The concept and essence of the law of large numbers. The Law of Large Numbers and its Importance in Statistics
The essence of the first element of statistical methodology is the collection of primary data about the object being studied. For example: during the census of a country, data is collected on every person living on its territory, which is entered into a special form.
The second element: summary and grouping is the division of the totality of data obtained during the observation stage into homogeneous groups by one or more signs. For example, as a result of grouping census materials, the population is divided into groups (by gender, age, population, education, etc.).
The essence of the third element of statistical methodology lies in the calculation and socio-economic interpretation of general statistical indicators:
1. Absolute
2. Relative
3. Medium
4. Variation indicators
5. Speakers
The three basic elements of statistical methodology also constitute the three stages of any statistical study.
3. The law of large numbers and statistical regularity.
The law of large numbers is important for 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 the combined actions 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 special properties mass phenomena. Mass phenomena, 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 the mass of phenomena as a whole. Under certain conditions, the value of a characteristic for an individual unit can be considered as random variable, taking into account that it obeys not only a general pattern, but is also formed under the influence of conditions independent of this pattern. It is for this reason that statistics widely use average indicators, which characterize the entire population with one number. Only with a large number of observations are random deviations from the main direction of development balanced, cancelled, and the statistical pattern appears more clearly. Thus, the essence of the law of large numbers is that in numbers that generalize 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 unified science, the following branches emerged and received a certain independence:
1. General theory of statistics, which develops the concept of categories and methods for measuring 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, agriculture, transport, communications, etc.).
All branches of statistics, by 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 population is a set of elements of the same type that are similar to each other in some ways and different in others. For example: this is a set of economic sectors, a set of universities, a set of cooperation between design bureaus, etc.
The individual elements of a statistical population are called its units. In the examples discussed above, the units of the population are, respectively, industries, a university (one) and an employee.
Units of a population usually have many characteristics.
A characteristic is a property of units of a population that expresses their essence and has the ability to vary, i.e. change. Characteristics that take on a single value in individual units of the population are called varying, and the values themselves are called variants.
Variable characteristics are divided into attributive or qualitative. A feature is called attributive or qualitative if its individual meaning (variants) are expressed in the form of a state or properties inherent in the phenomenon. Variants of attributive characteristics are expressed in verbal form. Examples of such signs include economic.
A characteristic is called quantitative if its individual value is expressed in the form of numbers. For example: salary, stipend, age, size of the pension fund.
According to the nature of variation, quantitative characteristics are divided into discrete and continuous.
Discrete are such quantitative characteristics that can only take on a very specific, usually integer, value.
Continuous are those characteristics that, within certain limits, can take on both whole and fractional values. For example: country's GNP, etc.
There are also differences between primary and secondary signs.
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 internal 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 displays the size and quantitative relationships of 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. Contents, 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 subject 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. For a comprehensive comprehensive characterization of the phenomena under study, a scientifically based system of statistical indicators is required. This 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 development of a market economy.
The main objectives of statistics in the context of the development of market relations in Russia are the following:
1. Improving accounting and reporting and reducing document flow on this basis.
Photo (c) LF Academy
Representatives of the Central Bank, the Ministry of Finance, Rosfinmontoring, the Ministry of Justice, as well as lawyers and scientists gathered on Thursday at the “Fintech and Law: Focusing” conference to discuss the regulation of new financial technologies and civil legal problems arising in connection with smart contracts, cryptocurrencies and blockchain.
The participants discussed the current state of regulation of these innovations in the financial sector in Russia and abroad, argued about the terms proposed in the bills (three relevant bills are currently being considered by the State Duma), and also raised the question of whether it is necessary to regulate cryptocurrencies and blockchain in general, since apologists for these technologies are of the opinion that these technologies themselves, without external control, ensure the trust of counterparties in each other.
The question has also been raised more than once whether the regulation of cryptocurrencies should be brought under already existing norms - for example, those that operate in the securities market (in the USA, this is what they did). The participants did not reach a consensus; the discussion will continue.
“The issue is not just at the stage of being worked out, the question is at the stage of being posed, first of all, from the point of view of law. A huge field for work, in fact, only a few bushes rise in this field,” said the moderator, State Secretary - Deputy Director of the Federal Service, summing up the main panel discussion Russian Federation on financial monitoring Pavel Livadny.
