You have to study the following main topics of the topic:
Connection of statistics with the theory and practice of market economy
Tasks of statistics
Concepts and methods of statistics
Law of large numbers, statistical regularity
Lesson 1. Introduction
1. The history of statistics
Statistics is an independent social science that has its own subject and method of research. It arose from the practical needs of social life. Already in ancient world there was a need to count the number of inhabitants of the state, to take into account people suitable for military affairs, to determine the number of livestock, the size of land and other property. Information of this kind was necessary for collecting taxes, waging wars, and so on. In the future, as social life develops, the range of phenomena taken into account gradually expands.
The volume of collected information has especially increased with the development of capitalism and world economic ties. The needs of this period compelled government bodies and capitalist enterprises to collect extensive and varied information on labor markets and the sale of goods and raw materials for practical purposes.
In the middle of the 17th century, a scientific direction arose in England, called "political arithmetic". This trend was initiated by William Petit (1623-1687) and John Graunt (1620-1674). "Political arithmetic", based on the study of information about mass social phenomena, sought to discover the patterns of social life and, thus, to point out the questions that arose in connection with the development of capitalism.
Along with the school of "political arithmetic" in England, a school of descriptive statistics or "state studies" developed in Germany. The emergence of this science dates back to 1660.
The development of political arithmetic and state science led to the emergence of the science of statistics.
The concept of "statistics" comes from the Latin word "status", which in translation means position, state, order of phenomena.
The term "statistics" was introduced into scientific circulation by Gottfried Achenwal (1719-1772), a professor at the University of Göttingen.
Depending on the object of study, statistics as a science is divided into social, demographic, economic, industrial, commercial, banking, financial, medical, etc. General properties of statistical data, regardless of their nature, and methods of their analysis are considered by mathematical statistics and the general theory of statistics.
The subject of statistics . Statistics deals primarily with the quantitative side of the phenomena and processes of social life. One of the characteristic features of statistics is that when studying the quantitative side of social phenomena and processes, it always reflects the qualitative features of the phenomena under study, i.e. studies quantity in inseparable connection, unity with quality.
Quality in the scientific and philosophical understanding is the properties inherent in an object or phenomenon that distinguish this object or phenomenon from others. Quality is what makes objects and phenomena certain. Using philosophical terminology, we can say that statistics studies social phenomena as the unity of their qualitative and quantitative certainty, i.e. studies the measure of social phenomena.
Statistical methodology . The most important constituent elements of the statistical methodology are:
mass surveillance
grouping, application of generalizing (summary) characteristics;
analysis and generalization of statistical facts and detection of regularities in the studied phenomena.
Let's take a closer look at these elements.
In order to characterize any mass phenomenon from a quantitative point of view, one must first collect information about its constituent elements. This is achieved with the help of mass observation, carried out on the basis of the rules and methods developed by statistical science.
The information collected in the process of statistical observation is subject to further summary (primary scientific processing), in the process of which characteristic parts (groups) are distinguished from the entire set of surveyed units. The selection of groups and subgroups of units from the entire surveyed mass is called in statistics grouping . Grouping in statistics is the basis for processing and analyzing the collected information. It is carried out on the basis of certain principles and rules.
In the process of processing statistical information, the totality of the surveyed units and its selected parts based on the use of the grouping method are characterized by a system of digital indicators: absolute and average values, relative values, dynamics indicators, etc.
3. Tasks of statistics
Complete and reliable statistical information is the necessary basis on which the process of economic management is based. Making managerial decisions at all levels, from a national or regional level to the level of an individual corporation or private firm, is impossible without official statistical support.
It is statistical data that make it possible to determine the volume of gross domestic product and national income, to identify the main trends in the development of economic sectors, to assess the level of inflation, to analyze the state of financial and commodity markets, to study the standard of living of the population and other socio-economic phenomena and processes.
Statistics is a science that studies the quantitative side of mass phenomena and processes in close connection with their qualitative side, a quantitative expression of the laws of social development in specific conditions of place and time.
To obtain statistical information, state and departmental statistics bodies, as well as commercial structures, conduct various kinds of statistical research. As already noted, the process of statistical research includes three main stages: data collection, their summary and grouping, analysis and calculation of generalizing indicators.
The results and quality of all subsequent work largely depend on how the primary statistical material is collected, how it is processed and grouped. Insufficient elaboration of the program-methodological and organizational aspects of statistical observation, the lack of logical and arithmetic control of the collected data, non-compliance with the principles of group formation can ultimately lead to absolutely erroneous conclusions.
No less complex, time-consuming and responsible is the final, analytical stage of the study. At this stage, average indicators and distribution indicators are calculated, the structure of the population is analyzed, the dynamics and relationships between the studied phenomena and processes are studied.
The techniques and methods of data collection, processing and analysis used at all stages of the study are the subject of study of the general theory of statistics, which is the basic branch of statistical science. The developed methodology is used in macroeconomic statistics, sectoral statistics (industry, agriculture, other trade), population statistics, social statistics, and other statistical fields. The great importance of statistics in society is explained by the fact that it is one of the most basic, one of the most important means by which an economic entity keeps records in the economy.
Accounting is a way to systematically measure and study generalized phenomena using quantitative methods.
For every study of quantitative relationships there is an account. Various quantitative relations between phenomena can be represented in the form of certain mathematical formulas, and this, in itself, will not yet be an account. One of the characteristic features of accounting is the calculation of INDIVIDUAL elements, INDIVIDUAL units that make up this or that phenomenon. Accounting uses various mathematical formulas, but their use is necessarily associated with counting elements.
Accounting is a means of control and generalization of the results obtained in the process of generalized development.
Thus, statistics is the most important tool for understanding and using economic and other laws of social development.
The economic reform poses qualitatively new tasks for statistical science and practice. In accordance with the state program for Russia's transition to the internationally accepted system of accounting and statistics, the system for collecting statistical information is being reorganized and the methodology for analyzing market processes and phenomena is being improved.
The System of National Accounts (SNA), which is widely used in world practice, corresponds to the peculiarities and requirements of market relations. Therefore, the transition to a market economy made it possible to introduce the SNA into statistical and accounting records, reflecting the functioning of the sectors of the market economy.
This is necessary for a comprehensive analysis of the economy at the macro level and for providing information to international economic organizations with which Russia cooperates.
Statistics play an important role in information and analytical support for the development of economic reform. The single purpose of this process is the assessment, analysis and forecasting of the state and development of the economy at the present stage.
The concept of the central limit theorem.
Inequality and Chebyshev's theorem.
The essence of the law of large numbers and its significance in statistics and economics.
Topic 8. The law of large numbers
The law of large numbers in probability theory is understood as a set of theorems in which a connection is established between the arithmetic mean of a sufficiently large number random variables and the arithmetic mean of their mathematical expectations.
