The essence of the law of large numbers.
Law of large numbers.
Topic 2.
Organization of state statistics in the Russian Federation.
Problems of statistics.
Statistics method.
Branches of statistics.
The general theory of statistics is related to other sciences.
| General theory of statistics | ||||||||||||
| 1. Demographic (social) statistics | 2. Economic statistics | 3. Education statistics | 4. Medical statistics | 5. Sports statistics | ||||||||
| 2.1 Labor statistics | 2.2 Salary statistics | 2.3 Statistics of mathematics and technology. supplies | 2.4 Transport statistics | 2.5 Communication statistics | 2.6 Financial credit statistics | |||||||
| 2.6.1 Higher financial computing | 2.6.2 Currency statistics | 2.6.3 Exchange rate statistics | Others | |||||||||
Statistics also develops the theory of observation.
The statistics method involves the following sequence of actions:
1. development of a statistical hypothesis,
2. statistical observation,
3. summary and grouping of statistical data,
4. data analysis,
5. interpretation of data.
The passage of each stage is associated with the use of special methods explained by the content of the work being performed.
1. Development of a system of hypotheses characterizing the development, dynamics, and state of socio-economic phenomena.
2. Organization of statistical activities.
3. Development of analysis methodology.
4. Development of a system of indicators for farm management at the macro and micro levels.
5. Make statistical observation data publicly available.
Principles:
1. centralized leadership,
2. unified organizational structure and methodology,
3. inextricable connection with government bodies.
The state statistics system has a hierarchical structure, consisting of federal, republican, regional, regional, district, city and district levels.
Goskomstat has departments, departments, and a computer center.
The massive nature of social laws and the uniqueness of their actions determine the extreme importance of studying aggregate data.
The law of large numbers is generated by the special properties of mass phenomena, which, on the one hand, differ from each other, and on the other, have something in common, due to their belonging to a certain class, type. Moreover, individual phenomena are more susceptible to the influence of random factors than their totality.
The law of large numbers is the definition of quantitative patterns of mass phenomena that appear only in a sufficiently large number of them.
However, its essence lies essentially in the fact that in the numbers obtained as a result of mass observation, certain correctness appears that is not found in a small number of facts.
The law of large numbers expresses the dialectic of the random and the extremely important. As a result of the mutual cancellation of random deviations, the average values calculated for values of the same type become typical, reflecting the effects of constant and significant facts in the conditions of place and time.
Tendencies and patterns revealed with the help of the law of large numbers are valid only as mass trends, but not as laws for each individual case.
The essence of the law of large numbers. - concept and types. Classification and features of the category "The essence of the law of large numbers." 2017, 2018.
Features of statistical methodology. Statistical population. Law of large numbers.
Law of Large Numbers
The massive nature of social laws and the uniqueness of their actions predetermine the need to study aggregate data.
The law of large numbers is generated by the special properties of mass phenomena. The latter, due to their individuality, on the one hand, differ from each other, and on the other, have something in common due to their belonging to a certain class or species. Moreover, individual 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 patterns 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 correctness appears 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 mutual cancellation of random deviations, average values calculated for quantities of the same type become typical, reflecting the effects of constant and significant facts in given conditions of place and time. Tendencies and patterns revealed with the help of the law of large numbers are valid only as mass trends, but not as laws for each individual case.
Statistics studies its subject using various methods:
· Mass observation method
· Method of statistical groupings
· Time series method
· Index analysis method
· Method of correlation-regression analysis of connections between indicators, etc.
Polit. arithmeticians studied general phenomena using numerical characteristics. Representatives of this school were Gratsite, who studied the patterns of mass phenomena, Petit, the creator of ecology. statistics, Galei - laid down the idea of the law of large numbers.
Statistical population- a multitude of single-quality, varying phenomena. The individual elements that make up the aggregate are the units of the aggregate. A statistical population is called homogeneous if the most essential features for each of its units of phenomena. basically identical and heterogeneous and, if different types of phenomena are combined. Frequency - repeatability of signs in the aggregate (in a distribution row).