Bills on cryptocurrencies
The following bills are currently being considered by the State Duma, none of them have yet passed a single reading.
According to the participants of the event, the controversy around these documents still does not subside (they were even called bills of “sleeping measures” - meaning the abundant references from these bills to other laws and regulations), it is possible that all three will be combined.
Positions of the Central Bank and the Ministry of Finance as the main regulators
In October last year, President Vladimir Putin, the government and the Central Bank determined the status of cryptocurrencies and regulated ICOs. According to the president, the use of cryptocurrencies carries serious risks, but he drew attention to the need to take advantage of the advantages that new technological solutions provide in the banking sector.
Let us recall that the Central Bank and the Ministry of Finance have disagreements on “On Digital Financial Assets” regarding the envisaged possibility of exchanging cryptocurrencies for rubles, foreign currency and/or other property. According to the Bank of Russia, such transactions should be allowed only in relation to tokens issued for the purpose of attracting financing (here the term “token” refers to crypto-protected digital obligations of the organization initiating the issuance of cryptocurrency, existing only in digital form - ed.).
On Thursday, the director of the legal department of the Central Bank presented their positions at the conference Alexey Guznov and Director of the Financial Policy Department of the Ministry of Finance Yana Pureskina.
According to the representative of the Central Bank, as the body that develops monetary policy, it is premature to introduce into the legal field the concepts of digital law, digital assets, and especially cryptocurrency as independent objects of civil law.
Guznov became interested in the recent history of cryptocurrencies - where they came from, “how they penetrated into our world.” One of the points of view is that cryptocurrencies originate from gamers who used cryptocurrencies to purchase gaming artifacts. Another one, not contradicting or alternative to the first: the philosophy of cryptocurrency was born among cryptopunks and inherits the philosophy of anarchism. The number of options is not limited to this.
“Cryptocurrency is not a currency, it is something that tries to call itself a currency,” Guznov said.
“We treat digital currencies as a legalized means of payment very carefully, but legally this is generally impossible,” he further noted and suggested that if the concept of digital currencies is introduced into the legal field, then at the level of “free will” transactions that do not require state support. In this case, there is really no need to regulate the circulation of cryptocurrencies.
Speaking about the positions of the Central Banks of other countries, he noted that digital currency is either prohibited or is treated with a certain degree of concern.
Guznov noted that banks feel the influence of fintech primarily in the fact that more and more clients do not visit the offices of credit institutions. But the opinion of a number of fintech representatives (expressed two or three years ago) that soon there will be no banks, but only fintech, is not shared by the representative of the Central Bank. “Now it turns out that banks are largely stimulating the development of fintech and involving it in standard banking.”
He called the acceptance of bank clients at the end of last year a big step. “Important tasks have been solved there that will make it possible, while preserving personal data, to provide access to financial technologies according to the formula 24/7/365.”
The representative of the Central Bank did not agree that an “uncontrollable wave of cryptocurrency transactions” could arise in the country. To regulate “data entities,” in his opinion, one can consider negotiability—whether it is free or limited. Now, in his opinion, the state has no other points where it can influence what is happening, except the point of transition - from one world [currency] to another [cryptocurrency, and vice versa].
The representative of the Ministry of Finance spoke briefly, since the plenary meeting was significantly behind schedule.
Yana Pureskina believes it is right to follow the path of regulation; she once again recalled the three legislative initiatives being considered by the State Duma. The Ministry of Finance believes that it is necessary to tune in to already existing legal structures, based on the assumption that cryptocurrency is a temporary phenomenon (and in this the position of the ministry is close to the position of the Ministry of Justice), new subjects [of regulation] will appear on the basis of new financial technologies, so determine the rules for every such case is impractical.
In particular, the dispute over whether cryptocurrency is an object of civil rights (that is, whether it can be recovered by creditors or included in the inheritance base) can be resolved by existing legislation. It states that to objects civil rights includes things, including cash and documentary securities, other property, including non-cash funds, uncertificated securities, property rights; results of work and provision of services; protected results intellectual activity and means of individualization equivalent to them (intellectual property); intangible benefits. Cryptocurrencies can easily be classified as “other property”.
