law of large numbers becomes the object of exact study in mathematics, it receives a completely general interpretation, which does not limit its action to any special area.

On this basis, the subject of statistics is generally distinguished from the subject of mathematics. The delimitation of objects cannot mean banishing from one science everything that has fallen into the field of view of another. It would be wrong, for example, to exclude from the presentation of physics everything related to the use of differential equations on the grounds that mathematics deals with them.

Why does the sex ratio at birth have certain proportions that have not undergone significant observation for many centuries?

As paradoxical as it may sound, it is death that is the main biological condition for reproduction and reproduction of new generations. In order to prolong the existence of a species, its individuals must leave behind offspring; otherwise, the view will disappear forever.

The problem of gender (who will be born a boy or a girl) includes many issues related not only to biological development, medical genetic characteristics, demographic data, but also in a broader aspect related to the psychology of sex, the behavior and aspirations of individuals of the opposite sex, with harmony or conflict between them.

The question of who will be born - a boy or a girl - and why this happens - is just a narrow circle of questions arising from a larger problem. Especially important theoretical and practical is the clarification of the question why the life expectancy of men is lower than the life expectancy of women. This phenomenon is common not only in humans, but also among numerous species of the animal world.

It is not enough to explain this only by the fact that the predominance of males at birth is due to their increased activity, and as a result of this - less “vitality”, is not enough. Biologists have long drawn attention to the shorter lifespan of males compared to females in most animals studied. The duration of life is opposed to its high pace, and this finds a biological justification.

The English researcher A. Comfort points out: “The organism must go through a fixed series of metabolic processes or stages of development, and the speed of their passage determines the observed life span.”

Ch. Darwin considered shorter life expectancy in males "as a natural and constitutional property, due only to sex."

The possibility of giving birth to a child of one sex or another in each particular case depends not only on the patterns inherent in this phenomenon, revealed in a large number of observations, but also on random attendant circumstances. Therefore, it is statistically impossible to determine in advance what gender each separately born child will be. Neither probability theory nor statistics does this, although in many cases the result of a single event is of great interest. Probability theory gives fairly definite answers when it comes to a large population of births. Incidental, external causes are random, but their totality reflects stable patterns. In the formation of sex, as is now known, even before conception, accidental causes may in some cases favor the emergence of male embryos, and in others - female. But this does not manifest itself in some regular order, but chaotically, randomly. The totality of factors that form certain sex ratios at birth is manifested only in a sufficiently large number of observations; and the more there are, the closer the theoretical probability approaches the actual results.

The probabilities of having boys are a number slightly greater than 0.5 (close to 0.51), and girls are less than 0.5 (close to 0.49). This very interesting fact posed a difficult task for biologists and statisticians - to explain the reason why the conception and birth of a boy or girl are not equally possible and corresponding to genetic prerequisites (Mendeleev's law of splitting by sex).

Satisfactory answers to these questions have not yet been received; it is only known that already from the moment of conception the proportion of boys is greater than the proportion of girls, and that during the period of intrauterine development these proportions gradually level out even by the time of birth, without, however, reaching equiprobable values. Boys are born approximately 5-6% more than girls.

Most of the species for which life tables have been compiled by biologists have a higher mortality among males. Geneticists explain this by the difference in the common chromosome complex between females and males.

C. Darwin considers the formed numerical ratio of sexes from representatives of various species as a result of evolutionary natural selection based on the principles of sexual selection. The genetic laws of sex formation were discovered later, and they are the missing link in the theoretical concepts of Ch. Darwin. Ch. Darwin's well-aimed observations deserve to be quoted here. The author observes that sexual selection would be a simple matter if the males greatly outnumbered the females. It is important to know the sex ratio not only at birth, but also during maturity, and this complicates the picture. With regard to people, the fact has been established that many more boys die than girls before birth, during childbirth and in the first years of childhood.

Two large groups of factors can be named that influence the ratio of mortality by sex and, in general, determine the excess mortality of men. These are exogenous, i.e. socio-economic factors, and endogenous factors associated with the genetic program of the viability of the male and female organism. Differences in mortality by sex can be explained by the constant interaction of these two groups of factors. These differences increase in direct proportion to the increase in life expectancy. The purely biological differences in the viability of men and women are superimposed by the impact of socio-economic conditions of life, the reaction to which the male and female organisms are different in terms of the ability to overcome them. bad influence at different age periods.

In the vast majority of countries of the world where more or less reliable and complete registration of mortality is carried out, the ratio of indicators by sex is confirmed by the position on the increase in male mortality, which has been repeatedly confirmed by practice - this pattern, as noted earlier, is inherent in the human population and not only it, but also many others. biological species.

Population statistics- a science that studies the quantitative patterns of phenomena and processes occurring in the population, in continuous connection with their qualitative side.

Population- an object of study and demography, which establishes the general patterns of their development, considering its life in all aspects: historical, political, economic, social, legal, medical and statistical. At the same time, it must be borne in mind that as knowledge about an object develops, new aspects of it open up, becoming a separate object of knowledge.

Population statistics studies its object in the specific conditions of place and time, revealing all new forms of its movement: natural, migratory, social.

Under natural movement population refers to the change in population due to births and deaths, i.e. occurring naturally. This also includes marriages and divorces, since they are counted in the same order as births and deaths.

migratory movement, or simply population migration, means the movement of people across the boundaries of certain territories, usually with a change of residence to long time or forever.

social movement population is understood as a change in the social conditions of life of the population. It is expressed in a change in the number and composition social groups people who have common interests, values ​​and norms of behavior that develop within the framework of a historically defined society.