Sign- a characteristic feature (property) or other feature of units of phenomena objects. Features are divided into: 1) quantitative (these features are expressed in numbers. They play a predominant role in statistics. These are features whose individual values differ in value); 2) qualitative ((attributive) are expressed in the form of concepts, definitions, expressing their essence, qualitative state); 3) alternative (qualitative features that can take only one of two opposite meanings). Features of individual units of the population take on separate meanings. Fluctuation of signs - variation.
Units of statistical population and variation of characteristics. Statistical indicators.
Phenomena and processes in the life of society are characterized by statistics using statistical indicators. A statistical indicator is a quantitative assessment of the properties of the phenomenon being studied. The statistical indicator reveals the unity of the qualitative and quantitative sides. If the qualitative side of a phenomenon is not determined, its quantitative side cannot be determined.
Statistics using stat. indicators characterizes: the size of the phenomena being studied; their peculiarity; patterns of development; their relationships.
Statistical indicators are divided into accounting, evaluation and analytical.
Accounting and evaluation indicators reflect the volume or level of the phenomenon being studied.
Analytical indicators are used to characterize the development features of a phenomenon, its prevalence in space, the relationship of its parts, and the relationship with other phenomena. The following analytical indicators are used: average values, indicators of structure, variations, dynamics, degree of crowding, etc. Variation- this is the diversity, variability of the value of a characteristic in individual units of the observation population.
Variation of the trait - gender - male, female.
Variation of salary - 10000, 100000, 1000000.
Individual characteristic values are called options this sign.
Each individual phenomenon subject to statistical study is called
Stages of statistical observation. Statistical observation. Goals and objectives of statistical observation. Basic concepts.
Statistical observation is the collection of necessary data on phenomena and processes of social life.
Any statistical study consists of the following stages:
· Statistical observation – collection of data about the phenomenon being studied.
· Summary and grouping – counting totals as a whole or by groups.
· Obtaining general indicators and their analysis (conclusions).
The task of statistical observation is to obtain reliable initial information and obtain it in the shortest possible time.
The tasks facing the manager determine the purpose of observation. It may stem from governmental regulations, regional administrations, and the company’s marketing strategy. The general purpose of statistical observation is to provide information support for management. It is specified depending on many conditions.
The object of observation is a set of units of the phenomena being studied about which data must be collected.
The unit of observation is the element of the object that has the characteristic being studied.
Signs may be:
- Quantitative
- Qualitative (attributive)
To register the collected data, it is used form- a specially prepared form, usually having a title, address and content parts. The title part contains the name of the survey, the organization conducting the survey, and by whom and when the form was approved. The address part contains the name, location of the research object and other details that allow it to be identified. Depending on the construction of the content part, two types of forms are distinguished:
§ Form card, which is compiled for each observation unit;
§ Form-list, which is compiled for a group of observation units.
Each form has its own advantages and disadvantages.
Blank card convenient for manual processing, but associated with additional costs in the design of the title and address books.
Blank list used for automatic processing and cost savings on the preparation of title and address parts.
To reduce costs for summarizing and entering data, it is advisable to use machines that read forms. The questions in the content part of the form must be formulated in such a way that they can be answered unambiguously, objectively. The best question is one that can be answered with “Yes” or “No.” You should not include in the form questions that are difficult or undesirable to answer. You cannot combine two different questions in one formulation. To assist respondents in correctly understanding the program and individual questions, instructions. They can be either on a form or in the form of a separate book.
To direct the respondent's answers in the right direction, statistical tips, that is, ready-made answer options. They are complete and incomplete. Incomplete ones give the respondent the opportunity to improvise.
Statistical tables. Subject and predicate of the table. Simple (list, territorial, chronological), group and combined tables. Simple and complex development of predicate statistical tables. Rules for constructing tables in statistics.
The results of the summary and grouping must be presented in such a way that they can be used.
There are 3 ways to present data:
1. data can be included in the text.
2. presentation in tables.
3. graphic method
A statistical table is a system of rows and columns in which statistical information about socio-economic phenomena is presented in a certain sequence.
