Properties of statistical data. Basic statistical characteristics of experimental data. Statistical analysis of specific data

Statistical methods

Statistical methods- methods of statistical data analysis. There are methods of applied statistics, which can be used in all areas of scientific research and any sectors of the national economy, and other statistical methods, the applicability of which is limited to one or another area. This refers to methods such as statistical acceptance control, statistical control of technological processes, reliability and testing, and planning of experiments.

Classification of statistical methods

Statistical methods of data analysis are used in almost all areas of human activity. They are used whenever it is necessary to obtain and justify any judgments about a group (objects or subjects) with some internal heterogeneity.

It is advisable to distinguish three types of scientific and applied activities in the field of statistical methods of data analysis (according to the degree of specificity of the methods associated with immersion in specific problems):

a) development and research of general-purpose methods, without taking into account the specifics of the field of application;

b) development and research of statistical models of real phenomena and processes in accordance with the needs of a particular area of ​​activity;

c) application of statistical methods and models for statistical analysis of specific data.

Applied Statistics

A description of the type of data and the mechanism for its generation is the beginning of any statistical study. Both deterministic and probabilistic methods are used to describe data. Using deterministic methods, it is possible to analyze only the data that is available to the researcher. For example, with their help, tables were obtained that were calculated by official state statistics bodies based on statistical reports submitted by enterprises and organizations. The obtained results can be transferred to a wider population and used for prediction and control only on the basis of probabilistic-statistical modeling. Therefore, only methods based on probability theory are often included in mathematical statistics.

We do not consider it possible to contrast deterministic and probabilistic-statistical methods. We consider them as sequential steps of statistical analysis. At the first stage, it is necessary to analyze the available data and present it in an easy-to-read form using tables and charts. Then it is advisable to analyze the statistical data on the basis of certain probabilistic and statistical models. Note that the possibility of deeper insight into the essence of a real phenomenon or process is ensured by the development of an adequate mathematical model.

In the simplest situation, statistical data are the values ​​of some characteristic characteristic of the objects being studied. Values ​​can be quantitative or provide an indication of the category to which the object can be classified. In the second case, they talk about a qualitative sign.

When measuring by several quantitative or qualitative characteristics, we obtain a vector as statistical data about an object. It can be thought of as a new kind of data. In this case, the sample consists of a set of vectors. There are part of the coordinates - numbers, and part - qualitative (categorized) data, then we are talking about a vector of different types of data.

One element of the sample, that is, one dimension, can be the function as a whole. For example, describing the dynamics of the indicator, that is, its change over time, is the patient’s electrocardiogram or the amplitude of the beat of the motor shaft. Or a time series describing the dynamics of a particular company’s performance. Then the sample consists of a set of functions.

Sample elements can also be other mathematical objects. For example, binary relationships. Thus, when surveying experts, they often use ordering (ranking) of objects of examination - product samples, investment projects, options for management decisions. Depending on the regulations of the expert study, the sampling elements can be various types of binary relations (ordering, partitioning, tolerance), sets, fuzzy sets, etc.

So, the mathematical nature of sample elements in various problems of applied statistics can be very different. However, two classes of statistical data can be distinguished - numerical and non-numerical. Accordingly, applied statistics is divided into two parts - numerical statistics and non-numerical statistics.

Numerical statistics are numbers, vectors, functions. They can be added and multiplied by coefficients. Therefore, in numerical statistics, various sums are of great importance. The mathematical apparatus for analyzing the sums of random elements of a sample is the (classical) laws of large numbers and central limit theorems.

Non-numerical statistical data are categorized data, vectors of different types of features, binary relations, sets, fuzzy sets, etc. They cannot be added and multiplied by coefficients. Therefore, it makes no sense to talk about sums of non-numeric statistics. They are elements of non-numerical mathematical spaces (sets). The mathematical apparatus for analyzing non-numerical statistical data is based on the use of distances between elements (as well as measures of proximity, indicators of difference) in such spaces. With the help of distances, empirical and theoretical averages are determined, the laws of large numbers are proved, nonparametric estimates of the probability distribution density are constructed, diagnostic problems and cluster analysis are solved, etc. (see).