AT Everyday life, business, scientific research, we are constantly faced with events and phenomena with an uncertain outcome. For example, a merchant does not know how many visitors will come to his store, a businessman does not know the dollar exchange rate in 1 day or a year; banker - will the loan be returned to him on time; insurance companies - when and to whom they will have to pay insurance premiums.
The development of any science involves the establishment of basic laws and cause-and-effect relationships in the form of definitions, rules, axioms, theorems.
Connecting link between probability theory and mathematical statistics are the so-called limit theorems, which include the law of large numbers. The law of large numbers defines the conditions under which the combined effect of many factors leads to a result that does not depend on chance. In its most general form, the law of large numbers was formulated by P.L. Chebyshev. A.N.Kolmogorov, A.Ya.Khinchin, B.V.Gnedenko, V.I.Glivenko made a great contribution to the study of the law of large numbers.
The limit theorems also include the so-called Central Limit Theorem of A. Lyapunov, which determines the conditions under which the sum of random variables will tend to a random variable with a normal distribution law. This theorem allows one to substantiate methods for testing statistical hypotheses, correlation-regression analysis, and other methods of mathematical statistics.
Further development of the central limit theorem is associated with the names of Lindenberg, S.N. Bernstein, A.Ya. Khinchin, P. Levy.
Practical use methods of probability theory and mathematical statistics is based on two principles, which are actually based on limit theorems:
the principle of the impossibility of the occurrence of an unlikely event;
the principle of sufficient confidence in the occurrence of an event, the probability of which is close to 1.
In the socio-economic sense, the law of large numbers is understood as general principle, due to which the quantitative regularities inherent in mass social phenomena are clearly manifested only in a sufficiently large number of observations. The law of large numbers is generated by the special properties of mass social phenomena. The latter, by virtue of their individuality, differ from each other, and also have something in common, due to their belonging to a certain species, class, to certain groups. Single phenomena are more affected by random and insignificant factors than the mass as a whole. In a large number of observations, random deviations from regularities cancel each other out. As a result of the mutual cancellation of random deviations, the averages calculated for quantities of the same type become typical, reflecting the action of constant and significant factors under given conditions of place and time. The trends and patterns revealed by the law of large numbers are massive statistical patterns.
The law of large numbers plays an important role in statistical methodology. In its most general form, it can be formulated as follows:
The law of large numbers is a general principle by virtue of which the cumulative action of a large number of random factors leads, under certain general conditions, to a result almost independent of chance.
The law of large numbers is generated by special properties of mass phenomena. The mass phenomena of the latter, in turn, on the one hand, due to their individuality, differ from each other, and on the other hand, they have something in common that determines their belonging to a certain class.
A single phenomenon is more susceptible to the influence of random and insignificant factors than a mass of phenomena as a whole. Under certain conditions, the value of a feature of an individual unit can be considered as a random variable, given that it obeys not only a general pattern, but is also formed under the influence of conditions that do not depend on this pattern. It is for this reason that statistics widely use averages, which characterize the entire population with one number. Only with a large number of observations, random deviations from the main direction of development are balanced, canceled out and the statistical regularity manifests itself more clearly. In this way, essence of the law of large numbers lies in the fact that in the numbers summarizing the result of mass statistical observation, the pattern of development of socio-economic phenomena is revealed more clearly than with a small statistical study.
LAW OF GREAT NUMBERS
Economy. Dictionary. - M.: "INFRA-M", Publishing house "Ves Mir". J. Black. General editorial staff: Doctor of Economics Osadchaya I.M. . 2000 .
Raizberg B.A., Lozovsky L.Sh., Starodubtseva E.B. . Modern economic dictionary. - 2nd ed., corrected. Moscow: INFRA-M. 479 p. . 1999
Economic dictionary. 2000 .
See what the "LAW OF GREAT NUMBERS" is in other dictionaries:
LAW OF GREAT NUMBERS- see LARGE NUMBERS LAW. Antinazi. Encyclopedia of Sociology, 2009 ... Encyclopedia of Sociology
Law of Large Numbers- the principle according to which the quantitative patterns inherent in mass social phenomena are most clearly manifested with a sufficiently large number of observations. Single phenomena are more susceptible to random and ... ... Glossary of business terms
LAW OF GREAT NUMBERS- claims that with a probability close to one, the arithmetic mean of a large number of random variables of approximately the same order will differ little from a constant equal to the arithmetic mean of the mathematical expectations of these variables. Diff. ... ... Geological Encyclopedia
law of large numbers- - [Ya.N. Luginsky, M.S. Fezi Zhilinskaya, Yu.S. Kabirov. English Russian Dictionary of Electrical Engineering and Power Industry, Moscow, 1999] Topics in electrical engineering, basic concepts EN law of averageslaw of large numbers ... Technical translator's guide
Law of Large Numbers- in probability theory asserts that the empirical mean (arithmetic mean) of a sufficiently large finite sample from a fixed distribution is close to the theoretical mean ( mathematical expectation) of this distribution. Depending ... Wikipedia
law of large numbers- didžiųjų skaičių dėsnis statusas T sritis fizika atitikmenys: engl. law of large numbers vok. Gesetz der großen Zahlen, n rus. law of large numbers, m pranc. loi des grands nombres, f … Fizikos terminų žodynas
LAW OF GREAT NUMBERS- a general principle, due to which the combined action of random factors leads, under certain very general conditions, to a result that is almost independent of chance. The convergence of the frequency of occurrence of a random event with its probability with an increase in the number ... ... Russian sociological encyclopedia
Law of Large Numbers- a law stating that the cumulative action of a large number of random factors leads, under some very general conditions, to a result that is almost independent of chance ... Sociology: a dictionary
LAW OF GREAT NUMBERS- statistical law expressing the relationship of statistical indicators (parameters) of the sample and the general population. The actual values of statistical indicators obtained from a certain sample always differ from the so-called. theoretical ... ... Sociology: Encyclopedia
LAW OF GREAT NUMBERS- the principle according to 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 ... Encyclopedic Dictionary of Economics and Law
Law of Large Numbers
Interacting daily in work or study with numbers and numbers, many of us do not even suspect that there is a very interesting law of large numbers, used, for example, in statistics, economics, and even psychological and pedagogical research. It refers to probability theory and says that the arithmetic mean of any large sample from a fixed distribution is close to the mathematical expectation of this distribution.
You probably noticed that it is not easy to understand the essence of this law, especially for those who are not particularly friendly with mathematics. Based on this, we would like to talk about it in simple language (as far as possible, of course), so that everyone can at least approximately understand for themselves what it is. This knowledge will help you better understand some mathematical patterns, become more erudite and positively influence the development of thinking.
Concepts of the law of large numbers and its interpretation
In addition to the above definition of the law of large numbers in probability theory, we can give its economic interpretation. In this case, it represents the principle that the frequency of a particular type of financial loss can be predicted with a high degree of certainty when there is a high level of losses of such types in general.