A distinction is made between the subject and predicate of the table.
The subject is an object characterized by numbers, usually the subject is given on the left side of the table.
A predicate is a system of indicators by which an object is characterized.
The general heading should reflect the content of the entire table and should be located above the table in the center.
Rule for compiling tables.
1. If possible, the table should be small in size and easily visible
2. The general title of the table should briefly express the size of its main content. content (territory, date)
3. numbering of columns and lines (subject) that are filled with data
4. when filling out tables you need to use symbols
5. compliance with the rules of rounding numbers.
Statistical tables are divided into 3 types:
1. simple tables do not contain the units of the statistical population being studied that are subject to systematization, but contain listings of the units of the population being studied. Depending on the nature of the material presented, these tables can be list, territorial and chronological. Tables whose subject contains a list of territories (districts, regions, etc.) are called listed territorial.
2. group statistical tables provide more informative material for the analysis of the phenomena being studied due to the formation of their subject groups according to an essential feature or the identification of connections between a number of indicators.
3. when constructing combination tables, each subject group, formed according to one characteristic, is divided into subgroups according to the second characteristic, every second group is divided according to the third characteristic, i.e. In this case, factor characteristics are taken in a certain combination. The combination table establishes the mutual effect on the effective characteristics and the significant connection between the factor groupings.
Depending on the research task and the nature of the initial information, the predicate of statistical tables can be simple And complex. In simple development, the indicators of the predicate are arranged sequentially one after another. By distributing indicators in a group according to one or more characteristics in a certain combination, a complex predicate is obtained.
Statistical graphs. Elements of a statistical graph: graphic image, graph field, spatial reference points, scale reference points, graph explication. Types of graphs according to the form of the graphic image and the image of construction.
Statistical chart - is a drawing in which statistical data is depicted using conventional geometric figures (lines, dots or other symbolic signs).
Basic elements of a statistical graph:
1. The graph field is the place where it is executed.
2. Graphic image - these are symbolic signs with the help of which stats are depicted. data (points, lines, squares, circles, etc.)
3. Spatial landmarks determine the placement of graphic images on the graph field. They are specified by a coordinate grid or contour lines and divide the graph field into parts, corresponding to the values of the indicators being studied.
4. Statistic scale guidelines. graphics give graphic images quantitative significance, which is conveyed using a system of scales. The scale of a graph is a measure of the conversion of a numerical value into a graphic one. A scale scale is a line whose individual points are read as a specific number. The graph scale can be rectilinear and curvilinear, uniform and uneven.
5. Operation of the graph is an explanation of its content, includes the title of the graph, an explanation of the scale scales, and explanations of individual elements of the graphic image. The title of the graph briefly and clearly explains the main content of the data depicted.
The graph also contains text that makes it possible to read the graph. The digital designations of the scale are complemented by an indication of the units of measurement.
Classification of graphs:
By construction method:
1. The diagram represents a drawing in which the stat. information is depicted through geometric shapes or symbolic signs. In stat. apply the following. types of charts:
§ linear
§ columnar
§ strip charts
§ circular
§ radial
2. A cartogram is a schematic (contour) map, or a terrain plan, in which individual territories, depending on the value of the depicted indicator, are indicated using graphic symbols (shading, colors, dots). The cartogram is divided into:
§ Background
§ Spot
In background cartograms, territories with different values of the studied indicator have different shading.
Dot cartograms use points of the same size located within certain territorial units as a graphic symbol.
3. Map diagrams (statistical maps) are a combination of a contour map (plan) of an area with a diagram.
According to the form of the graphic images used:
1. In dot plots as graphs. images, a set of points is used.
2. In line graphs, the graph. the images are lines.
3. For planar graphs, graph. The images are geometric shapes: rectangles, squares, circles.
4. Figure graphs.
By the nature of the graphics problems being solved:
Distribution series; structures stat. aggregates; dynamics series; communication indicators; task completion indicators.
Variation of a trait. Absolute indicators of variation: range of variation, average linear deviation, dispersion, standard deviation. Relative measures of variation: coefficients of oscillation and variation.