Applied research uses various types of statistical data. This is due, in particular, to the methods of obtaining them. For example, if testing of some technical devices continues until a certain point in time, then we get the so-called. censored data consisting of a set of numbers - the duration of operation of a number of devices before failure, and information that the remaining devices continued to operate at the end of the test. Censored data is often used in assessing and monitoring the reliability of technical devices.

Typically, statistical methods for analyzing data of the first three types are considered separately. This limitation is caused by the fact noted above that the mathematical apparatus for analyzing data of a non-numerical nature is significantly different than for data in the form of numbers, vectors and functions.

Probabilistic-statistical modeling

When applying statistical methods in specific fields of knowledge and sectors of the national economy, we obtain scientific and practical disciplines such as “statistical methods in industry”, “statistical methods in medicine”, etc. From this point of view, econometrics is “statistical methods in economics”. These disciplines of group b) are usually based on probabilistic-statistical models built in accordance with the characteristics of the field of application. It is very instructive to compare probabilistic-statistical models used in various fields, to discover their similarities and at the same time to note some differences. Thus, one can see the similarity of problem statements and statistical methods used to solve them in such areas as scientific medical research, specific sociological research and marketing research, or, in short, in medicine, sociology and marketing. These are often grouped together under the name "sample studies".

The difference between sample studies and expert studies is manifested, first of all, in the number of objects or subjects surveyed - in sample studies we are usually talking about hundreds, and in expert studies - about tens. But the technology of expert research is much more sophisticated. The specificity is even more pronounced in demographic or logistic models, when processing narrative (text, chronicle) information or when studying the mutual influence of factors.

Issues of reliability and safety of technical devices and technologies, queuing theory are discussed in detail in a large number of scientific works.

Statistical analysis of specific data

The application of statistical methods and models for statistical analysis of specific data is closely tied to the problems of the relevant field. The results of the third of the identified types of scientific and applied activities are at the intersection of disciplines. They can be considered as examples of the practical application of statistical methods. But there are no less reasons to attribute them to the corresponding field of human activity.

For example, the results of a survey of instant coffee consumers are naturally attributed to marketing (which is what they do when giving lectures on marketing research). The study of the dynamics of price growth using inflation indices calculated from independently collected information is of interest primarily from the point of view of economics and management of the national economy (both at the macro level and at the level of individual organizations).

Development prospects

The theory of statistical methods is aimed at solving real problems. Therefore, new formulations of mathematical problems for the analysis of statistical data constantly arise in it, and new methods are developed and justified. Justification is often carried out by mathematical means, that is, by proving theorems. A major role is played by the methodological component - how exactly to set problems, what assumptions to accept for the purpose of further mathematical study. The role of modern information technologies, in particular, computer experiments, is great.

An urgent task is to analyze the history of statistical methods in order to identify development trends and apply them for forecasting.

Literature

2. Naylor T. Machine simulation experiments with models of economic systems. - M.: Mir, 1975. - 500 p.

3. Kramer G. Mathematical methods of statistics. - M.: Mir, 1948 (1st ed.), 1975 (2nd ed.). - 648 p.

4. Bolshev L. N., Smirnov N. V. Tables of mathematical statistics. - M.: Nauka, 1965 (1st ed.), 1968 (2nd ed.), 1983 (3rd ed.).

5. Smirnov N. V., Dunin-Barkovsky I. V. Course in probability theory and mathematical statistics for technical applications. Ed. 3rd, stereotypical. - M.: Nauka, 1969. - 512 p.