Population statistics solves a number of problems:

Her most important task- determination of the population. But it is often required to know the population of individual continents and their parts, various countries, economic regions of countries, administrative regions. At the same time, not a simple arithmetic, but a special - statistical account - an account of categories of the population is maintained. The number of births, deaths, marriages, divorce cases, the number of incoming and outgoing migrants is statistically established, i.e. the volume of the population is determined.

Second task- establishing the structure of the population, demographic processes. Attention here is primarily drawn to the division of the population according to sex, age, level of education, professional, production characteristics, according to belonging to urban and rural areas.

Structure of the population by sex can be characterized by an equal number of sexes, male or female predominance and the degree of this preponderance.

Population structure by age can be represented by one-year data and age groups, as well as a trend in age composition, such as aging or rejuvenation.

Educational structure shows the proportion of the literate population with a certain degree of education in different territories and different environments.

Professional- Distribution of people by professions acquired in the process of training, by occupation.

Production- by sectors of the national economy.

Territorial location of the population or its resettlement. Here, a distinction is made between the degree of urbanization, the definition of the density of the entire population, a different understanding of density and its state.

Third task consists in the study of the interrelations that take place in the population itself between its various groups and the study of the dependence of the processes occurring in the population on the environmental factors in which these processes take place.

The fourth task consists of considering the dynamics of demographic processes. In this case, the characteristics of the dynamics can be given as a change in the population size and as a change in the intensity of the processes occurring in the population in time and space.

Fifth task- Population statistics are opened with forecasts of its size and composition for the future. Providing data on the population forecast for the near and far future.

Research methods used in population statistics

Method in the most general sense means a way to achieve the goal, regulation of activity. The method of concrete science is a set of methods of theoretical and practical knowledge of reality. For an independent science, not only the presence of a subject of research special from other sciences is necessary, but also the existence of its own own methods studying this subject. The totality of research methods used in any science is methodology this science.

Since population statistics is sectoral statistics, the basis of its methodology is statistical methodology.

The most important method included in the statistical methodology is obtaining information about the processes and phenomena being studied - statistical observation . It serves as the basis for data collection both in current statistics and in censuses, monographic and sample studies of the population. Here, the full use of the provisions of theoretical statistics on the establishment of the object of the unit of observation, the introduction of concepts of the date and moment of registration, the program, organizational issues of observation, systematization and publication of its results. Statistical methodology also contains the principle of independent assignment of each enumerated person to a certain group - the principle of self-determination.

The next step in the statistical study of socio-economic phenomena is the determination of their structure, i.e. selection of parts and elements that make up the totality. We are talking about the method of groupings and classifications, which in population statistics are called typological and structural.

To understand the structure of the population, it is necessary, first of all, to single out the sign of grouping and classification. Any feature that has been observed can also serve as a grouping feature. For example, on the question of the attitude towards the person recorded first in the census form, it is possible to determine the structure of the population being enumerated, where it seems likely to distinguish a significant number of groups. This attribute is attributive, therefore, when developing census questionnaires on it, it is necessary to compile in advance a list of classifications (groupings according to attribute characteristics) needed for analysis. When compiling classifications with a large number of attribute records, the assignment to certain groups is justified in advance. So, according to their occupation, the population is divided into several thousand species, which statistics reduce to certain classes, which is recorded in the so-called dictionary of occupations.

When studying the structure by quantitative characteristics, it becomes possible to use such statistical generalizing indicators as the mean, mode and median, distance measures or variation indicators to characterize different parameters of the population. The considered structures of phenomena serve as the basis for studying the connection in them. In the theory of statistics, functional and statistical relationships are distinguished. The study of the latter is impossible without dividing the population into groups and then comparing the value of the effective feature.

Grouping according to a factor attribute and comparing it with changes in the attribute of an effective one allows you to establish the direction of the relationship: it is direct or reverse, as well as to give an idea of ​​its form. broken regression . These groupings make it possible to construct a system of equations necessary to find regression equation parameters and determining the tightness of the connection by calculating the correlation coefficients. Groupings and classifications serve as the basis for using dispersion analysis of relationships between indicators of population movement and the factors that cause them.

Statistical methods are widely used in the study of the population. dynamics research , graphic study of phenomena , index , selective And balance . It can be said that population statistics use the entire arsenal to study its object. statistical methods and examples. In addition, methods developed only for the study of the population are also used. These are the methods real generation (cohorts) And conditional generation . The first allows us to consider changes in the natural movement of peers (born in the same year) - a longitudinal analysis; the second considers the natural movement of peers (living at the same time) - a cross-sectional analysis.

It is interesting to use averages and indices when taking into account the characteristics and comparing the processes occurring in the population, when the conditions for comparing data are not equal to each other. Using different weightings when calculating generalizing averages, a standardization method has been developed that allows eliminating the influence of different age characteristics of the population.

Probability theory as mathematical science studies the properties of the objective world with the help of abstractions , the essence of which consists in a complete abstraction from qualitative certainty and in highlighting their quantitative side. Abstraction is the process of mental abstraction from many aspects of the properties of objects and at the same time the process of isolating, isolating any aspects of interest to us, properties and relations of the objects under study. The use of abstract mathematical methods in population statistics makes it possible statistical modeling processes occurring in the population. The need for modeling arises when it is impossible to study the object itself.

The largest number of models used in population statistics has been developed to characterize its dynamics. Among them stand out exponential And logistics. Of particular importance in the population forecast for future periods are models stationary And stable population, which determine the type of population that has developed under these conditions.

If the construction of models of the exponential and logistic population uses data on the dynamics of the absolute population for the past period, then the models of the stationary and stable population are built on the basis of the characteristics of the intensity of its development.