In addition, depending on the level of convergence of features, we can distinguish the weak and strengthened laws of large numbers. We are talking about weak when convergence exists in probability, and about strong when convergence exists in almost everything.
If we interpret it a little differently, then we should say this: one can always find such a finite number of trials where, with any pre-programmed probability less than one, the relative frequency of occurrence of some event will differ very little from its probability.
Thus, the general essence of the law of large numbers can be expressed as follows: the result of the complex action of a large number of identical and independent random factors will be such a result that does not depend on chance. And speaking in even simpler language, then in the law of large numbers, the quantitative laws of mass phenomena will clearly manifest themselves only when there are a large number of them (that is why the law of large numbers is called the law).
From this we can conclude that the essence of the law is that in the numbers that are obtained by mass observation, there are some correctness, which is impossible to detect in a small number of facts.
The essence of the law of large numbers and its examples
The law of large numbers expresses the most general patterns of the accidental and the necessary. When random deviations "extinguish" each other, the averages determined for the same structure take on the form of typical ones. They reflect the operation of essential and permanent facts under the specific conditions of time and place.
Regularities defined by the law of large numbers are strong only when they represent mass tendencies, and they cannot be laws for individual cases. Thus, the principle of mathematical statistics comes into force, which says that the complex action of a number of random factors can cause a non-random result. And the most striking example of the operation of this principle is the convergence of the frequency of occurrence of a random event and its probability when the number of trials increases.
Let's remember the usual coin toss. Theoretically, heads and tails can fall out with the same probability. This means that if, for example, a coin is tossed 10 times, 5 of them should come up heads and 5 should come up heads. But everyone knows that this almost never happens, because the ratio of the frequency of heads and tails can be 4 to 6, and 9 to 1, and 2 to 8, etc. However, with an increase in the number of coin tosses, for example, up to 100, the probability that heads or tails will fall out reaches 50%. If, theoretically, an infinite number of such experiments are carried out, the probability of a coin falling out on both sides will always tend to 50%.
How exactly the coin will fall is influenced by a huge number of random factors. This is the position of the coin in the palm of your hand, and the force with which the throw is made, and the height of the fall, and its speed, etc. But if there are many experiments, regardless of how the factors act, it can always be argued that the practical probability is close to the theoretical probability.
And here is another example that will help to understand the essence of the law of large numbers: suppose we need to estimate the level of earnings of people in a certain region. If we consider 10 observations, where 9 people receive 20 thousand rubles, and 1 person - 500 thousand rubles, the arithmetic mean will be 68 thousand rubles, which, of course, is unlikely. But if we take into account 100 observations, where 99 people receive 20 thousand rubles, and 1 person - 500 thousand rubles, then when calculating the arithmetic mean, we get 24.8 thousand rubles, which is already closer to the real state of affairs. By increasing the number of observations, we will force the average value to tend to the true value.
It is for this reason that in order to apply the law of large numbers, it is first necessary to collect statistical material in order to obtain truthful results by studying a large number of observations. That is why it is convenient to use this law, again, in statistics or social economics.
Summing up
The importance of the fact that the law of large numbers works is difficult to overestimate for any field of scientific knowledge, and especially for scientific developments in the field of the theory of statistics and methods of statistical knowledge. The action of the law is also of great importance for the objects under study themselves with their mass regularities. Almost all methods of statistical observation are based on the law of large numbers and the principle of mathematical statistics.
But, even without taking into account science and statistics as such, we can safely conclude that the law of large numbers is not just a phenomenon from the field of probability theory, but a phenomenon that we encounter almost every day in our lives.
We hope that now the essence of the law of large numbers has become more clear to you, and you can easily and simply explain it to someone else. And if the topic of mathematics and probability theory is interesting to you in principle, then we recommend reading about Fibonacci numbers and the Monty Hall paradox. Also get acquainted with approximate calculations in life situations and the most popular numbers. And, of course, pay attention to our cognitive science course, because after passing it, you will not only master new thinking techniques, but also improve your cognitive abilities in general, including mathematical ones.
1.1.4. Statistics Method
Statistics Method involves the following sequence of actions:
development of a statistical hypothesis,
summary and grouping of statistical data,
The passage of each stage is associated with the use of special methods, explained by the content of the work performed.
1.1.5. Tasks of statistics
Development of a system of hypotheses characterizing the development, dynamics, state of socio-economic phenomena.
Organization of statistical activity.
Development of analysis methodology.
Development of a system of indicators for managing the economy at the macro and micro levels.
To popularize the data of statistical observation.
1.1.6. The law of large numbers and its role in the study of statistical regularities
The mass nature of social laws and the originality of their actions predetermine the need for the study of aggregate data.
The law of large numbers is generated by special properties of mass phenomena. The latter, by virtue of their individuality, on the one hand, differ from each other, and on the other hand, they have something in common, due to their belonging to a certain class, species. Moreover, single phenomena are more susceptible to the influence of random factors than their totality.
The law of large numbers in its simplest form states that the quantitative regularities of mass phenomena are clearly manifested only in a sufficiently large number of them.
Thus, its essence lies in the fact that in the numbers obtained as a result of mass observation, certain regularities appear that cannot be detected in a small number of facts.
The law of large numbers expresses the dialectic of the accidental and the necessary. As a result of the mutual cancellation of random deviations, the average values calculated for a value of the same type become typical, reflecting the actions of constant and significant facts under given conditions of place and time.
The tendencies and regularities revealed by the law of large numbers are valid only as mass tendencies, but not as laws for each individual case.
The manifestation of the operation of the law of large numbers can be seen in many areas of the phenomena of social life studied by statistics. For example, the average output per worker, the average unit cost of a product, the average wage, and other statistical characteristics express patterns common to a given mass phenomenon. Thus, the law of large numbers contributes to the disclosure of the patterns of mass phenomena as an objective necessity for their development.
1.1.7. The main categories and concepts of statistics: statistical population, population unit, sign, variation, statistical indicator, system of indicators
Since statistics deals with mass phenomena, the main concept is the statistical totality.
Population - this is a set of objects or phenomena studied by statistics that have one or more common features and differ from each other in other ways. So, for example, when determining the volume of retail trade turnover, all trade enterprises selling goods to the population are considered as a single statistical aggregate - “retail trade”.
E population unit – this is the primary element of the statistical population, which is the carrier of the signs to be registered, and the basis of the account maintained during the survey.
For example, in a census of commercial equipment, the observation unit is the trade enterprise, and the population unit is their equipment (counters, refrigeration units, etc.).
sign — This is a characteristic property of the phenomenon under study, which distinguishes it from other phenomena. Signs can be characterized by a number of statistical values.
In different branches of statistics, different features are studied. So, for example, the object of study is an enterprise, and its features are the type of product, output volume, number of employees, etc. Or the object is a separate person, and the signs are gender, age, nationality, height, weight, etc.
Thus, statistical features, i.e. there are a lot of properties, qualities of objects of observation. All their diversity is usually divided into two large groups: signs of quality and signs of quantity.
