When studying various economic phenomena, we are constantly faced with causal relationships, when some phenomena, called causes, give rise to another phenomenon, called a consequence (result). The causes will be called factor signs or simply factors, and the result will be called an effective sign. The study and measurement of relationships between causes and effects are carried out using statistical methods.
The main task of correlation analysis is to measure the tightness of the relationship between variables (random variables) by point and interval estimates of the corresponding coefficients (characteristics).
With the help of correlation analysis, the selection of factors that have the most significant impact on the resulting attribute (based on the degree of connection between them), the discovery of previously unknown causation.
Correlation does not directly reveal causal relationships between variables, but establishes the numerical value of the closeness of these relationships and the reliability of judgments about their presence.
Let it be required to study the impact on the economic indicator Y factors X 1 , X m .
Considering the relationship between the performance indicator Y and factor signs X 1 , X m , two categories of relationships can be identified:
1) Functional dependence;
2) Correlation dependence;
Functional relationships are characterized by full correspondence between the change in factor characteristics and the change in the effective value, that is, each specific set of factor values corresponds to a certain value of the effective characteristic.
In economics, as a rule, we deal with phenomena and processes where there are no such rigid ties. The causation of economic phenomena is associated with a huge set of interdependent circumstances. The number of circumstances (factors) that affect the studied economic indicator reaches several hundred.
The relationship between causes and effects is multi-valued and has a probabilistic character. In this case, there is a correlation dependence.
In the correlations between the measurement of factors and the resulting attribute, there is no complete correspondence. The influence of individual factors is manifested only on average during the mass observation of actual data. The fact is that the identified factors are not the only reason for the change in the performance indicator. Along with him on the magnitude Y influenced by many other reasons.
Therefore, for the same set of factor values, the value Y may turn out to be different. Thus, the simultaneous effect on the studied trait Y a large number a wide variety of factors leads to the fact that one set of factor values corresponds to the whole distribution of values of the effective feature Y .
When comparing functional and correlation dependences, it should be borne in mind that in the presence of a functional dependence, it is possible, knowing the value of the factors, to accurately determine the value Y . In the presence of a correlation dependence, only the trend of change is established Y when factors change.
When studying correlation dependencies, it is necessary:
1) Establish the fact of the existence of a connection, determine its direction and form;
2) Measure the degree of closeness of the relationship between the signs;
3) Find an analytical expression of the relationship, that is, build a regression model;
4) Assess the adequacy of the model and give its interpretation.
In order for the results of the correlation analysis to give the desired result, certain requirements must be met in relation to the selection of the object of study and signs-factors. One of essential conditions the correct application of the methods of correlation analysis are the requirements of the one-sidedness of those objects that are being studied. Another important requirement that ensures the reliability of the conclusions of the correlation analysis is the requirement for a sufficient number of observations. Besides, great importance has a selection of factors influencing the performance indicator. The factors-signs included in the consideration should be as independent as possible from each other, since the presence of a close relationship between them indicates that they characterize the same aspects of the phenomenon under study and largely duplicate each other.
It should be noted that all the main provisions of the correlation analysis are developed on the assumption of the normal nature of the distribution of the considered features (random variables). In fact, we are faced with certain deviations from the initial assumptions. But this does not mean that the use of correlation analysis methods should be abandoned.
In correlation analysis, the following variants of dependencies are distinguished:
1) Pair correlation - the relationship between two signs (effective and factorial or two factorial);
2) Partial correlation - the relationship between the effective and one factor signs with fixed values of other factor signs;
3) Multiple correlation - the relationship between the effective and two or more factor characteristics.
End of work -
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All topics in this section:
Monitoring, summarizing and grouping statistics
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The statistical distribution of the sample
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Determining the size of the interval. Sturgess formula
The value of the interval is the difference between the largest and smallest values of the attribute in each group, called the boundaries of the interval.