So the statistical methodology of studying the population has at its disposal a number of methods of the general theory of statistics, mathematical methods and special methods developed in population statistics themselves.

Population statistics, using the methods discussed above, develops a system of generalizing indicators, indicates the necessary information, methods for calculating them, the cognitive capabilities of these indicators, the conditions for use, the order of recording and meaningful interpretation.

The importance of generalizing statistical indicators in solving the most important problems when considering demographic policy is necessary for balanced population growth, in studying population migration, which forms the basis of inter-district redistribution of the labor force and the achievement of uniformity in its distribution.

Since the population in a certain aspect studies many other sciences - health care, pedagogy, sociology, etc., it is necessary to use the experience of these sciences, develop their methods in relation to the needs of statistics.

The tasks of renewal facing our country should also affect the solution of demographic problems. Development of comprehensive programs for economic and social development should include sections on demographic programs; their solution should contribute to the development of the population with the least demographic losses.

Bibliography

Kildishev et al. “Population statistics with the basics of demography” M .: Finance and Statistics, 1990 - 312 p.

Poor M.S. “Boys girls? Medico-demographic analysis” M.: Statistics, 1980 – 120 p.

Andreeva B.M., Vishnevsky A.G. “Longevity. Analysis and Modeling” M.: Statistics, 1979 – 157 p.

Boyarsky A.Ya., Gromyko G.L. “General theory of statistics” M.: ed. Moscow Universities, 1985 - 372 p.

Vasilyeva E.K. “Socio-demographic portrait of a student” M.: Thought, 1986 - 96 p.

Bestuzhev-Lada I.V. “The World of Our Tomorrow” M.: Thought, 1986 – 269 p.

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Law of Large Numbers

The practice of studying random phenomena shows that although the results of individual observations, even those carried out under the same conditions, can differ greatly, at the same time, the average results for a sufficiently large number of observations are stable and weakly depend on the results of individual observations. The theoretical justification for this remarkable property of random phenomena is the law of large numbers. The general meaning of the law of large numbers is that the joint action of a large number of random factors leads to a result that is almost independent of chance.

Central limit theorem

Lyapunov's theorem explains the wide distribution of the normal distribution law and explains the mechanism of its formation. The theorem allows us to assert that whenever a random variable is formed as a result of adding a large number of independent random variables, the variances of which are small compared to the variance of the sum, the law of distribution of this random variable turns out to be practically a normal law. And since random variables are always generated by an infinite number of causes, and most often none of them has a variance comparable to the variance of the random variable itself, most of the random variables encountered in practice are subject to the normal distribution law.

Let us dwell in more detail on the content of the theorems of each of these groups.

In practical research, it is very important to know in what cases it is possible to guarantee that the probability of an event will be either sufficiently small or arbitrarily close to unity.

Under law of large numbers and is understood as a set of sentences in which it is stated that with a probability arbitrarily close to one (or zero), an event will occur that depends on a very large, indefinitely increasing number of random events, each of which has only a slight influence on it.

More precisely, the law of large numbers is understood as a set of sentences in which it is stated that with a probability arbitrarily close to one, the deviation of the arithmetic mean of a sufficiently large number of random variables from a constant value, the arithmetic mean of their mathematical expectations, will not exceed a given arbitrarily small number.

Separate, single phenomena that we observe in nature and in social life often appear as random (for example, a registered death, the sex of a born child, air temperature, etc.) due to the fact that many factors that are not related to the essence of the emergence or development of a phenomenon. It is impossible to predict their total effect on the observed phenomenon, and they manifest themselves differently in individual phenomena. Based on the results of one phenomenon, nothing can be said about the patterns inherent in many such phenomena.

However, it has long been noted that the arithmetic mean of the numerical characteristics of certain features (the relative frequency of the occurrence of an event, the results of measurements, etc.) with a large number of repetitions of the experiment is subject to very slight fluctuations. In the middle one, as it were, the regularity inherent in the essence of phenomena manifests itself; in it, the influence of individual factors, which made the results of individual observations random, is mutually canceled out. Theoretically, this behavior of the average can be explained using the law of large numbers. If some very general conditions regarding random variables are met, then the stability of the arithmetic mean will be a practically certain event. These conditions constitute the most important content of the law of large numbers.

The first example of the operation of this principle can be the convergence of the frequency of occurrence of a random event with its probability with an increase in the number of trials - a fact established in Bernoulli's theorem (Swiss mathematician Jacob Bernoulli(1654-1705)). Bernoull's theorem is one of the simplest forms of the law of large numbers and is often used in practice. For example, the frequency of occurrence of any quality of the respondent in the sample is taken as an estimate of the corresponding probability).

Outstanding French mathematician Simeon Denny Poisson(1781-1840) generalized this theorem and extended it to the case when the probability of events in a trial varies independently of the results of previous trials. He was also the first to use the term "law of large numbers".

Great Russian mathematician Pafnuty Lvovich Chebyshev(1821 - 1894) proved that the law of large numbers operates in phenomena with any variation and also extends to the regularity of the average.

A further generalization of the theorems of the law of large numbers is connected with the names A.A.Markov, S.N.Bernshtein, A.Ya.Khinchin and A.N.Kolmlgorov.

The general modern formulation of the problem, the formulation of the law of large numbers, the development of ideas and methods for proving theorems related to this law belong to Russian scientists P. L. Chebyshev, A. A. Markov and A. M. Lyapunov.

CHEBYSHEV'S INEQUALITY

Let us first consider auxiliary theorems: the lemma and Chebyshev's inequality, which can be used to easily prove the law of large numbers in the Chebyshev form.

Lemma (Chebyshev).