Qualitative sign (attributive) - a sign, the individual meanings of which are expressed in the form of concepts, names.
Profession - turner, locksmith, technologist, teacher, doctor, etc.
Quantitative sign - a sign, certain values of which have quantitative expressions.
Height - 185, 172, 164, 158.
Weight - 105, 72, 54, 48.
Each object of study may have a number of statistical features, but from object to object, some features change, others remain unchanged. Changing features from one object to another are called variable. It is these features that are studied in statistics, since it is not interesting to study an unchanging feature. Suppose that there are only men in your group, everyone has one attribute (gender - male) and there is nothing more to say on this basis. And if there are women, then it is already possible to calculate their percentage in the group, the dynamics of changes in the number of women by months of the academic year, etc.
Variation sign - this is the diversity, the variability of the value of the attribute in individual units of the observation population.
Variation of the trait - gender - male, female.
Salary variation - 10000, 100000, 1000000.
The individual characteristic values are called options this sign.
Phenomena and processes in the life of society are studied by statistics through statistical indicators.
statistic - this is a generalizing characteristic of some property of the statistical population or its part. In this it differs from a sign (property inherent in a unit of the population). For example, the semester GPA for a group of students is a statistic. A score in some subject of a particular student is a sign.
Statistical indicator system is a set of interconnected statistical indicators that comprehensively reflect the processes of social life in certain conditions of place and time.
The law of large numbers. statistical regularity
The concept of statistics and its main provisions
Statistics as a Population Parameter
The law of large numbers. statistical regularity
Boy or girl
Research methods used in population statistics
Bibliography
Word statistics in the middle of the XVIII century. began to designate a set of various kinds of factual information about states (from the Latin “status” - state). Such information included data on the size and movement of the population of states, their territorial division and administrative structure, economy, etc.
Currently, the term "statistics" has several related meanings. One of them closely corresponds to the above. Statistics is often referred to as a set of facts about a particular country. The main ones are systematically published in special editions in the prescribed form.
However, modern statistics in the considered sense of the word is distinguished from the “state of reference” of past centuries not only by the vastly increased completeness and versatility of the information contained in it. With regard to the nature of the information, it now includes only what is received quantitative expression. So, statistics do not include information about whether a given state is a monarchy or a republic. What language is adopted in it as the state language, etc.
But it includes quantitative data on the number of people who use this or that language as their spoken language. Statistics do not include the list and location on the map of individual territorial parts of the state, but include quantitative data on the distribution of the population, industry, etc. on them.
A common feature of the information that makes up statistics is that they always do not refer to one single (individual) phenomenon, but cover the summary characteristics of a number of such phenomena, or, as they say, their totality. An individual phenomenon differs from the totality by its indecomposability into independently existing and similar constituent elements. The totality consists of just such elements. The disappearance of one of the elements of the aggregate does not destroy it as such.
Thus, the population of a city remains its population even after one of its members has died or moved to another.
Different aggregates and their units in reality are combined and intertwined with each other, sometimes in very complex complexes. A specific feature of statistics is that in all cases its data refer to the population. The characteristics of individual individual phenomena fall into its field of vision only as a basis for obtaining summary characteristics of the totality.
For example, the registration of a marriage has a certain meaning for a given individual couple entering into it; certain rights and obligations follow from it for each spouse. Statistics include only summary data on the number of marriages, on the composition of those who entered into them - by age, by source of livelihood, etc. Individual cases of marriage are of interest to statistics only in so far as, based on information about them, it is possible to obtain summary data.
Statistics as a Population Parameter
AT recent times the term "statistics" has often come to be understood in a somewhat narrower, but more precisely defined sense, connected with the processing of the results of a series of individual observations.
Let's imagine that as a result of observations we got the numbers x 1 , x 2 . x n. These numbers are considered as one of the possible realizations of the set n quantities in their combination.
A statistic is a parameter f depending on x 1 , x 2 . x n. Since these quantities are, as noted, one of their possible realizations, the value of this parameter also turns out to be one of a number of possible ones. Therefore, each statistic in this sense has its own probability distribution (i.e. for any given number a there is a possibility that the f will be no more than a).
Compared with the content invested in the term “statistics” in the sense discussed above, here, firstly, we mean its narrowing each time to one value - a parameter, which does not exclude the joint consideration of several parameters (several statistics) in one complex problem . Secondly, it emphasizes the presence of a mathematical rule (algorithm) for obtaining the parameter value from the totality of observation results: calculate their arithmetic mean, take the maximum of the delivered values, calculate the ratio of the number of some of their special group to the total number, etc.
Finally, in the indicated sense, the term "statistics" is applied to a parameter obtained from the results of observations in any field of phenomena - social and others. It could be the average yield, or the average span of the pine trees in the forest, or the average result of repeated measurements of the parallax of some star, and so on. in this sense, the term "statistics" is used mainly in mathematical statistics, which, like any branch of mathematics, cannot be limited to one or another area of phenomena.
Statistics is also understood as the process of its “keeping”, i.e. the process of collecting and processing information about the facts necessary to obtain statistics in both considered senses.
At the same time, the information necessary for statistics can be collected for the sole purpose of obtaining generalized characteristics for the mass of cases of a given kind, i.e. It is natural for the purposes of statistics. Such, for example, is the information collected during population censuses.
The law of large numbers. Statistical regularity.
The main generalization of the experience of studying any mass phenomena is the law of large numbers. A separate individual phenomenon, considered as one of the phenomena of this kind, contains an element of chance: it could be or not be, be this or that. When a large number of such phenomena are combined into general characteristics in their entire mass, chance disappears to the greater extent, the more individual phenomena are connected.
Mathematics, in particular the theory of probability, considered in a purely quantitative aspect, the law of large numbers, expresses it with a whole chain of mathematical theorems. They show under what conditions and to what extent one can count on the absence of randomness in the characteristics covering the mass, how this is connected with the number of individual phenomena included in them. Statistics is based on these theorems in the study of each specific mass phenomenon.
regularity, which manifested itself only in a large mass of phenomena through overcoming the randomness inherent in its individual elements, is called statistical regularity .
In some cases, statistics is faced with the task of measuring its manifestations, while its very existence is theoretically clear in advance.
In other cases, a regularity can be found empirically by statistics. In this way, for example, it was found that with an increase in family income in its budget, the percentage of expenses for food decreases.
Thus, whenever statistics in the study of a phenomenon reaches generalizations and finds a regularity operating in it, this latter immediately becomes the property of that particular science, to the circle of interests of which this phenomenon belongs. Therefore, for each statistic acts as a method.
Considering the results of mass observation, statistics finds similarities and differences in them, combines elements into groups, revealing different types, differentiating the entire observed mass according to these types. The results of observation of individual elements of the mass are used, further, to obtain the characteristics of the entire population and the special parts distinguished in it, i.e. in order to obtain general indicators.