Indicators of variation of averaged static characteristics: range of variation, average linear deviation, average quadratic deviation (dispersion), coefficient of variation. Calculation formulas and procedure for calculating variation indicators.
Application of variation indicators in the analysis of statistical data in the activities of enterprises and organizations, BR institutions, macroeconomic indicators.
The average indicator gives a generalizing, typical level of the attribute, but does not show the degree of its variability and variation.
Therefore, average indicators must be supplemented with indicators of variation. The reliability of averages depends on the size and distribution of inclinations.
It is important to know the main indicators of variation, to be able to calculate and use them correctly.
The main indicators of variation are: range of variation, average linear deviation, dispersion, standard deviation, coefficient of variation.
Formulas for variation indicators:
1. range of variation.
X μαχ - maximum value of the characteristic
X min - minimum value of the attribute.
The range of variation can only serve as an approximate measure of the variation of a trait, because it is calculated on the basis of its two extreme values, and the rest are not taken into account; in this case, the extreme values of a characteristic for a given population can be purely random.
2. average linear deviation.
Means that deviations are taken without taking into account their sign.
Average linear deviation is rarely used in economic statistical analysis.
3. Dispersion.
Index method for comparing complex sets and its elements: indexed value and co-measurer (weight). Statistical index. Classification of indices according to the object of study: indices of prices, physical volume, cost and labor productivity.
The word "index" has several meanings:
Index,
Pointer,
Inventory, etc.
This word, as a concept, is used in mathematics, economics and other sciences. In statistics, an index is understood as a relative indicator that expresses the ratio of the magnitudes of a phenomenon in time and space.
The following tasks are solved using indexes:
1. Measuring the dynamics of a socio-economic phenomenon over 2 or more periods of time.
2. Measuring the dynamics of the average economic indicator.
3. Measuring the ratio of indicators across different regions.
According to the object of study, indices are:
Labor productivity
Cost
Physical volume of products, etc.
P1 - unit price of goods in the current period
P0 - unit price of goods in the base period
2. the physical volume index shows how the volume of production has changed in the current period compared to the base
q1- quantity of goods sold or produced in the current period
q0-quantity of goods sold or produced in the base period
3. The cost index shows how the cost per unit of production has changed in the current period compared to the base period.
Z1 - unit cost of production in the current period
Z0 - unit cost of production in the base period
4. The labor productivity index shows how the labor productivity of one worker has changed in the current period compared to the base period
t0 - labor intensity of the total worker for the base period
t1 - labor intensity of one worker for the current period
By selection method
Repeated
Non-repetitive sampling type
At resampling the total number of units in the general population remains unchanged during the sampling process. The unit included in the sample after registration is again returned to the general population - “selection according to the returned ball scheme.” Resampling is rare in socioeconomic life. Usually the sample is organized according to a non-repetitive sampling scheme.
At non-repetitive sampling a population unit included in the sample is returned to the general population and does not participate in the sample in the future (selection according to the unreturned ball scheme). Thus, with non-repetitive sampling, the number of units in the general population is reduced during the research process.
3. according to the degree of coverage of population units:
Large samples
Small samples (small sample (n<20))
Small sample in statistics.
A small sample is understood as a non-continuous statistical survey in which the sample population is formed from a relatively small number of units in the general population. The volume of a small sample usually does not exceed 30 units and can reach 4-5 units.
In trade, a small sample is used when a large sample is either impossible or impractical (for example, if the research involves damage or destruction of the samples being examined).
The magnitude of the error of a small sample is determined by formulas different from the formulas of sample observation with a relatively large sample size (n>100). The average error of a small sample is calculated using the formula:
The marginal error of a small sample is determined by the formula:
T - confidence coefficient depending on the probability (P) with which the maximum error is determined
μ is the average sampling error.
In this case, the value of the confidence coefficient t depends not only on the given confidence probability, but also on the number of sampling units n.
Using a small sample in trade, a number of practical problems are solved, first of all, establishing the limit within which the general average of the characteristic being studied is located.