Graphical display of statistical data
A graphical way of displaying statistical data is their conditional representation using points, lines, planes, geometric shapes and conventional signs. Graphs in statistics apply
Results of summary and grouping of materials statistical observation are drawn up in the form of statistical distribution series. The statistical distribution series is an ordered
Goals and objectives of studying the topic
study absolute and relative values; average values (the concept of an average value, the power mean formula, the geometric mean formula, the property of the majorance of the means, mode, median, pho
Absolute and relative values
As a result of statistical observation, summary and grouping of the collected statistical material, versatile information about the processes and phenomena under study was obtained. Summary data for the studied
Average values
The average value is a generalized characteristic of a set of homogeneous phenomena for any one quantitatively variable attribute. Averages play an important role
Indicators of trait variation
Variation in statistics is understood as such quantitative changes in the value of the trait under study within a homogeneous population, which are due to the intersecting influence of the action of different
There are two types of generalizing indicators that characterize the quantitative side of the phenomena and processes under study: absolute and relative. Absolute indicators - named numbers, them
Laws of distribution of random variables
Numerical characteristics of random variables
The distribution law fully characterizes the random variable. However, it is often unknown. In some cases, it is even more convenient to use numbers that describe a random variable in total. Such chi
Economic indicators, as a rule, are random variables. A random variable is a variable that, as a result of an experiment (test), can take one and only one possibility.
Law of uniform density
In practice, there are continuous random variables, about which it is known in advance that their possible values lie within a certain certain interval. Moreover, it is known that within
exponential distribution
The exponential (exponential) is the probability distribution of the value X, which is described by the density
Normal distribution law
normal law distribution (Gaussian law) is characterized by density.
Truncated distribution laws
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Types and forms of correlation relationships between phenomena
Before proceeding to the study of the relationship between phenomena, it is necessary to find out the type of relationship between the factor and resultant features. In statistics, a functional relationship and stochastic dependence are distinguished.functional call such a connection in which a certain value factor attribute corresponds to only one value of the resultant attribute. If a causal dependence does not appear in each individual case, but in general, on average, with a large number of observations, then such a dependence is calledstochastic . A special case of a stochastic connection iscorrelation a connection in which the change in the average value of the effective attribute is due to the change in factor signs.
Depending on the direction of action, a connection is distinguisheddirect and reverse . With a direct connection, the direction of change in the resulting attribute coincides with the direction of the attribute-factor, i.e. with an increase in the factor attribute, the resultant one also increases and vice versa.
According to the analytical expression (form), the connections can berectilinear and curvilinear. With a straight-line relationship with an increase in the value of the factor attribute, there is a continuous increase or decrease in the value of the effective attribute. Mathematically, such a relationship is represented by the equation of a straight line y \u003d a o + a 1 x, and graphically - a straight line.
With a curvilinear relationship with an increase in the value of the factor attribute, the increase (or decrease) of the effective attribute occurs unevenly or its direction reverses. Geometrically, such connections are represented by curved lines (hyperbola, parabola, etc.).
Another important characteristic of relationships is in terms of interacting factors. If a relationship between two features is characterized, then it is called steam room. If more than two variables are being studied - multiple.
Relationships between phenomena established on the basis of theoretical analysis can be studied, measured and quantified using various statistical methods. Balance and index methods are used to study functional relationships. To study the correlations between attributive features - the method of mutual contingency, for quantitatively varying features - the method of parallel series, graphical method, method of analytical groupings, correlation and regression analysis.
2. Pair correlation and pair regression
In the most general form, the task of statistics in the field of studying relationships is to quantify their presence and direction, as well as to characterize the strength and form of influence of some factors on others. The tasks of the correlation analysis itself are reduced to measuring the closeness of the relationship between varying features, determining unknown causal relationships, and evaluating the factors that have the greatest impact on the resulting feature. The tasks of regression analysis lie in the field of establishing the form of dependence, determining the regression function, using an equation to estimate unknown values of the dependent variable.
Paired regression characterizes the relationship between two features: resultant and factorial. The analytical relationship between them is described by the equations:
Straight at X = a O + a 1 X
Hyperbolas
Parabolas 
etc.
You can determine the type of equation by examining the dependence graphically. However, there are more general indications that allow you to identify the relationship equation without resorting to a graphical representation, if the resulting and factor signs increase equally, approximately arithmetic progression, then this indicates that the relationship between them is linear, and in the case of feedback, it is hyperbolic. If the factor sign increases in arithmetic progression, and the resultant sign increases much faster, then parabolic or power regression is used.
The evaluation of the parameters of the regression equations is carried out by the least squares method. The essence of this method is to find the parameters of the model, at which the sum of squared deviations of the empirical values of the resulting feature from the theoretical ones is minimized.