If there are no negative values ​​of the random variable X, then the probability that it will take on some value that exceeds the positive number A is not greater than a fraction, the numerator of which is the mathematical expectation of the random variable, and the denominator is the number A:

Proof.Let the distribution law of the random variable X be known:

(i = 1, 2, ..., ), and we consider the values ​​of the random variable to be arranged in ascending order.

In relation to the number A, the values ​​of the random variable are divided into two groups: some do not exceed A, while others are greater than A. Suppose that the first group includes the first values ​​of the random variable ().

Since , then all terms of the sum are non-negative. Therefore, discarding the first terms in the expression, we obtain the inequality:

Because the

,

That

Q.E.D.

Random variables can have different distributions with the same mathematical expectations. However, for them, Chebyshev's lemma will give the same estimate of the probability of one or another test result. This shortcoming of the lemma is related to its generality: it is impossible to achieve a better estimate for all random variables at once.

Chebyshev's inequality .

The probability that the deviation of a random variable from its mathematical expectation will exceed in absolute value a positive number

Proof.Since a random variable that does not take negative values, we apply the inequality from the Chebyshev lemma for a random variable for :

Q.E.D.

Consequence. Because the

,

That

- another form of Chebyshev's inequality

We accept without proof the fact that the lemma and Chebyshev's inequality are also true for continuous random variables.

Chebyshev's inequality underlies the qualitative and quantitative statements of the law of large numbers. It defines the upper bound on the probability that the deviation of the value of a random variable from its mathematical expectation is greater than some given number. It is remarkable that the Chebyshev inequality gives an estimate of the probability of an event for a random variable whose distribution is unknown, only its mathematical expectation and variance are known.

Theorem. (Law of large numbers in Chebyshev form)

If the variances of independent random variables are limited by one constant C, and their number is large enough, then the probability is arbitrarily close to unity that the deviation of the arithmetic mean of these random variables from the arithmetic mean of their mathematical expectations will not exceed the given positive number in absolute value, no matter how small it is neither was:

.

We accept the theorem without proof.

Consequence 1. If independent random variables have the same, equal, mathematical expectations, their variances are limited by the same constant C, and their number is large enough, then, no matter how small the given positive number is, the probability that the deviation of the mean is arbitrarily close to unity arithmetic of these random variables from will not exceed in absolute value .

The fact that the approximate value of an unknown quantity is taken as the arithmetic mean of the results of a sufficiently large number of measurements made under the same conditions can be justified by this theorem. Indeed, the measurement results are random, since they are affected by a lot of random factors. The absence of systematic errors means that the mathematical expectations of individual measurement results are the same and equal. Consequently, according to the law of large numbers, the arithmetic mean of a sufficiently large number of measurements will practically arbitrarily differ little from the true value of the desired value.

(Recall that errors are called systematic if they distort the measurement result in the same direction according to a more or less clear law. These include errors that appear as a result of the imperfection of the instruments (instrumental errors), due to the personal characteristics of the observer (personal errors) and etc.)

Consequence 2 . (Bernoulli's theorem.)

If the probability of the occurrence of event A in each of the independent trials is constant, and their number is sufficiently large, then the probability is arbitrarily close to unity that the frequency of the occurrence of the event differs arbitrarily little from the probability of its occurrence:

Bernoulli's theorem states that if the probability of an event is the same in all trials, then with an increase in the number of trials, the frequency of the event tends to the probability of the event and ceases to be random.

In practice, experiments are relatively rare in which the probability of an event occurring in any experiment is unchanged, more often it is different in different experiences. Poisson's theorem refers to a test scheme of this type:

Corollary 3 . (Poisson's theorem.)

If the probability of occurrence of an event in a -test does not change when the results of previous trials become known, and their number is large enough, then the probability that the frequency of occurrence of an event differs arbitrarily little from the arithmetic mean probabilities is arbitrarily close to unity:

Poisson's theorem states that the frequency of an event in a series of independent trials tends to the arithmetic mean of its probabilities and ceases to be random.

In conclusion, we note that none of the considered theorems gives either an exact or even an approximate value of the desired probability, but only its lower or upper bound is indicated. Therefore, if it is required to establish the exact or at least approximate value of the probabilities of the corresponding events, the possibilities of these theorems are very limited.

Approximate probabilities for large values ​​can only be obtained using limit theorems. In them, either additional restrictions are imposed on random variables (as is the case, for example, in the Lyapunov theorem), or random variables of a certain type are considered (for example, in the Moivre-Laplace integral theorem).

The theoretical significance of Chebyshev's theorem, which is a very general formulation of the law of large numbers, is great. However, if we apply it to the question of whether it is possible to apply the law of large numbers to a sequence of independent random variables, then, if the answer is yes, the theorem will often require that there be much more random variables than is necessary for the law of large numbers to come into force. This shortcoming of the Chebyshev theorem is explained general character her. Therefore, it is desirable to have theorems that would more accurately indicate the lower (or upper) bound on the desired probability. They can be obtained by imposing on random variables some additional restrictions, which are usually satisfied for random variables encountered in practice.

REMARKS ON THE CONTENT OF THE LAW OF LARGE NUMBERS

If the number of random variables is large enough and they satisfy some very general conditions, then, no matter how they are distributed, it is practically certain that their arithmetic mean deviates arbitrarily small from a constant value - the arithmetic mean of their mathematical expectations, i.e. is practically constant. Such is the content of the theorems relating to the law of large numbers. Consequently, the law of large numbers is one of the expressions of the dialectical connection between chance and necessity.

One can give many examples of the emergence of new qualitative states as manifestations of the law of large numbers, primarily among physical phenomena. Let's consider one of them.