Selective observation. General and sample populations. Registration and representativeness errors. Sampling bias. Average and maximum sampling errors. Extension of the results of sample observation to the general population.
In any static research, two types of errors occur:
1. Registration errors can be random (unintentional) and systematic (tendentious) in nature. Random errors usually balance each other out, since they do not have a predominant tendency towards exaggerating or understating the value of the characteristic being studied. Systematic errors are directed in one direction due to deliberate violation of selection rules. They can be avoided with proper organization and monitoring.
2. Representativeness errors are inherent only in selective observation and arise due to the fact that the sample population does not completely reproduce the general population.
sample share
general variance
general standard deviation
sample variance
sample standard deviation
During selective observation, randomness in the selection of units must be ensured.
Sample proportion is the ratio of the number of units in the sample population to the number of units in the general population.
Sample proportion (or frequency) is the ratio of the number of units possessing the studied characteristic m to the total number of units in the sample population n.
To characterize the reliability of sample indicators, a distinction is made between the average and the maximum sampling error.
1. average sampling error during rotational sampling
For a share, the maximum error during rotational selection is equal to:
Percentage for non-repetitive selection:
The value of the Laplace integral is the probability (P) for different t are given in a special table:
at t=1 P=0.683
at t=2 P=0.954
at t=3 P=0.997
This means that with a probability of 0.683 it is possible to guarantee that the deviation of the general average from the sample average will not exceed a single average error
Cause-and-effect relationships between phenomena. Stages of studying cause-and-effect relationships: qualitative analysis, building a connection model, interpreting the results. Functional connection and stochastic dependence.
The study of objectively existing connections between phenomena is the most important task of the theory of statistics. In the process of statistical research of dependencies, cause-and-effect relationships between phenomena are revealed, which makes it possible to identify factors (signs)
having a major influence on the variation of the studied phenomena and processes. Cause-effect relationships are such a connection between phenomena and processes when a change in one of them - the cause - leads to a change in the other - the effect.
Signs according to their significance for studying the relationship are divided into two classes. Signs that cause changes in other signs associated with them are called factorial, or simply factors. Characteristics that change under the influence of factor characteristics are called
effective.
The concept of the relationship between various characteristics of the phenomena being studied. Signs-factors and effective signs. Types of relationships: functional and correlation. Correlation field. Direct and feedback. Linear and nonlinear connections.
Direct and backward connections.
Depending on the direction of action, functional and stochastic connections can be direct and reverse. With a direct connection, the direction of change in the resulting characteristic coincides with the direction of change in the factor characteristic, i.e. with an increase in the factor attribute, the effective attribute also increases, and, conversely, with a decrease in the factor attribute, the effective attribute also decreases. Otherwise, there are feedback connections between the quantities under consideration. For example, the higher the worker’s qualifications (grade), the higher the level of labor productivity - a direct relationship. And the higher the labor productivity, the lower the cost per unit of production - feedback.
Straight and curvilinear connections.
According to the analytical expression (form), connections can be rectilinear or curvilinear. In a linear relationship, with an increase in the value of a factor characteristic, there is a continuous increase (or decrease) in the values of the resulting characteristic. Mathematically, such a relationship is represented by a straight line equation, and graphically by a straight line. Hence its shorter name - linear connection.
With curvilinear relationships, with an increase in the value of a factor characteristic, the increase (or decrease) of the resulting characteristic occurs unevenly or the direction of its change is reversed. Geometrically, such connections are represented by curved lines (hyperbola, parabola, etc.).
Subject and tasks of statistics. Law of large numbers. Main categories of statistical methodology.
Currently, the term “statistics” is used in 3 meanings:
· By “statistics” we mean a branch of activity that is engaged in the collection, processing, analysis, and publication of data on various phenomena of social life.
· Statistics refers to digital material used to characterize general phenomena.
· Statistics is a branch of knowledge, an academic subject.
The subject of statistics is the quantitative side of mass general phenomena in inextricable connection with their qualitative side. Statistics studies its subject using definitions. categories:
· Statistical aggregate – a totality of social-ec. objects and phenomena generally. Life, united. Some quality. The basis, for example, is a set of enterprises, firms, families.