6. Norman Draper, Harry Smith Applied regression analysis. Multiple Regression = Applied Regression Analysis. - 3rd ed. - M.: “Dialectics”, 2007. - P. 912. - ISBN 0-471-17082-8

See also

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See what “Statistical methods” are in other dictionaries:

STATISTICAL METHODS- STATISTICAL METHODS scientific methods for describing and studying mass phenomena that allow quantitative (numerical) expression. The word “statistics” (from Igal. stato state) has a common root with the word “state”. Initially it... ... Philosophical Encyclopedia

STATISTICAL METHODS –- scientific methods of describing and studying mass phenomena that allow quantitative (numerical) expression. The word “statistics” (from Italian stato – state) has a common root with the word “state”. Initially it related to the science of management and... Philosophical Encyclopedia

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statistical methods- (in psychology) (from the Latin status state) certain methods of applied mathematical statistics, used in psychology mainly for processing experimental results. The main purpose of using S. m. is to increase the validity of conclusions in ... ... Great psychological encyclopedia

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STATISTICAL METHODS- some methods of applied mathematical statistics used to process experimental results. A number of statistical methods have been developed specifically for testing the quality of psychological tests, for use in professional... ... Vocational education. Dictionary

STATISTICAL METHODS- (in engineering psychology) (from the Latin status state) some methods of applied statistics used in engineering psychology to process experimental results. The main purpose of using S. m. is to increase the validity of conclusions in ... ... Encyclopedic Dictionary of Psychology and Pedagogy

Groupings in criminal legal statistics allow us to give the most complete and comprehensive criminological and criminal legal characteristics based on a wide variety of criteria:

• Ш by type - articles of the Criminal Code,
• Sh on the object of the attack,
• Ш on a territorial basis - district, region, region, republic,
• W ratio of mercenary and violent crimes,
• Ш according to the time of commission of crimes, etc.),
• The identity of the criminals (by gender, age, education, social status, place of residence, etc.),
• Ш causes and conditions conducive to the commission of crimes, as well as measures of social and legal control over them.

At the same time, it is very important to compare various groupings from criminal law statistics not only with each other, but also with groupings from other branches of statistics (demographic, socio-economic, etc.), reflecting interrelated phenomena.

Differences in the purpose of the grouping and the tasks that they solve in statistical analysis are expressed in their existing classification: typological, structural, analytical.

The most important task of groupings in statistics is to divide the studied mass of population units into characteristic types, i.e. into groups homogeneous in essential characteristics. This problem is solved using a typological grouping.

Typological groupings- this is the differentiation of the studied population into homogeneous groups, types according to an essential qualitative feature.

The main goal of a typological grouping is to distinguish one type of phenomenon from another by statistical means. This type of grouping is largely determined by established ideas about what types of phenomena constitute the content of the population being studied.

In legal statistics, these are three types of legal relations: criminal law, administrative law and civil law, which define its sections.

In criminal statistics, in particular, this could be, for example, the gender distribution of persons who committed crimes.

This grouping according to a qualitative characteristic, when there are only two values ​​of this characteristic, and one of them excludes the other, is called alternative in statistics.

The sequence of actions for carrying out this type of grouping is elementary:

• 1) the type of phenomenon that should be highlighted is determined - in our case, registered crimes;
• 2) a grouping characteristic is selected as the basis for describing the type - in our case, the gender of the persons who committed the crimes;
• 3) the boundaries of the intervals are established (in our case, for all persons identified as having committed crimes);
• 4) the grouping is drawn up in a table, the selected groups (based on a combination of grouping characteristics) are combined into the intended types, and the number (specific gravity) of each of them is determined.

When typological grouping, that is, when summing up units into qualitatively homogeneous categories, these categories should, as noted, be determined on the basis of the provisions of the relevant science and the norms of the law. For example, the grouping of punishments by type is carried out by criminal law (judicial) statistics in full accordance with Art. 43-59 of the Criminal Code, establishing with exhaustive completeness the exact qualitative characteristics of their individual types (fine, correctional labor, imprisonment, etc.