By modern ideas gases are made up of individual particles - molecules, which are in chaotic motion, and it is impossible to say exactly where it will be at a given moment, and at what speed this or that molecule will move. However, observations show that the total effect of molecules, such as the pressure of a gas on

vessel wall, manifests itself with amazing constancy. It is determined by the number of blows and the strength of each of them. Although the first and second are a matter of chance, the instruments do not detect fluctuations in the pressure of a gas under normal conditions. This is explained by the fact that due to the huge number of molecules, even in the smallest volumes

a change in pressure by a noticeable amount is almost impossible. Therefore, the physical law that states the constancy of gas pressure is a manifestation of the law of large numbers.

The constancy of pressure and some other characteristics of a gas at one time served as a weighty argument against the molecular theory of the structure of matter. Subsequently, they learned to isolate a relatively small number of molecules, ensuring that the influence of individual molecules still remained, and thus the law of large numbers could not manifest itself sufficiently. Then it was possible to observe fluctuations in gas pressure, confirming the hypothesis of the molecular structure of matter.

The law of large numbers underlies various types of insurance (human life insurance for various periods, property, livestock, crops, etc.).

When planning the range of consumer goods, the demand for them from the population is taken into account. In this demand, the operation of the law of large numbers is manifested.

The sampling method widely used in statistics finds its scientific justification in the law of large numbers. For example, the quality of wheat brought from the collective farm to the procurement point is judged by the quality of grains accidentally captured in a small measure. There are few grains in the measure compared to the whole batch, but in any case, the measure is chosen such that there are quite enough grains in it for

manifestation of the law of large numbers with an accuracy that satisfies the need. We have the right to take the corresponding indicators in the sample as indicators of weediness, moisture content and the average weight of grains of the entire batch of incoming grain.

Further efforts of scientists to deepen the content of the law of large numbers were aimed at obtaining the most general conditions for the applicability of this law to a sequence of random variables. For a long time there were no fundamental successes in this direction. After P. L. Chebyshev and A. A. Markov, only in 1926 did the Soviet academician A. N. Kolmogorov manage to obtain the conditions necessary and sufficient for the law of large numbers to be applicable to a sequence of independent random variables. In 1928, the Soviet scientist A. Ya. Khinchin showed that sufficient condition the applicability of the law of large numbers to a sequence of independent identically distributed random variables is the existence of their mathematical expectation.

For practice, it is extremely important to fully clarify the question of the applicability of the law of large numbers to dependent random variables, since phenomena in nature and society are mutually dependent and mutually determine each other. Much work has been devoted to elucidating the restrictions that must be imposed

into dependent random variables so that the law of large numbers can be applied to them, the most important ones being those of the outstanding Russian scientist A. A. Markov and the great Soviet scientists S. N. Bernshtein and A. Ya. Khinchin.

The main result of these papers is that the law of large numbers is applicable to dependent random variables, if only strong dependence exists between random variables with close numbers, and between random variables with distant numbers, the dependence is sufficiently weak. Examples of random variables of this type are the numerical characteristics of the climate. The weather of each day is noticeably influenced by the weather of the previous days, and the influence noticeably weakens with the distance of the days from each other. Consequently, the long-term average temperature, pressure and other characteristics of the climate of a given area, in accordance with the law of large numbers, should practically be close to their mathematical expectations. The latter are objective characteristics of the local climate.

In order to experimentally verify the law of large numbers in different time the following experiments were carried out.

1. Experience Buffon. The coin is flipped 4040 times. The coat of arms fell 2048 times. The frequency of its occurrence was equal to 0.50694 =

2. Experience Pearson. The coin is flipped 12,000 and 24,000 times. The frequency of the loss of the coat of arms in the first case turned out to be 0.5016, in the Second - 0.5005.

H. Experience Vestergaard. From the urn, in which there were equally white and black balls, 5011 white and 4989 black balls were obtained with 10,000 extractions (with the return of the next drawn ball to the urn). The frequency of white balls was 0.50110 = (), and black - 0.49890.

4. Experience of V.I. Romanovsky. Four coins are thrown 21160 times. Frequencies and frequencies of various combinations of coat of arms and grating were distributed as follows:

Combinations of the number of coat of arms and tails	Frequencies	Frequencies
		empirical	Theoretical
4 and 0	1 181	0,05858	0,0625
3 and 1	4909	0,24350	0,2500
2 and 2	7583	0,37614	0,3750
1 and 3	5085	0,25224	0,2500
1 and 4		0,06954	0,0625
Total	20160	1,0000	1,0000

The results of experimental tests of the law of large numbers convince us that the experimental frequencies are close to the probabilities.

CENTRAL LIMIT THEOREM

It is easy to prove that the sum of any finite number of independent normally distributed random variables is also distributed according to the normal law.

If independent random variables are not distributed according to the normal law, then some very loose restrictions can be imposed on them, and their sum will still be normally distributed.

This problem was posed and solved mainly by Russian scientists P. L. Chebyshev and his students A. A. Markov and A. M. Lyapunov.

Theorem (Lyapunov).

If independent random variables have finite mathematical expectations and finite variances , their number is large enough, and with an unlimited increase

,

where are the absolute central moments of the third order, then their sum with a sufficient degree of accuracy has a distribution

(In fact, we present not Lyapunov's theorem, but one of its corollaries, since this corollary is quite sufficient for practical applications. Therefore, the condition , which is called the Lyapunov condition, is a stronger requirement than is necessary for the proof of Lyapunov's theorem itself.)

The meaning of the condition is that the action of each term (random variable) is small compared to the total action of all of them. Many random phenomena that occur in nature and in social life proceed exactly according to this pattern. In this regard, the Lyapunov theorem is of exceptionally great importance, and normal law distribution is one of the basic laws in probability theory.