· Population unit – the primary element of a statistical population.
· Sign – quality. Features of a unit of aggregation.
· Statistical indicator – the concept reflects quantities. characteristics (dimensions) of signs in general. phenomena.
· Statistical system Indicators – a set of statistical data. indicators reflecting the relationships between creatures. between phenomena.
The main objectives of statistics are:
1. comprehensive study of deep transformations of ecology. and social processes based on scientific evidence. indicator systems.
2. generalization and forecasting of development trends, etc. sectors of the economy as a whole
3. timely provision. reliability of information state, household, eq. authorities and the general public
The theoretical basis of statistics is materialist dialectics, which requires consideration of social phenomena in interconnection and interdependence, in continuous development (in dynamics), in historical conditionality; it indicates the transition of quantitative changes to qualitative ones.
The specific techniques with which statistics studies its subject form statistical methodology. It includes methods:
statistical observation – collection of primary statistical material, registration of facts. This is the first stage of statistical research;
summary and grouping of observation results into certain aggregates. This is the second stage of statistical research;
methods for analyzing the obtained summary and grouped data using special techniques (the third stage of statistical research): using absolute, relative and average values, statistical coefficients, indicators of variation, index method, indicators of time series, correlation-regression method. At this stage, the relationships between phenomena are identified, patterns of their development are determined, and forecast estimates are given.
Statistical methods are used as a research tool in many other sciences: economic theory, mathematics, sociology, marketing, etc.
1.4. Objectives of statistics in a market economy.
The main tasks of statistics in modern conditions are:
development and improvement of statistical methodology, methods for calculating statistical indicators based on the needs of a market economy and the SNA introduced into statistical accounting, ensuring the comparability of statistical information in international comparisons;
study of ongoing economic and social processes based on a scientifically based system of indicators;
generalization and forecasting of trends in the development of modern society, including the economy, at the macro and micro levels;
providing information to legislative and executive structures, government bodies, economic bodies, and the public;
improvement of the practical statistical accounting system: reduction of reporting, its unification, transition from continuous reporting to non-continuous types of observation (one-time, sample surveys).
1.5. The essence of the law of large numbers.
The patterns studied by statistics—the forms of manifestation of a causal relationship—are expressed in the recurrence of events with a certain regularity with a fairly high degree of probability. In this case, the condition must be met that the factors giving rise to events change slightly or do not change at all. A statistical pattern is discovered based on the analysis of mass data and is subject to the law of large numbers.
The essence of the law of large numbers is that in summary statistical characteristics (the total number obtained as a result of mass observation), the effects of the elements of chance are extinguished, and certain correctness (trends) appear in them, which cannot be detected on a small number of facts.
The law of large numbers is generated by the connections of mass phenomena. It must be remembered that trends and patterns revealed with the help of the law of large numbers are valid only as mass trends, but not as laws for individual units, for individual cases.
You will study the following main issues of the topic:
The connection between statistics and the theory and practice of market economics
Objectives of statistics
Concepts and methods of statistics
Law of large numbers, statistical regularity
Lesson 1. Introduction
1. History of statistics
Statistics is an independent social science with its own subject and research method. It arose from the practical needs of social life. Already in the ancient world, there was a need to count the number of inhabitants of the state, take into account people suitable for military affairs, determine the number of livestock, the size of land and other property. Information of this kind was necessary for collecting taxes, waging wars, etc. Subsequently, as social life develops, the range of phenomena taken into account gradually expands.
The volume of information collected has especially increased with the development of capitalism and world economic relations. The needs of this period forced government authorities and capitalist enterprises to collect for practical needs extensive and varied information about labor markets and the sale of goods and raw materials.
In the middle of the 17th century, a scientific direction arose in England, called “political arithmetic.” This direction was started by William Petit (1623-1687) and John Graunt (1620-1674). “Political arithmetics,” based on the study of information about mass social phenomena, sought to discover the laws of social life and, thus, address 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 science” developed in Germany. The emergence of this science dates back to 1660.