Mass observation, grouping and summary of its results, calculation and analysis of generalizing indicators - these are the main features of the statistical method.
Statistics as a science takes care of and is reduced to mathematical statistics. In mathematics, the tasks of characterizing mass phenomena are considered only in a purely quantitative aspect, divorced from the qualitative content (which is mandatory for mathematics, as a science in general). Statistics, even in the study of the general laws of mass phenomena, proceeds not only from quantitative generalizations of these phenomena, but, above all, from the mechanism of the emergence of the mass phenomenon itself.
At the same time, from what has been said about the role of quantitative measurement for statistics, it follows great importance for it, mathematical methods in general, specially adapted for solving problems that arise in the study of mass phenomena (probability theory and mathematical statistics). Moreover, the role of mathematical methods here is so great that an attempt to exclude them from the course of statistics (due to the presence in the plans of a separate subject - mathematical statistics) significantly impoverishes statistics.
Refusal of this attempt, however, should not mean the opposite extreme, namely, the absorption by statistics of the entire theory of probability and mathematical statistics. If, for example, in mathematics, the average value for a series of distributions (probabilities or empirical frequencies) is considered, then statistics also cannot bypass the appropriate techniques, but here this is one of the aspects, along with which a number of others arise (general and group averages, the occurrence and the role of averages in the information system, the material content of the system of weights, chronological averages, average and relative values, etc.).
Or another example: the mathematical theory of sampling focuses all attention on the error of representativeness - for different systems of selection, different characteristics, etc. System error, i.e. the error not absorbed in the average value, it eliminates in advance by constructing the so-called unbiased estimates free of it. In statistics, perhaps the main question in this matter is the question of how to avoid this systemic error.
In the study of the quantitative side of mass phenomena, a number of problems of a mathematical nature arise. To solve them, mathematics develops appropriate techniques, but for this it must consider them in a general form, for which the qualitative content of a mass phenomenon is indifferent. So the manifestation of the law of large numbers was first noticed precisely in the socio-economic field and almost simultaneously in gambling (the very distribution of which was explained by the fact that they were a cast from the economy, in particular, developing commodity-money relations). From the moment, however, when the law of large numbers becomes the object of exact study in mathematics, it receives a completely general interpretation, which does not limit its action to any special area.
On this basis, the subject of statistics is generally distinguished from the subject of mathematics. The delimitation of objects cannot mean banishing from one science everything that has fallen into the field of view of another. It would be wrong, for example, to exclude from the presentation of physics everything connected with the application of differential equations on the grounds that they are dealt with by mathematics.
Why does the sex ratio at birth have certain proportions that have not undergone significant observation for many centuries?
As paradoxical as it may sound, it is death that is the main biological condition for reproduction and reproduction of new generations. In order to prolong the existence of a species, its individuals must leave behind offspring; otherwise, the view will disappear forever.
The problem of gender (who will be born a boy or a girl) includes many questions related not only to biological development, medical and genetic characteristics, with demographic data, but also in a broader aspect are associated with the psychology of sex, with the behavior and aspirations of individuals of the opposite sex, with harmony or conflicts between them.
The question of who will be born - a boy or a girl - and why this happens - is just a narrow circle of questions arising from a larger problem. Especially important theoretical and practical is the clarification of the question why the life expectancy of men is lower than the life expectancy of women. This phenomenon is common not only in humans, but also among numerous species of the animal world.
It is not enough to explain this only by the fact that the predominance of males at birth is due to their increased activity, and as a result of this - less “vitality”, is not enough. Biologists have long drawn attention to the shorter lifespan of males compared to females in most animals studied. The duration of life is opposed to its high pace, and this finds a biological justification.
The English researcher A. Comfort points out: “The organism must go through a fixed series of metabolic processes or stages of development, and the speed of their passage determines the observed life span.”
Ch. Darwin considered shorter life expectancy in males "as a natural and constitutional property, due only to sex."
The possibility of giving birth to a child of one sex or another in each particular case depends not only on the patterns inherent in this phenomenon, revealed in a large number of observations, but also on random attendant circumstances. Therefore, it is statistically impossible to determine in advance what gender each separately born child will be. Neither probability theory nor statistics does this, although in many cases the result of a single event is of great interest. Probability theory gives fairly definite answers when it comes to a large population of births. Incidental, external causes are random, but their totality reflects stable patterns. In the formation of sex, as is now known, even before conception, accidental causes may in some cases favor the emergence of male embryos, and in others - female. But this does not manifest itself in some regular order, but chaotically, randomly. The totality of factors that form certain sex ratios at birth is manifested only in a sufficiently large number of observations; and the more there are, the closer the theoretical probability approaches the actual results.
The probabilities of having boys are a number slightly greater than 0.5 (close to 0.51), and girls are less than 0.5 (close to 0.49). This very interesting fact posed a difficult task for biologists and statisticians - to explain the reason why the conception and birth of a boy or girl are not equally possible and corresponding to genetic prerequisites (Mendeleev's law of splitting by sex).
Satisfactory answers to these questions have not yet been received; it is only known that already from the moment of conception the proportion of boys is greater than the proportion of girls and that during the period of intrauterine development these proportions gradually level out and by the time of birth, without, however, reaching equiprobable values. Boys are born approximately 5-6% more than girls.
Most of the species for which life tables have been compiled by biologists have a higher mortality among males. Geneticists explain this by the difference in the common chromosome complex between females and males.
Ch. Darwin considers the formed numerical ratio sexes from representatives of various species, as a result of evolutionary natural selection based on the principles of sexual selection. The genetic laws of sex formation were discovered later, and they are the missing link in the theoretical concepts of Ch. Darwin. Ch. Darwin's well-aimed observations deserve to be quoted here. The author observes that sexual selection would be a simple matter if the males greatly outnumbered the females. It is important to know the sex ratio not only at birth, but also during maturity, and this complicates the picture. With regard to people, the fact has been established that many more boys die than girls before birth, during childbirth and in the first years of childhood.
Two large groups of factors can be named that influence the ratio of mortality by sex and, in general, determine the excess mortality of men. These are exogenous, i.e. socio-economic factors, and endogenous factors associated with the genetic program of the viability of the male and female organism. Differences in mortality by sex can be explained by the constant interaction of these two groups of factors. These differences increase in direct proportion to the increase in life expectancy. The purely biological differences in the viability of men and women are superimposed by the impact of socio-economic conditions of life, the reaction to which the male and female organisms are different in terms of the ability to overcome their negative impact at different age periods.
In the vast majority of countries of the world where more or less reliable and complete registration of mortality is carried out, the ratio of indicators by sex is confirmed by the position on the increase in male mortality, which has been repeatedly confirmed by practice - this pattern, as noted earlier, is inherent in the human population and not only it, but also many others. biological species.
Population statistics- a science that studies the quantitative patterns of phenomena and processes occurring in the population, in continuous connection with their qualitative side.