Let, for example, measurement some size . Various deviations of the observed values ​​from its true value (mathematical expectation) are obtained as a result of the influence of a very large number of factors, each of which generates a small error , and . Then the total measurement error is a random variable, which, according to the Lyapunov theorem, must be distributed according to the normal law.

At gun shooting under the influence of a very large number of random causes, shells are scattered over a certain area. Random effects on the projectile trajectory can be considered independent. Each cause causes only a small change in the trajectory compared to the total change due to all causes. Therefore, it should be expected that the deviation of the projectile rupture site from the target will be a random variable distributed according to the normal law.

By Lyapunov's theorem, we have the right to expect that, for example, adult male height is a random variable distributed according to the normal law. This hypothesis, as well as those considered in the previous two examples, is in good agreement with observations. To confirm, we present the distribution by height of 1000 adult male workers and the corresponding theoretical numbers of men, i.e., the number of men who should have the growth of these groups, based on the distribution assumption growth of men according to the normal law.

Height, cm	number of men
	experimental data	theoretical forecasts
143-146
146-149
149-152
152-155
155-158
158- 161
161- 164
164-167
167-170
170-173
173-176
176-179
179 -182
182-185
185-188

It would be difficult to expect a more accurate agreement between the experimental data and the theoretical ones.

One can easily prove, as a corollary of Lyapunov's theorem, a proposition that will be needed in what follows to justify the sampling method.

Offer.

The sum of a sufficiently large number of identically distributed random variables with absolute central moments of the third order is distributed according to the normal law.

The limit theorems of the theory of probability, the theorems of Moivre-Laplace explain the nature of the stability of the frequency of occurrence of an event. This nature consists in the fact that the limiting distribution of the number of occurrences of an event with an unlimited increase in the number of trials (if the probability of an event in all trials is the same) is a normal distribution.

System of random variables.

The random variables considered above were one-dimensional, i.e. were determined by one number, however, there are also random variables that are determined by two, three, etc. numbers. Such random variables are called two-dimensional, three-dimensional, etc.

Depending on the type of random variables included in the system, systems can be discrete, continuous or mixed if the system includes different types of random variables.

Let us consider systems of two random variables in more detail.

Definition. distribution law system of random variables is called a relation that establishes a relationship between the areas of possible values ​​of the system of random variables and the probabilities of the occurrence of the system in these areas.

Example. From an urn containing 2 white and 3 black balls, two balls are drawn. Let be the number of drawn white balls, and the random variable is defined as follows:

Let's make a distribution table of the system of random variables:

Since is the probability that no white balls are taken out (hence, two black balls are taken out), while , then

.

Probability

.

Probability

Probability is the probability that no white balls are taken out (and, therefore, two black balls are taken out), while , then

Probability is the probability that one white ball (and, therefore, one black) is drawn, while , then

Probability - the probability that two white balls are drawn (and, therefore, no black ones), while , then

.

Thus, the distribution series of a two-dimensional random variable has the form:

Definition. distribution function system of two random variables is called a function of two argumentsF( x, y) , equal to the probability of joint fulfillment of two inequalitiesX< x, Y< y.

We note the following properties of the distribution function of a system of two random variables:

1) ;

2) The distribution function is a non-decreasing function with respect to each argument:

3) The following is true:

4)

5) The probability of hitting a random point ( X , Y ) into an arbitrary rectangle with sides parallel to the coordinate axes, is calculated by the formula:

Distribution density of a system of two random variables.

Definition. Joint distribution density probabilities of a two-dimensional random variable ( X , Y ) is called the second mixed partial derivative of the distribution function.

If the distribution density is known, then the distribution function can be found by the formula:

The two-dimensional distribution density is non-negative and the double integral with infinite limits of the two-dimensional density is equal to one.

From the known joint distribution density, one can find the distribution density of each of the components of a two-dimensional random variable.

; ;

Conditional laws of distribution.

As shown above, knowing the joint distribution law, one can easily find the distribution laws for each random variable included in the system.

However, in practice, the inverse problem is more often - according to the known laws of distribution of random variables, find their joint distribution law.

In the general case, this problem is unsolvable, because the distribution law of a random variable says nothing about the relationship of this variable with other random variables.

In addition, if random variables are dependent on each other, then the distribution law cannot be expressed in terms of the distribution laws of the components, since should establish a connection between the components.

All this leads to the need to consider conditional distribution laws.

Definition. The distribution of one random variable included in the system, found under the condition that another random variable has taken a certain value, is called conditional distribution law.

The conditional distribution law can be specified both by the distribution function and by the distribution density.

The conditional distribution density is calculated by the formulas:

The conditional distribution density has all the properties of the distribution density of one random variable.

Conditional mathematical expectation.

Definition. conditional mathematical expectation discrete random variable Y at X = x (x is a certain possible value of X) is called the product of all possible values Y on their conditional probabilities.

For continuous random variables:

,

Where f( y/ x) is the conditional density of the random variable Y when X = x .

Conditional expectationM( Y/ x)= f( x) is a function of X and called regression function X on Y.

Example.Find the conditional expectation of the component Y at

X=x1 =1 for a discrete two-dimensional random variable given by the table:

Y
	x1=1	x2=3	x3=4	x4=8
y1=3	0,15	0,06	0,25	0,04
y2=6	0,30	0,10	0,03	0,07

The conditional variance and conditional moments of the system of random variables are defined similarly.

Dependent and independent random variables.

Definition. Random variables are called independent, if the distribution law of one of them does not depend on what value the other random variable takes.