The development of political arithmetic and government 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, condition, 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, trade, banking, financial, medical, etc. The general properties of statistical data, regardless of their nature and methods of their analysis are considered by mathematical statistics and the general theory of statistics.
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 being studied, i.e. studies quantity in inextricable 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 components of statistical methodology are:
mass surveillance
grouping, application of generalizing (summary) characteristics;
analysis and generalization of statistical facts and detection of patterns in the phenomena being studied.
Let's take a closer look at these elements.
To characterize any mass phenomenon quantitatively, it is first necessary collect information about its constituent elements. This is achieved through mass observation, carried out on the basis of rules and methods developed by statistical science.
The information collected during the statistical observation process is subsequently subjected to summary (primary scientific processing), during which characteristic parts (groups) are identified from the entire population of surveyed units. The identification 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 collected information. It is carried out on the basis of certain principles and rules.
In the process of processing statistical information, the set of surveyed units and its selected parts based on the application of the grouping method are characterized by a system of digital indicators: absolute and average values, relative values, dynamics indicators, etc.
3. Objectives of statistics
Complete and reliable statistical information is the necessary basis on which the process of economic management is based. Making management decisions at all levels, from national or regional to the level of an individual corporation or private firm, is impossible without official statistical support.
It is statistical data that makes it possible to determine the volume of gross domestic product and national income, identify the main trends in the development of economic sectors, estimate the level of inflation, analyze the state of financial and commodity markets, 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 inextricable connection with their qualitative side, the quantitative expression of the laws of social development in specific conditions of place and time.
To obtain statistical information, state and departmental statistics bodies, as well as commercial structures, conduct various types 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 general 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 elaboration of programmatic, methodological and organizational aspects of statistical observation, lack of logical and arithmetic control of the collected data, non-compliance with the principles of group formation can ultimately lead to completely erroneous conclusions.
The final, analytical stage of the study is no less complex, time-consuming and responsible. At this stage, average indicators and distribution indicators are calculated, the structure of the population is analyzed, and the dynamics and relationships between the phenomena and processes being studied are studied.
The techniques and methods of collecting, processing and analyzing data used at all stages of the study are the subject of study of the general theory of statistics, which is a basic branch of statistical science. The developed methodology is used in macroeconomic statistics, sectoral statistics (industry, agriculture, trade, etc.), population statistics, social statistics, and other statistical sectors. The great importance of statistics in society is explained by the fact that it represents 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 of systematically measuring and studying generalized phenomena using quantitative methods.
For every study of quantitative relationships there is an accounting. Various quantitative relationships between phenomena can be represented in the form of certain mathematical formulas, and this, in itself, will not be taken into 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 use is necessarily associated with the calculation of elements.
Accounting is a means of monitoring and summarizing 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.
Economic reform poses qualitatively new challenges for statistical science and practice. In accordance with the state program for Russia’s transition to an accounting and statistics system accepted in international practice, 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), widely used in world practice, meets the characteristics and requirements of market relations. Therefore, the transition to a market economy made it possible to introduce the SNA into statistical and accounting accounting, reflecting the functioning of sectors of the market economy.
This is necessary for a comprehensive analysis of the economy at the macro level and providing information to international economic organizations with which Russia cooperates.
Statistics play a large role in information and analytical support for the development of economic reform. The single goal of this process is to assess, analyze and forecast the state and development of the economy at the present stage.
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 (mathematical expectation) of this distribution. Depending on the type of convergence, a distinction is made between the weak law of large numbers, when convergence occurs in probability, and the strong law of large numbers, when convergence occurs almost everywhere.
There is always a finite number of trials in which, with any given advance probability, there is less 1 the relative frequency of occurrence of some event will differ as little as possible 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 finite sample analysis are based on this property. A clear example is the forecast of election results based on a survey of a sample of voters.