The main idea of the new regulation is to provide protection for parties involved in cryptocurrency transactions: “The phenomenon is happening, it is growing in volume, and in the Digital Financial Assets Bill we are addressing this basic objective [of protection].” The goal is to ensure that if controversial situations arise, the parties to the conflict - ICO participants - will be able to apply to the courts for legal protection.
“It is necessary to find a balance between the needs to provide the economy with new ways to attract investment, and there are such needs in the economy now, to facilitate the attraction of investments for small and medium-sized businesses, which are now less accessible to bank lending, and for which it is difficult to access the exchange infrastructure” , said Pureskina. According to her, the issue of taxation of mining and the transfer of cryptocurrencies to fiat money still remains open.
Plenary session(from left to right): Alexey Guznov, Pavel Livadny, German Klimenko, Nikolay Chernogor, Yana Pureskina. Photo (c) Tatyana Kostyleva Dissenting opinion
We also present the most interesting opinions of other participants in the discussion.
Pavel Livadny(Rosfinmonitoring): “Blockchain evangelists say that everyone sits and sees everything. Let’s assume that I didn’t sell my apartment, but my blockchain showed that I sold it. An hour later I went to the computer and saw this, and another 10-15 transactions were made with the apartment. How can I prove that I didn't do this? Especially considering that blockchain apologists do not want government regulation. Blockchain is a false idea.”
MICEX representative reported that the exchange is not yet ready to organize a cryptocurrency section.
Director of the Department of Information and Communication Technologies and Analytical Implementation of External State Audit (Control) of the Accounts Chamber Apparatus Alexey Sklyar: “In the public sector, blockchain technology can be used in very limited areas where there can be complete openness between government agencies - property accounting, for the generation of budget reporting.”
Deputy Director of the Institute of Legislation and Comparative Law under the Government of the Russian Federation Nikolay Chernogor: “The emergence of fintech is a manifestation of the desire to escape from strict government regulation. Now the law seeks to invade every nook and cranny of social interaction.”
Associate Professor, Department of Theory and History of Law, Faculty of Law High school economics, legal advisor at IBM Alexander Savelyev, on the definition of digital law proposed in the bill. “The hallmark of [digital rights] is the ability to familiarize yourself with the description of an object at any time. Let's remember that now, and many resources are lying around, so at any time you may not be able to familiarize yourself with them. It makes sense to clarify a number of points [in the bill]. It turns out that if at least one requirement is not met, there is no judicial protection.”
Second section - lawyers argue about the problems of terminology, the implementation of rights and the fulfillment of duties of citizens Don't lose it. Subscribe and receive a link to the article in your email.
Interacting daily with figures and figures in work or study, 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 mathematical expectation this distribution.
You've probably noticed that understanding the essence of this law is not easy, especially for those who are not particularly good at mathematics. Based on this, we would like to talk about it in simple 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 have a positive impact on.
Concepts of the law of large numbers and its interpretation
In addition to the definition of the law of large numbers in probability theory discussed above, we can also give its economic interpretation. In this case, it represents the principle that the frequency of financial losses of a particular type can be predicted from high degree reliability when it is observed high level losses of these types in general.
In addition, depending on the level of convergence of signs, weak and strong laws of large numbers can be distinguished. 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, we should say this: it is always possible to find a finite number of trials where, with any probability programmed in advance less than one the relative frequency of occurrence of an 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 a result that does not depend on chance. And to put it in even simpler terms, then in the law of large numbers, the quantitative patterns of mass phenomena will clearly manifest themselves only when their number is large (that is why the law is called the law of large numbers).
From this we can conclude that the essence of the law is that in the numbers that are obtained through mass observation, there are some correctnesses that cannot be detected 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 accidental and necessary. When random deviations “cancel out” each other, the average indicators determined for the same structure take on the form of typical ones. They reflect the actions of essential and permanent facts in specific conditions of time and place.
Patterns defined by the law of large numbers are strong only when they represent mass trends, and they cannot be laws for individual cases. Thus, the principle of mathematical statistics comes into force, saying 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 with the same probability. This means that if, for example, you flip a coin 10 times, 5 of them should come up heads and 5 of them 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, 9 to 1, 2 to 8, etc. However, as the number of coin tosses increases, for example to 100, the probability of getting heads or tails reaches 50%. If, theoretically, an infinite number of similar experiments are carried out, the probability of a coin falling out on both sides will always tend to 50%.