Population- an object of study and demography, which establishes the general patterns of their development, considering its life in all aspects: historical, political, economic, social, legal, medical and statistical. At the same time, it must be borne in mind that as knowledge about an object develops, new aspects of it open up, becoming a separate object of knowledge.
Population statistics studies its object in the specific conditions of place and time, revealing all new forms of its movement: natural, migratory, social.
Under natural movement population refers to the change in population due to births and deaths, i.e. occurring naturally. This also includes marriages and divorces, since they are counted in the same order as births and deaths.
migratory movement, or simply population migration, means the movement of people across the boundaries of certain territories, usually with a change of residence for a long time or permanently.
social movement population is understood as a change in the social conditions of life of the population. It is expressed in a change in the size and composition of social groups of people who have common interests, values and norms of behavior that develop within the framework of a historically defined society.
Population statistics solves a number of problems:
Her most important task- determination of the population. But often it is required to know the population of individual continents and their parts, various countries, economic regions of countries, administrative regions. At the same time, not a simple arithmetic, but a special - statistical account - an account of categories of the population is maintained. The number of births, deaths, marriages, divorce cases, the number of incoming and outgoing migrants is statistically established, i.e. the volume of the population is determined.
Second task- establishing the structure of the population, demographic processes. Attention here is primarily drawn to the division of the population according to sex, age, level of education, professional, production characteristics, according to belonging to urban and rural areas.
Structure of the population by sex can be characterized by an equal number of sexes, male or female predominance and the degree of this preponderance.
Population structure by age can be represented by one-year data and age groups, as well as a trend in age composition, such as aging or rejuvenation.
Educational structure shows the proportion of the literate population with a certain degree of education in different territories and different environments.
Professional- Distribution of people by professions acquired in the process of training, by occupation.
Production- by sectors of the national economy.
Territorial location of the population or its resettlement. Here, a distinction is made between the degree of urbanization, the definition of the density of the entire population, a different understanding of density and its state.
Third task consists in the study of the interrelations that take place in the population itself between its various groups and the study of the dependence of the processes occurring in the population on the environmental factors in which these processes take place.
The fourth task consists of considering the dynamics of demographic processes. In this case, the characteristics of the dynamics can be given as a change in the population size and as a change in the intensity of the processes occurring in the population in time and space.
Fifth task- Population statistics are opened with forecasts of its size and composition for the future. Providing data on the population forecast for the near and far future.
Research methods used in population statistics
Method in the most general sense means a way to achieve the goal, regulation of activity. The method of concrete science is a set of methods of theoretical and practical knowledge of reality. For an independent science, it is necessary not only to have a subject of study that is special from other sciences, but also to have its own methods for studying this subject. The totality of research methods used in any science is methodology this science.
Since population statistics is sectoral statistics, the basis of its methodology is statistical methodology.
The most important method included in the statistical methodology is obtaining information about the processes and phenomena being studied - statistical observation . It serves as the basis for data collection both in current statistics and in censuses, monographic and sample studies of the population. Here, the full use of the provisions of theoretical statistics on the establishment of the object of the unit of observation, the introduction of concepts of the date and moment of registration, the program, organizational issues of observation, systematization and publication of its results. Statistical methodology also contains the principle of independent assignment of each enumerated person to a certain group - the principle of self-determination.
The next step in the statistical study of socio-economic phenomena is the determination of their structure, i.e. selection of parts and elements that make up the totality. We are talking about the method of groupings and classifications, which in population statistics are called typological and structural.
To understand the structure of the population, it is necessary, first of all, to single out the sign of grouping and classification. Any feature that has been observed can also serve as a grouping feature. For example, on the question of the attitude towards the person recorded first in the census form, it is possible to determine the structure of the population being enumerated, where it seems likely to distinguish a significant number of groups. This attribute is attributive, therefore, when developing census questionnaires on it, it is necessary to compile in advance a list of classifications (groupings according to attribute characteristics) needed for analysis. When compiling classifications with a large number of attribute records, the assignment to certain groups is justified in advance. So, according to their occupation, the population is divided into several thousand species, which statistics reduce to certain classes, which is recorded in the so-called dictionary of occupations.
When studying the structure by quantitative characteristics, it becomes possible to use such statistical generalizing indicators as the mean, mode and median, distance measures or variation indicators to characterize different parameters of the population. The considered structures of phenomena serve as the basis for studying the connection in them. In the theory of statistics, functional and statistical relationships are distinguished. The study of the latter is impossible without dividing the population into groups and then comparing the value of the effective feature.
Grouping according to a factor attribute and comparing it with changes in the attribute of an effective one allows you to establish the direction of the relationship: it is direct or reverse, as well as to give an idea of its form. broken regression . These groupings make it possible to construct a system of equations necessary to find regression equation parameters and determining the tightness of the connection by calculating the correlation coefficients. Groupings and classifications serve as the basis for using dispersion analysis of relationships between indicators of population movement and the factors that cause them.
Statistical methods are widely used in the study of the population. dynamics research , graphic study of phenomena , index , selective and balance . It can be said that population statistics use the entire arsenal to study its object. statistical methods and examples. In addition, methods developed only for the study of the population are also used. These are the methods real generation (cohorts) and conditional generation . The first allows us to consider changes in the natural movement of peers (born in the same year) - a longitudinal analysis; the second considers the natural movement of peers (living at the same time) - a cross-sectional analysis.
It is interesting to use averages and indices when taking into account the characteristics and comparing the processes occurring in the population, when the conditions for comparing data are not equal to each other. Using different weightings when calculating generalizing averages, a standardization method has been developed that allows eliminating the influence of different age characteristics of the population.
Probability theory as a mathematical science studies the properties of the objective world with the help of abstractions , the essence of which consists in a complete abstraction from qualitative certainty and in highlighting their quantitative side. Abstraction is the process of mental abstraction from many aspects of the properties of objects and at the same time the process of isolating, isolating any aspects of interest to us, properties and relations of the objects under study. The use of abstract mathematical methods in population statistics makes it possible statistical modeling processes occurring in the population. The need for modeling arises when it is impossible to study the object itself.
The largest number of models used in population statistics has been developed to characterize its dynamics. Among them stand out exponential and logistics. Of particular importance in the population forecast for future periods are models stationary and stable population, which determine the type of population that has developed under these conditions.
If the construction of models of the exponential and logistic population uses data on the dynamics of the absolute population for the past period, then the models of the stationary and stable population are built on the basis of the characteristics of the intensity of its development.
So the statistical methodology for studying the population has at its disposal a number of methods of the general theory of statistics, mathematical methods and special methods developed in the population statistics themselves.
Population statistics, using the methods discussed above, develops a system of generalizing indicators, indicates the necessary information, methods for calculating them, the cognitive capabilities of these indicators, the conditions for use, the order of recording and meaningful interpretation.