Structural groupings- this is the distribution of typically homogeneous groups according to quantitative characteristics that can change (vary). In the scientific literature, this type of grouping is sometimes called variational. With their help, criminal statistics study, for example, the structure of criminals according to varying characteristics: age, number of convictions, terms of imprisonment, wages and other quantitative characteristics.

Structural, or variational, grouping of statistics can be done to examine the changing structure of typically homogeneous groups of crimes, offenders, civil claims, and other indicators. For the structural grouping of material, it is necessary to have homogeneous aggregates, divided according to the value of the changing (varying) characteristic.

If the typological grouping is based on qualitative characteristics, then the variational grouping is based on quantitative ones (proportion of crimes, persons, cases, age of offenders, sentence terms, number of convictions, number of completed classes, amount of damage, amount of claim, terms of investigation and consideration of criminal or civil cases). affairs, etc.) .

Quantitative shifts in the structure of the phenomena under study over several years indicate changes in objective trends and patterns, investigative or judicial practice, and the effectiveness of the activities of law enforcement or other legal bodies. Taking, for example, absolute and relative conviction rates over many years, we will identify trends in judicial practice and its relationship with actual crime. Having studied the dynamics of the absolute numbers of recorded crimes of a certain type, the dynamics of its share in the structure of all crime, we will discover trends in the development of this act.

Structural groupings can be built on the basis of the share distribution of crimes by areas and objects of criminal encroachment, subjects of the Federation, regions and territories

Structural differences in this case may reveal the peculiarities of the criminological situation in a particular region.

Structural (variational) groupings are adjacent to the rows of distribution of population units according to varying characteristics.

Analytical groupings- this is a distribution according to dependence, the relationship between two or more heterogeneous groups of phenomena or their characteristics (for example, the distribution of thefts by place and time of their commission; those convicted of motor vehicle crimes - by the driver’s work experience, etc.).

Analytical groupings are of great importance for all branches of legal statistics. They make it possible to identify many hidden dependencies and relationships, which is important for making practical decisions and the development of legal science. Other types of groupings, as well as other statistical techniques, also have analytical potential, but the analytical grouping itself directly pursues the establishment of dependencies between the phenomena under study. By the nature of their tasks, correlational groupings are close to the analytical grouping, when the dependence between the phenomena or processes under study can be measured relatively accurately.

All types of considered groupings are usually used together when analyzing socio-legal, tortological and criminological aspects. For example, to establish the social danger and severity of the crimes committed, we can divide their totality into categories of acts and forms of guilt (typological grouping). To determine the effectiveness of the fight against crime of various law enforcement agencies (internal affairs, drug control, customs service, prosecutor's office, security service), we can study the variation in the detection rate of crimes in the mentioned departments (variation grouping).

In order to establish the causes and conditions of growth or (decrease in crime in a city, region, country), a number of analytical groups should be applied.

The subject of statistics has changed throughout the history of the development of statistical science; until now, scientists have not come to an unambiguous answer on this issue.

The subject of statistics is the study of social phenomena and their analysis.

Thus, the English statisticians J.E. Yula and M.J. Kendal believe: “Regardless of the branch of knowledge in which numerical data is obtained, they have a certain kind of properties, the identification of which may require a special kind of scientific method of processing. The latter is known as statistical method or statistics.”

The versatility of statistics as a science stems from the fact that it deals with methods of measurement and interpretation, both in the social sciences and in the natural sciences. Statistics is recognized as a special method used in various fields of activity in solving a variety of problems, defined as “the collection, presentation and interpretation of numerical data.”

Statistical methodology and practice are inextricably linked, complement and develop each other. Statistical theory generalizes the experience of practical work, develops new ideas and methods that enrich practical statistical activity. Statistical practice is scientifically organized work.

Thus, statistics– a science that studies the quantitative side of mass social phenomena in order to establish patterns in inextricable connection with their qualitative side in specific conditions of place and time in their interrelation and interdependence (N.N. Ryauzovsky “General Theory of Statistics”).