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Let's 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 define it for accuracy, and then we’ll talk about intuition. Let's take a random variable, for example X. Let's say we know its mathematical expectation or the average for the population. The Law of Large Numbers simply says that if we take an example of the nth number of observations of a random variable and take the average of all those observations... Let's take a variable. Let's call it X with a subscript n and a bar at the top. This is the arithmetic mean of the nth number of observations of our random variable. Here's my first observation. I do the experiment once and make this observation, then I do it again and make this observation, and I do it again and get this. I conduct this experiment the nth number of 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 expected value of the random variable. Or I can also write that my sample mean will approach the population mean for the nth quantity tending 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, I will 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 expected value and probability and all that jazz. But I think it is often unclear why this happens. And before I begin to explain why this is so, let me give a specific 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 a fair 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 success of any trial. So this is equal to 50. That is, the law of large numbers says that if we take a sample, or if I average these trials, I will get. .. The first time I do a test, I'll toss a coin 100 times, or I'll take a box with a hundred coins, shake it, and then count how many heads I get, and I'll get, say, the number 55. That would be X1. Then I shake the box again and get the number 65. Then again and I get 45. And I do this n number of times, and then divide it by the number of trials. The law of large numbers tells us that this average (the average of all my observations) will approach 50 as n approaches infinity. Now I would like to talk a little about why this happens. Many people believe that if after 100 trials my result is above average, then according to the laws of probability I should get more or fewer heads in order to, so to speak, compensate for the difference. That's not exactly what's going to happen. This is often called the "gambler's fallacy." Let me show you the difference. I'll 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 do. And my Y axis will be the sample mean. We know that the mathematical expectation of this arbitrary variable is 50. Let me draw this. This is 50. Let's return to our example. If n is... During my first test I got 55, that's my average. I only have one data entry point. Then after two tests I get 65. So my average would be 65+55 divided by 2. That's 60. And my average has gone up a little. Then I got 45, which again lowered my arithmetic average. I'm not going to plot 45. Now I need to average all of this. 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 back down to 55. We can continue these tests. After we've done three trials and gotten that average, many people think that the probability gods will make sure that we get fewer heads in the future, that the next few trials will have lower scores to lower the average. But it is not always the case. In the future, the probability always remains the same. There will always be a 50% chance that I will get heads. It’s not that I initially get a certain number of heads, more than I expect, and then suddenly I have to get tails. This is the gambler's fallacy. Just because you get a disproportionately large number of heads does not mean that at some point you will start to get a disproportionately large number of tails. This is not entirely true. The law of large numbers tells us that it doesn't matter. Let's say that after a certain finite number of tests, your average... The probability of this is quite small, but, nevertheless... Let's say your average has reached this mark - 70. You think, "Wow, we've moved away from the expected value." But the law of large numbers says it doesn't care how many tests we do. We still have an endless number of challenges ahead. The mathematical expectation of this infinite number of trials, especially in a situation like this, would be as follows. When you come to a finite number that expresses some large 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. It doesn't tell us that if we get a lot of heads, then somehow the probability of getting tails will increase to compensate. This law tells us that it doesn't matter what the outcome is on a finite number of trials as long as you still have an infinite number of trials left. And if you do enough of them, you'll end up back at the expected value again. This is an important point. Think about it. But this is not used every day 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, naturally, in a short time, with a small sample, then a few people will hit the jackpot. But over a long period of time, the casino will always win due to the parameters of the games they invite you to play. This is an important principle of probability 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 tend to the true mean. And to be more specific, the arithmetic mean of your sample will converge with the mathematical expectation of the 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 (sequential 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 . Let us denote by the sample average of the first n (\displaystyle n) members:
.
Then X ¯ n → P μ (\displaystyle (\bar (X))_(n)\to ^(\!\!\!\!\!\!\mathbb (P) )\mu ).
That is, for any 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.}Strengthened 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) ). Let us denote by X ¯ n (\displaystyle (\bar (X))_(n)) sample mean of 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 certain assumptions that can only be met with some degree of accuracy. For example, successive test conditions often cannot be maintained indefinitely and with absolute accuracy. In addition, the law of large numbers only speaks about improbability significant deviation of the average value from the mathematical expectation.