The importance of generalizing statistical indicators is great in solving the most important problems when considering demographic policy, it is necessary for balanced population growth, in studying population migration, which forms the basis of inter-district redistribution of the labor force and the achievement of uniformity in its distribution.
Since the population in a certain aspect studies many other sciences - health care, pedagogy, sociology, etc., it is necessary to use the experience of these sciences, to develop their methods in relation to the needs of statistics.
The tasks of renewal facing our country should also affect the solution of demographic problems. The development of comprehensive programs for economic and social development should include sections on demographic programs; their solution should contribute to the development of the population with the least demographic losses.
Bibliography
Kildishev et al. “Population statistics with the basics of demography” M .: Finance and Statistics, 1990 - 312 p.
Poor M.S. “Boys girls? Medico-demographic analysis” M.: Statistics, 1980 – 120 p.
Andreeva B.M., Vishnevsky A.G. “Longevity. Analysis and Modeling” M.: Statistics, 1979 – 157 p.
Boyarsky A.Ya., Gromyko G.L. “General theory of statistics” M.: ed. Moscow Universities, 1985 - 372 p.
Vasilyeva E.K. “Socio-demographic portrait of a student” M.: Thought, 1986 - 96 p.
Bestuzhev-Lada I.V. “The World of Our Tomorrow” M.: Thought, 1986 – 269 p.
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The essence of the law of large numbers.
The law of large numbers.
Topic 2
Organization of state statistics in the Russian Federation.
Tasks of statistics.
statistics method.
Branches of statistics.
The general theory of statistics is connected with 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 Wage statistics | 2.3 Statistics math.-tech. supplies | 2.4 Transport statistics | 2.5 Communication statistics | 2.6 Financial credit statistics | |||||||
| 2.6.1 Higher financial computing | 2.6.2 Statistics monetary circulation | 2.6.3 Exchange rate statistics | Other | |||||||||
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. data interpretation.
The passage of each stage is associated with the use of special methods, explained by the content of the work performed.
1. Development of a system of hypotheses characterizing the development, dynamics, state of socio-economic phenomena.
2. Organization of statistical activities.
3. Development of analysis methodology.
4. Development of a system of indicators for managing the economy at the macro and micro levels.
5. Make statistical observation data publicly available.
Principles:
1. centralized management,
2. unified organizational structure and methodology,
3. inseparable connection with government bodies.
The system of state statistics has a hierarchical structure, consisting of federal, republican, territorial, regional, district, city and district levels.
The State Statistics Committee has departments, departments, and a computer center.
The massive nature of social laws and the originality of their actions predetermine the extreme importance of the study of 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 hand, have something in common, due to their belonging to a certain class, type. Moreover, single phenomena are more susceptible to the influence of random factors than their totality.
The law of large numbers is the definition of the quantitative laws of mass phenomena, which manifest themselves only in a sufficiently large number of them.
Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, its essence lies essentially in the fact that in the numbers obtained as a result of mass observation, certain regularities appear that are not found in a small number of facts.
The law of large numbers expresses the dialectic of the accidental and the extremely important. As a result of the mutual cancellation of random deviations, the average values calculated for a value of the same type become typical, reflecting the actions of constant and significant facts in terms of place and time.
The tendencies and regularities revealed by the law of large numbers are valid only as mass tendencies, but not as laws for each individual case.
The 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.
Ministry of Education and Science
State educational institution
Higher professional education
"Samara State University"
Faculty of Law
Department__________________
__________________
__________________
TEST
on the course: "Legal statistics"
Option number 3
Is done by a student
3 courses of the correspondence department
Faculty of Law
09303.30 groups
Nesmeyanova Daria Sergeevna
SAMARA 2011
1 The law of large numbers and its significance in legal statistics 3
2 Static tables and their types 6
Task 1 8
Task 2 9
List of used literature 10
1 The law of large numbers and its significance in legal statistics
In the decision the most important task- establishing and quantifying the regularities and interdependence of social phenomena, statistical science relies on the law of large numbers (LLN), the meaning of which is that the correctness and regularities of social phenomena can be detected only with their mass observation.
Of course, every science, each in its own field, deals with mass phenomena, because the law reflects the mass-like, essential, and necessary. And although any regularity is general, and therefore of a mass character, in statistics the concept of mass character is specific. It becomes obvious if we recall the division of regularities into dynamic and statistical. Statistics operates not with generic, but with group concepts, in which we are talking about average results, and while in generic - about each unit included in it. Therefore, in legal statistics, knowledge about the offense as a statistical aggregate is not at the same time knowledge about the specific crimes included in it. Although in this case the statistician does not deal with purely random phenomena, but with individual ones, which are characterized by random deviations.
This is the specificity of the statistical quantitative analysis of social processes, in which the meaning of the law of large numbers is manifested: the conclusions drawn on its basis, the discovered trend, the regularity refer to the totality (“large number”) as such. That is, the ZBN underlies the very logic of statistical reasoning; on the basis of the ZBN, a mass regularity is revealed.
Statistical regularities are characterized by a complex interweaving of internal and external causes, necessary and accidental.
And these regularities are by no means formed in the course of a "game of chance", but primarily as a result of the action of internal necessary causes. Many variations and random deviations are smoothed out (eliminated) in the mass, which leads to the formation of statistical patterns. The manifestation of such a regularity is the result of the operation of the law of large numbers, which consists in the fact that the totality of a large number of random phenomena has certain characteristics that do not depend on the case, expressed by quantitative indicators. That is, the idea of the ZBH and its action cannot be separated from the idea of a statistical regularity as a form in which the regularity of a mass phenomenon is clothed, studied by statistics from a quantitative side. Moreover, the ZBH manifests itself more clearly, the larger the statistical population.
Mass regularities, and with them the ZBCh, are manifested in the most diverse areas of reality. They are especially evident in demographics, in criminal statistics. Thus, in countries with a market economy in the working environment, births and deaths are inversely proportional to the level of wages; in all countries with high life expectancy, women live longer than men; the mortality of men in all age cohorts, from children to the most elderly, is 2-3 times higher than the mortality of women; a constant value is the number of marriages, the sexual distribution of criminals, motives, murder weapons; significant stability of accidents is found in certain periods of the year and hours of the day; according to Russian postal and telegraph statistics, a significant stability was noted for every million letters taken out of mailboxes (1906-1910) without specifying the addressee (25-27) or without specifying the destination (21-29), etc. In a small number observations (for example, individual crimes), random factors do not make it possible to detect a pattern. On the contrary, when a large number of individual phenomena are summed up, chances paralyze each other, which makes it possible to establish laws that, on small scales, are masked by individual deviations. Statistical regularity is not a special form of matter movement, but only an external manifestation of this movement in statistical distributions and generalizing statistical characteristics. Statistically established correctness in changes in quantitative indicators, the repeatability and stability of facts only testify to the fact that in the studied mass phenomenon there is a certain regularity, the opening of which is the task of the corresponding science (for example, criminology).