The essence of this definition is related to six main points:

1. Not all phenomena are studied, but only social and socio-economic ones. These phenomena are complex, diverse (for example: labor production, healthcare, cultural activities, population, etc.), differ from natural phenomena, which are relatively stable in nature and repeatable over time.

2. Mass socio-economic phenomena are studied, not individual ones, since development patterns are manifested through many facts, when data are generalized with a sufficiently large number of units (law of large numbers).

3. Phenomena are given a quantitative assessment, on the basis of which their qualitative content is revealed (for example: for a quantitative analysis of unemployment, the employment indicator and the unemployment rate are used).

4. The numerical characteristics of the same phenomenon are different in space and time.

5. Socio-economic phenomena are studied in dynamics in order to identify trends and direction of development, and forecast future situations.

6. Study of phenomena in relationship and interdependence.

Thus, when using statistical methods, it is important to remember the unity of the quantitative and qualitative aspects of the phenomenon being studied.

So, statistics deals with the study of mass phenomena or aggregates.

Totality- is a group homogeneous by any characteristic, which consists of a core and the phenomena surrounding it (“layer”). The core is a concentrated expression of all the specific properties of a given group that distinguish one set from others. “Layer” - units with an incomplete set of specific properties that belong to a given population with a certain probability.

For example: the population is students, among the students there are:

- “ideal student” - an excellent student, reads a lot, actively participates in extracurricular activities - this is the core.

A student for whom only “interesting”, specialized knowledge is important; - this is one layer.

A student who is only interested in extracurricular life, etc. – this is another layer.

Thus, the “quality” of some students can be almost unmistakably attributed to one type or another, while others are quite difficult.

The ratio of the core and its environment in different aggregates is different, and depends on the conditions of existence of the aggregate: duration, stability, interaction with other aggregates, etc. However, the core should constitute the majority of units of the aggregate, since it determines its characteristic features.

Since statistics deals with the study of phenomena at a specific point in place and time, it has a limited number of data.

Statistical population- this is a set of objectively existing units of the phenomenon being studied, united by a single qualitative basis, a common connection, but differing from each other in individual characteristics. (For example, a set of households, a set of families, a set of enterprises, firms, associations, etc.).

The totality must be distinguished from the system and structure, since in the totality there is no order, here all the elements are separated.

Sign – it is a qualitative feature of a unit of the aggregate.

According to the nature of the display of the properties of the units of the population under study, the signs are divided into two main groups:

1. Quantitative – characteristics that have direct quantitative expression, that is, they can be added (for example: age, income, number of children, number of years of education, work experience, etc.). They assume a more-less relationship.

2. Quality – characteristics that do not have a direct quantitative expression, that is characteristics that cannot be added (for example: gender, profession, nature of work, attitude towards something). They assume relations of “equality-inequality”. (!do not allow more-less relationships.)

All quality characteristics are divided into:

Attributive - which is a feature of a given phenomenon (for example: profession, nature of work, etc.)

Alternative – options that are opposite in meaning (for example: the product is good or bad, for representatives of certain age groups there is a probability of surviving or not surviving to the next age group; each person may be married or not, a man or a woman, etc.).

In addition, signs in statistics can be divided into different groups, depending on the basis. The main classifications of characteristics are presented in Figure 1.2.

Classifications of characteristics in statistics

Descriptive- characteristics expressed verbally (form of ownership of the enterprise, type of raw materials used, profession, etc.) Descriptive characteristics are divided into nominal, which cannot be ordered or ranked (nationality, industry of the enterprise, etc.) and ordinal, which can be ranked (tariff category , student performance score, company ratings, etc.).

Quantitative characteristics - those whose individual values ​​have a numerical expression (the area of ​​the region, the value of the enterprise's assets, the price of goods, etc.).

Primary characteristics characterize the unit of the population as a whole. They can be measured, counted, weighed and exist on their own, regardless of their statistical study (number of city residents, gross grain harvest, amount of insurance payments).