The concept of dependence of random variables is very important in probability theory.

Conditional distributions of independent random variables are equal to their unconditional distributions.

Let us define the necessary and sufficient conditions for the independence of random variables.

Theorem. Y are independent, it is necessary and sufficient that the distribution function of the system ( X, Y) was equal to the product of the distribution functions of the components.

A similar theorem can be formulated for the distribution density:

Theorem. In order for the random variables X and Y are independent, it is necessary and sufficient that the joint distribution density of the system ( X, Y) was equal to the product of the distribution densities of the components.

The following formulas are practically used:

For discrete random variables:

For continuous random variables:

The correlation moment serves to characterize the relationship between random variables. If the random variables are independent, then their correlation moment is zero.

The correlation moment has a dimension equal to the product of the dimensions of the random variables X and Y . This fact is a disadvantage of this numerical characteristic, since with different units of measurement, different correlation moments are obtained, which makes it difficult to compare the correlation moments of different random variables.

In order to eliminate this shortcoming, another characteristic is applied - the correlation coefficient.

Definition. Correlation coefficient rxy random variables X and Y is the ratio of the correlation moment to the product of the standard deviations of these quantities.

The correlation coefficient is a dimensionless quantity. For independent random variables, the correlation coefficient is zero.

Property: The absolute value of the correlation moment of two random variables X and Y does not exceed the geometric mean of their dispersions.

Property: The absolute value of the correlation coefficient does not exceed unity.

Random variables are called correlated if their correlation moment is nonzero, and uncorrelated if their correlation moment is zero.

If random variables are independent, then they are uncorrelated, but from uncorrelation one cannot conclude that they are independent.

If two quantities are dependent, then they can be either correlated or uncorrelated.

Often, according to a given distribution density of a system of random variables, one can determine the dependence or independence of these variables.

Along with the correlation coefficient, the degree of dependence of random variables can also be characterized by another quantity, which is called coefficient of covariance. The coefficient of covariance is determined by the formula:

Example. The distribution density of the system of random variables X andindependent. Of course, they will also be uncorrelated.

Linear regression.

Consider a two-dimensional random variable ( X , Y ), where X and Y are dependent random variables.

Let us represent approximately one random variable as a function of another. An exact match is not possible. We assume that this function is linear.

To determine this function, it remains only to find the constant values a And b.

Definition. Functiong( X) called best approximation random variable Y in the sense of the least squares method, if the mathematical expectation

Takes on the smallest possible value. Also functiong( x) called mean square regression Y to X .

Theorem. Linear mean square regression Y on X is calculated by the formula:

in this formula mx= M( X random variable Yrelative to random variable X. This value characterizes the magnitude of the error resulting from the replacement of a random variableYlinear functiong( X) = aX +b.

It is seen that if r= ± 1, then the residual variance is zero, and hence the error is zero and the random variableYis exactly represented by a linear function of the random variable X.

Direct Root Mean Square Regression X onYis determined similarly by the formula: X and Yhave linear regression functions in relation to each other, then we say that the quantities X AndYconnected linear correlation dependence.

Theorem. If a two-dimensional random variable ( X, Y) is normally distributed, then X and Y are connected by a linear correlation dependence.

E.G. Nikiforova

Law of large numbers in probability theory states that the empirical mean (arithmetic mean) of a sufficiently large finite sample from a fixed distribution is close to the theoretical mean (expectation) of this distribution. Depending on the type of convergence, the weak law of large numbers is distinguished, when convergence in probability takes place, and the strong law of large numbers, when convergence almost everywhere takes place.

There is always a finite number of trials for which, with any given probability, less than 1 the relative frequency of occurrence of some event will differ arbitrarily little from its probability.

The general meaning of the law of large numbers: the joint action of a large number of identical and independent random factors leads to a result that, in the limit, does not depend on chance.

Methods for estimating probability based on the analysis of a finite sample are based on this property. good example is a prediction of election results based on a poll of a sample of voters.