The regularity of a mass phenomenon, the objective connections inherent in this phenomenon, find their expression not in individual indicators, but in an average value, in the nature of the distribution. The arithmetic mean of a large number of random variables is practically not a random value, but a necessary, regular one. This is what the action of the ZBCh consists of, if we approach its interpretation from a philosophical and methodological position. Therefore, sometimes the LLN is also called the law of averages.
At the same time, consideration of the ZLN as one of the laws of objective reality excludes its relation to the level of generalizing statistical characteristics stated by it. This level is determined by the conditions arising from the very nature of the mass phenomenon. It is correctly noted that the ZBN does not create levels, but only regulates random deviations from the levels set by the nature of this phenomenon1.
From what has been said, it is clear that the ZBN is based on the concept of randomness and probability - a decrease in the degree of randomness and an increase in the degree of probability of the presence of a certain feature occurs as the statistical population increases. This can be illustrated by the following example: if it is known that the population of a city is represented by a ratio of 48% men and 52% women, then a small population of people (for example, visitors to the theater, football match, etc.) can deviate significantly from these characteristics; if you increase the studied population, then an approximation to the indicated characteristics will follow.
The natural-science justification, exact formulation and conditions for the applicability of the LSP are given in the theory of probability. In other words, the theory of probability is the mathematical foundation of the ZBN. It is used to calculate the chances of a possible occurrence of a random event.
Probability is a mathematical, numerical characteristic of the degree of possibility of the occurrence of any particular event in certain conditions that can be repeated an unlimited number of times2.
Probability is usually denoted by the letter P. For example, the expression P(L) = 0.5 means that the probability of the event A occurring is 0.5.
Probability is usually classified according to the following scale:
0.00 - completely excluded
0.10 - highly uncertain.
0.20 - very implausible
0.30-0.40 - implausible
0.60 - probably
0.70 - very likely
0.80-0.90 - highly likely
1.00 - completely reliable.
Thus, the probability receives a certain quantitative expression, despite the fact that the presence of one or another sign or its fluctuation is random.
If black and white balls are placed in an urn, then either of them can equally be found during the extraction. In this case, alternative variability is manifested, which consists in the possibility of only two outcomes: only a white ball or only a black ball can be taken out of the urn. The same thing happens when you toss a coin. This circumstance of the same possibility of falling out of any side of the coin is called equipoise. An event is said to be equally likely if there are no reasons that make one of these events more likely than the other. An event is said to be incompatible if the occurrence of one makes the occurrence of the other impossible.
With repeated tossing of a coin or with repeated removal of balls from an urn, a set of single experiments is formed, which has the properties of a statistical set. In a separate experiment, the result can be different - an eagle or tails, a black or white ball, and in the totality of experiments a certain pattern is manifested in the ratio between the number of coats of arms and tails that fell out or the number of black and white balls taken out.
The result of each single experiment with a coin or balls also depends on two groups of factors: the main ones associated with the properties of the phenomenon, and random ones not related to these properties. However, the convenience of the coin or urn model is, firstly, that it is easy to separate the main causes and properties of the phenomenon from side ones; secondly, on this model it is easy to trace how each group of causes operates and what is the result of each of them.
In the examples under consideration, the main property of the coin is its symmetry, due to which, when tossing, the chances of getting a coat of arms or tails are completely equal; The main property of an urn with balls is the ratio between the number of black and white balls. If, for example, there are 100 black and 100 white balls in an urn, then when one ball is drawn, the chances of a black or white ball appearing are exactly the same, and if the urn contains twice as many blacks as whites, then the chances of drawing a black ball are correspondingly greater.
To a priori, i.e. before experiment, to determine the probability of occurrence of any random phenomenon, you need to know the number of chances that favor its occurrence, as well as the number of all possible chances (both favorable and unfavorable). The ratio of the first value to the second is called mathematical probability. It is expressed as a fraction, where the numerator is the number of favorable chances, and the denominator is the number of all possible chances. For example, when tossing a coin, there are two possible outcomes. If we consider the loss of an eagle as a favorable outcome, then its probability is equal to 1/2. If we consider the appearance of a black ball from an urn containing 70 black balls and 30 white balls as a favorable outcome, then the probability of a favorable outcome in the extraction of one ball is 70/100, and the probability of an unfavorable outcome is 30/100.
If the probability of a favorable outcome is denoted by p, and the probability of an unfavorable outcome by q, then in all cases of alternative variability, i.e. when only two outcomes are possible, p + q= 1. In the experiment with balls 70/100 + 30/100 = 1, in the experiment with a coin 1/2 + 1/2 = 1.
Probability is an assessment of the degree of objective possibility of a particular result when selecting one unit for luck from the entire population.
This definition of probability, given by P.S. Laplace, is the definition of the simplest, so-called classical probability, applicable to a very narrow range of phenomena. For mass (for example, offenses), a statistical or frequency concept of probability is more suitable, defined as a constant number around which frequencies fluctuate.
The application of the theory of probability to social phenomena, in particular to crime, is due, along with the independence of individual events (the irregularity of crimes), also to their well-known stability.
Crime is a typical statistical aggregate with relatively stable characteristics that make it possible to specifically study it and even predict its changes. Therefore, “it is impossible to speak of a certain probability of a crime as an “unshakable regularity”. It changes as conditions change. But as long as these certain conditions are valid, this or that certain probability is also valid. This makes it possible to study these phenomena on the basis of methods of mathematical statistics.” If conditions remain unchanged for certain reasons, then the number of crimes is also stable on average, which makes it possible to establish the probability with which they are committed.
2 Statistical tables and their types
A special place in statistics is occupied by the tabular method, which is of universal importance. With the help of statistical tables, the data of the results of statistical observation, summaries and groupings are presented. Therefore, a statistical table is usually defined as a form of compact visual presentation of statistical data.
The analysis of tables makes it possible to solve many problems in the study of changes in phenomena over time, the structure of phenomena and their interrelations. Thus, statistical tables play the role of a universal means of rational representation, generalization and analysis of statistical information.
Externally, a statistical table is a system of horizontal rows and vertical columns built in a special way, having a common heading, headings of columns and lines, at the intersection of which statistical data is recorded.
Each figure in the statistical tables is a specific indicator that characterizes the size or levels, dynamics, structure or relationships of phenomena in specific conditions of place and time, that is, a certain quantitative and qualitative characteristic of the phenomenon under study.
If the table is not filled with numbers, that is, it has only a general heading, column and row headings, then we have a layout of a statistical table. It is with its development that the process of compiling statistical tables begins.
The main elements of the statistical table are the subject and predicate of the table.
The subject of the table is the object of statistical study, that is, individual units of the population, their groups, or the entire population as a whole.
The predicate of the table is the statistical indicators characterizing the object under study.
The subject and indicators of the predicate of the table must be determined very precisely. As a rule, the subject is located on the left side of the table and makes up the content of the lines, and the predicate is on the right side of the table and makes up the content of the columns.