A huge number of random factors influence exactly how the coin will fall. This is the position of the coin in the palm of your hand, the force with which the throw is made, the height of the fall, its speed, etc. But if there are a lot of experiments, regardless of how the factors influence, it can always be argued that the practical probability is close to the theoretical probability.
Here’s another example that will help you 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 receives 500 thousand rubles, the arithmetic average 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 receives 500 thousand rubles, then when calculating the arithmetic average we get 24.8 thousand rubles, which is 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 true results by studying a large number of observations. That is why it is convenient to use this law, again, in statistics or social economics.
Let's sum it up
The importance of the fact that the law of large numbers works is difficult to overestimate for any field scientific knowledge, and especially for scientific developments in the field of statistics theory and methods of statistical cognition. The effect of the law is also of great importance for the objects under study themselves with their mass patterns. 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 clearer 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 and. Also check out and. And, of course, pay attention to ours, because after completing it, you will not only master new thinking techniques, but also improve your cognitive abilities in general, including mathematical ones.
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 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 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 with the help of 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 set 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 dissimilar and, if combined different types phenomena. Frequency - repeatability of characteristics in the aggregate (in a distribution series).
Sign- 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 aspects. 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 decrees of government bodies, 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.” Questions that are difficult or undesirable to answer should not be included in the form. 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 source information, the predicate of statistical tables can be simple And complex. Indicators of the predicate in simple development 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 contour map(plan) of the 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. B line graphs 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:
Indicator
Pointer,
Inventory, etc.
This word, as a concept, is used in mathematics, economics and other sciences. In statistics, an index means relative indicator, which expresses the relationship between 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 total number of units 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 directed in one direction due to a deliberate violation of the 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, random 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. Traits that cause changes in other related traits 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 feedbacks.
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 the branch of activity that is engaged in the collection, processing, analysis, and publication of data on various phenomena public 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. Basis e.g., a set of enterprises, firms, families.
· Population unit – the primary element of a statistical population.
· Sign – quality. Feature of a unit of aggregation.
· Statistical indicator – the concept reflects quantities. characteristics (dimensions) of signs in general. phenomena.
· Statistical system Indicators are 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 LAW OF LARGE NUMBERS is the principle by which 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.…
THE LAW OF LARGE NUMBERS -- in probability theory, states that the empirical mean (arithmetic mean) of a finite sample from a fixed distribution is close to the theoretical mean (mathematical expectation) of that distribution.
STRONG LAW OF LARGE NUMBERS -- The Law of Large Numbers in probability theory states that the empirical mean (arithmetic mean) of a finite sample from a fixed distribution is close to the theoretical mean (mathematical expectation) of that distribution.
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 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 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.
The principle of mathematical statistics, according to which the joint action of a set of random factors can lead to a non-random (deterministic) result. The first example of the operation of this principle is the convergence of the frequency of occurrence of a random event with its probability as the number of trials increases.
The simplest example is the coin toss experiment. Theoretically, getting heads or tails is equally likely. This means that if you toss a coin 10 times, it should come up heads 5 times and tails 5 times. However, it is generally known that the likelihood of this is very low. With the same success, you can get 9 to 1, 3 to 5, etc. However, if you increase the number of trials to, say, 100, the probability of getting heads or tails approaches 50%. In the limit, if we direct the number of experiments to infinity, then the probability of heads and tails will asymptotically tend to 50%.
Which side the coin will fall depends on many random factors: how it will lie in the experimenter’s palm, the force of the throw, the height of the fall, speed, etc. However, with a sufficiently large number of experiments, regardless of the effect of these factors, we can always assert that the empirical (experimental) probability will be close to the theoretical one.
For example, suppose you need to estimate the income of the population in a certain region. If we look at 10 observations in which 9 respondents had incomes of about 20,000, and one had incomes of 500,000, then calculating a simple average will show an income of 68,000, which, generally speaking, does not reflect the real picture. If we consider 100 observations, of which 99 show an income of 20,000 and only one - 500,000, then the average will be about 28,000, which more adequately reflects the real situation. As the number of observations increases, the average will tend to its true value.
It is the law of large numbers when analyzing data that requires what is called “collecting statistics,” that is, using as many observations as possible to obtain reliable results.