Secondary characteristics are obtained by calculation through the ratio of primary characteristics. Secondary features are products of human consciousness, the results of cognition of the object being studied.

Direct signs are properties inherent in the object that they characterize.

Indirect signs are properties inherent not to the object being studied itself, but to other aggregates related to the object.

Alternative signs - those that take only the bottom of the meaning (gender of a person, place of residence (urban-rural), signs of possession or non-possession of something.

Discrete signs. have only integer values.

Continuous signs - capable of taking on any values, both integer and fractional. Continuous include all secondary characteristics.

Momentary signs - characteristics of a state, the presence of something at a certain point in time.

Interval signs - characteristics of a process for a certain period of time: year, half-year, quarter, month, day, etc.

A feature of statistical research is that it studies only varying characteristics, i.e. characteristics that take on different meanings (for attributive, alternative characteristics) or have different quantitative levels in individual units of the population.

A significant property of a statistical population is variation.

Variation– this is a property of a statistical population, reflecting the ability to change, due to both external and internal factors, both related to the essence of the object under study and not related to it.

Statistical pattern- this is a pattern established through the law of large numbers in mass variable phenomena combined into a statistical totality.

Statistical patterns are evident in trends.

Statistics functions:

1. Descriptive - with the help of figures and numbers, a characteristic of a particular situation, process, phenomenon is given

2. Explanatory – cause-and-effect relationships between phenomena and processes are identified; factors that determine certain connections are identified.

The nature of statistical data is determined by 3 main properties:

1. Uncertainty of statistics

2. Probabilistic nature of statistical data (the attribute may or may not take this value)

3. Abstractness of statistical data.

Eliseeva I.I. Workshop on the general theory of statistics. M.: Finance and Statistics, 2008. P.8.

It is clear that it is quite possible to turn the values ​​of the characteristics, the names of which are given in the “Company Indicators” column, into numbers, but this transition will depend on the researcher and will carry an inevitable touch of subjectivity.

Sometimes it is not possible to clearly classify data as categorized or quantitative. For example, in the Old Testament, in the Fourth Book of Moses, “Numbers,” the number of warriors in different tribes is indicated. On the one hand, this is typical categorized data, the gradations are the names of the tribes. On the other hand, these data can be considered quantitative, as a sample; it is quite natural to add them up, calculate the arithmetic mean, etc.

The situation described is typical. There are many different types of statistics. This is due, in particular, to the methods of obtaining them. For example, if testing of some technical devices continues until a certain point, then we get the so-called censored data consisting of a set of numbers - the duration of operation of a number of devices before failure, and information that the remaining devices continued to operate at the end of the test. This kind of data is often used in assessing and monitoring the reliability of technical devices.

A description of the type of data and, if necessary, the mechanism for generating it is the beginning of any statistical study.

In the simplest case, statistical data are the values ​​of some characteristic characteristic of the objects being studied. Values ​​can be quantitative or represent an indication of the category to which the object can be classified. In the second case, they talk about a qualitative sign. More complex features are also used, the list of which will expand as the presentation unfolds in the textbook.

Non-numerical statistical data are categorized data, vectors of different types of features, binary relations, sets, fuzzy sets, etc. They cannot be added and multiplied by coefficients. Therefore, it makes no sense to talk about sums of non-numeric statistics. They are elements of non-numerical mathematical spaces (sets). The mathematical apparatus for analyzing non-numerical statistical data is based on the use of distances between elements (as well as measures of proximity, indicators of difference) in such spaces. Using distances, empirical and theoretical averages are determined, the laws of large numbers are proved, nonparametric estimates of the probability distribution density are constructed, diagnostic problems are solved, and cluster analysis, etc. (cm. "Non-numeric data statistics").

Let us summarize information about the main areas of applied statistics in table 1.2. Note that models for generating censored data are included in each of the areas under consideration.