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Subtitles

Let's take a look at the law of large numbers, which is perhaps the most intuitive law in mathematics and probability theory. And because it applies to so many things, it is sometimes used and misunderstood. Let me first give it a definition for accuracy, and then we'll talk about intuition. Let's take a random variable, say X. Let's say we know its mathematical expectation or population mean. The law of large numbers simply says that if we take the example of n-th number of observations of a random variable and average the number of all those observations... Let's take a variable. Let's call it X with a subscript n and a dash at the top. This is the arithmetic mean of the nth number of observations of our random variable. Here is my first observation. I do the experiment once and I make this observation, then I do it again and I make this observation, I do it again and I get this. I run this experiment n times and then divide by the number of my observations. Here is my sample mean. Here is the average of all the observations I made. The law of large numbers tells us that my sample mean will approach the mean of the random variable. Or I can also write that my sample mean will approach the population mean for the nth number going to infinity. I won't make a clear distinction between "approximation" and "convergence", but I hope you intuitively understand that if I take a fairly large sample here, then I get the expected value for the population as a whole. I think most of you intuitively understand that if I do enough tests with a large sample of examples, eventually the tests will give me the values ​​I expect, taking into account the mathematical expectation, probability and all that. But I think it's often unclear why this happens. And before I start explaining why this is so, let me give you a concrete example. The law of large numbers tells us that... Let's say we have a random variable X. It is equal to the number of heads in 100 tosses of the correct coin. First of all, we know the mathematical expectation of this random variable. This is the number of coin tosses or trials multiplied by the odds of any trial succeeding. So it's equal to 50. That is, the law of large numbers says that if we take a sample, or if I average these trials, I get. .. The first time I do a test, I toss a coin 100 times, or take a box with a hundred coins, shake it, and then count how many heads I get, and get, say, the number 55. This will be X1. Then I shake the box again and I get the number 65. Then again - and I get 45. And I do this n times, and then I divide it by the number of trials. The law of large numbers tells us that this average (the average of all my observations) will tend to 50 while n will tend to infinity. Now I would like to talk a little about why this happens. Many believe that if, after 100 trials, my result is above average, then according to the laws of probability, I should have more or less heads in order to, so to speak, compensate for the difference. This is not exactly what will happen. This is often referred to as the "gambler's fallacy". Let me show you the difference. I will use the following example. Let me draw a graph. Let's change the color. This is n, my x-axis is n. This is the number of tests I will run. And my y-axis will be the sample mean. We know that the mean of this arbitrary variable is 50. Let me draw this. This is 50. Let's go back to our example. If n is... During my first test, I got 55, which is my average. I have only one data entry point. Then after two trials, I get 65. So my average would be 65+55 divided by 2. That's 60. And my average went up a bit. Then I got 45, which lowered my arithmetic mean again. I won't plot 45 on the chart. Now I need to average it all out. What is 45+65 equal to? Let me calculate this value to represent the point. That's 165 divided by 3. That's 53. No, 55. So the average goes down to 55 again. We can continue these tests. After we have done three trials and come up with this average, many people think that the gods of probability will make it so that we get fewer heads in the future, that the next few trials will be lower in order to reduce the average. But it is not always the case. In the future, the probability always remains the same. The probability that I will roll heads will always be 50%. Not that I initially get a certain number of heads, more than I expect, and then suddenly tails should fall out. This is the "player's fallacy". If you get a disproportionate number of heads, it does not mean that at some point you will start to fall a disproportionate number of tails. This is not entirely true. The law of large numbers tells us that it doesn't matter. Let's say, after a certain finite number of trials, your average... The probability of this is quite small, but, nevertheless... Let's say your average reaches this mark - 70. You're thinking, "Wow, we've gone way beyond expectation." But the law of large numbers says it doesn't care how many tests we run. We still have an infinite number of trials ahead of us. The mathematical expectation of this infinite number of trials, especially in a situation like this, will be as follows. When you come up with a finite number that expresses some great value, an infinite number that converges with it will again lead to the expected value. This is, of course, a very loose interpretation, but this is what the law of large numbers tells us. It is important. He doesn't tell us that if we get a lot of heads, then somehow the odds of getting tails will increase to compensate. This law tells us that it doesn't matter what the result is with a finite number of trials as long as you still have an infinite number of trials ahead of you. And if you make enough of them, you'll be back to expectation again. This important point. Think about it. But this is not used daily in practice with lotteries and casinos, although it is known that if you do enough tests... We can even calculate it... what is the probability that we will seriously deviate from the norm? But casinos and lotteries work every day on the principle that if you take enough people, of course, for short term, with a small sample, then a few people will hit the jackpot. But over the long term, the casino will always benefit from the parameters of the games they invite you to play. This is an important probability principle that is intuitive. Although sometimes, when it is formally explained to you with random variables, it all looks a little confusing. All this law says is that the more samples there are, the more the arithmetic mean of those samples will converge towards the true mean. And to be more specific, the arithmetic mean of your sample will converge with the mathematical expectation of a random variable. That's all. See you in the next video!

Weak law of large numbers

The weak law of large numbers is also called Bernoulli's theorem, after Jacob Bernoulli, who proved it in 1713.

Let there be an infinite sequence (consecutive enumeration) of identically distributed and uncorrelated random variables . That is, their covariance c o v (X i , X j) = 0 , ∀ i ≠ j (\displaystyle \mathrm (cov) (X_(i),X_(j))=0,\;\forall i\not =j). Let . Denote by the sample mean of the first n (\displaystyle n) members:

.

Then X ¯ n → P μ (\displaystyle (\bar (X))_(n)\to ^(\!\!\!\!\!\!\mathbb (P) )\mu ).

That is, for every positive ε (\displaystyle \varepsilon )

lim n → ∞ Pr (| X ¯ n − μ |< ε) = 1. {\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\bar {X}}_{n}-\mu |<\varepsilon \,\right)=1.}

Strong law of large numbers

Let there be an infinite sequence of independent identically distributed random variables ( X i ) i = 1 ∞ (\displaystyle \(X_(i)\)_(i=1)^(\infty )) defined on one probability space (Ω , F , P) (\displaystyle (\Omega ,(\mathcal (F)),\mathbb (P))). Let E X i = μ , ∀ i ∈ N (\displaystyle \mathbb (E) X_(i)=\mu ,\;\forall i\in \mathbb (N) ). Denote by X¯n (\displaystyle (\bar(X))_(n)) sample mean of the first n (\displaystyle n) members:

X ¯ n = 1 n ∑ i = 1 n X i , n ∈ N (\displaystyle (\bar (X))_(n)=(\frac (1)(n))\sum \limits _(i= 1)^(n)X_(i),\;n\in \mathbb (N) ).

Then X ¯ n → μ (\displaystyle (\bar (X))_(n)\to \mu ) almost always.

Pr (lim n → ∞ X ¯ n = μ) = 1. (\displaystyle \Pr \!\left(\lim _(n\to \infty )(\bar (X))_(n)=\mu \ right)=1.) .

Like any mathematical law, the law of large numbers can only be applied to the real world under known assumptions, which can only be met with some degree of accuracy. So, for example, the conditions of successive tests often cannot be maintained indefinitely and with absolute accuracy. In addition, the law of large numbers only speaks of improbability significant deviation of the mean value from the mathematical expectation.