The systems of normal equations for finding the regression parameters have the form:
For a linear relationship 
Hyperbolas 
Parabolas 
The parameter a o in the regression equations is a constant value u, usually makes no economic sense. Other parameters at x are called regression coefficients, which show how many units on average change y when x changes by one unit.
Quantitatively, the dependence of the change in the theoretical value of y x on the change in x, which is expressed by the regression coefficients, is often more convenient to express in relative terms. To do this, calculate the coefficient of elasticity (E). It characterizes by how many percent increases in x when x increases by one percent and is calculated by the formula:
To quantify the tightness of the connection with a linear form, it is widely used linear correlation coefficient:
,
Where n is the number of observations.
The correlation coefficient takes values in the range from -1 to +1. It is generally accepted that if r0.3, then the connection is weak; at r=(0.3-0.7) - average; at r> 0.7 - strong, or close. When r= 1 - the connection is functional.
In the case of a linear and non-linear relationship between two features, the so-called correlation ratio or correlation index () is used to measure the closeness of the relationship. The correlation index is based on comparing the difference between two variances 
And 
. - dispersion, which measures deviations of actual (empirical) values (y) from theoretical ones (y x), and characterizes the residual variation due to other factors, Dispersion 
measures the variation due to the factor x.
The correlation index ranges from 0 to 1 and is suitable for measuring the closeness of the connection in any form. Moreover, by aligning the values of y with respect to different functions, it is possible, by the magnitude of the variance characterizing the residual variation 
to judge which function best aligns the empirical line of communication.
3. Multiple regression and correlation
The study of the relationship between two or more related features is called multiple (multifactorial) regression. When studying dependencies using multiple regression methods, the problem is formulated in the same way as when using paired regression, i.e. it is required to determine the analytical expression of the relationship between the resulting feature and factor features.
The most difficult problem is the choice of the form of communication. The difficulty lies in the fact that from an infinite set of functions it is required to find one that will express the real-life relationships between the studied indicators and factors better than others. The choice of the function type can be based on theoretical knowledge about the phenomenon under study or on the experience of previous similar studies. The form of communication can be determined by enumeration of functions different types. But in most practical cases, any function of many variables can be reduced to a linear form, i.e. the multiple regression equation can be built in a linear form:
Each coefficient of this equation shows the degree of influence of the corresponding factor on the analyzed indicator at a fixed position (at the average level) of the other factors: with a change in each factor by one, the indicator changes by the corresponding regression coefficient.
In case of inadequacy of the linear multiple regression equation, it is recommended to increase the order of the equation.
The problem of selecting factor features for building relationship models can be solved on the basis of heuristic or multivariate statistical methods of analysis.
Equation parameters can be determined by graphical method, least squares method, etc. For example, for a two-way linear least squares regression, you need to solve the following system of normal equations:
With the help of multivariate correlation analysis, various kinds of characteristics of the closeness of the relationship between the studied indicator and factors are found: paired, partial and multiple correlation coefficients, multiple coefficient of determination.
To study the closeness of the relationship between two of the variables under consideration (without taking into account their interaction with other variables), we use paired correlation coefficients. The methodology for calculating such coefficients is similar to the linear correlation coefficient.
^ Partial correlation coefficients characterize the degree of influence of one of the arguments on the function, provided that the remaining independent variables are fixed at a constant level. Depending on the number of variables whose influence is excluded, they can be of the first order (if the influence of one variable is excluded), the second order (if the influence of two variables is excluded), etc. For example, the partial correlation coefficient of the first order between features y and x 1 with the exclusion of the influence of x 2 is calculated by the formula:
Where r - paired correlation coefficients between the corresponding features.
An indicator of the closeness of the connection established between the effective and two or more factor characteristics is cumulative multiple correlation coefficient. In the case of a linear two-factor relationship, it can be calculated using the formula:
The value of R 2 is called the cumulative coefficient of multiple determination. It shows what proportion of the variation of the studied indicator is explained by the influence of the factors included in the multiple regression equation.
The values of R and R 2 are between 0 and 1.
In order to determine which of the factors has the greatest impact on the indicator under study, partial elasticity coefficients (E i) are calculated, with the help of which the difference in units of measurement is eliminated. They are calculated according to the formula:
4. Nonparametric methods for estimating the relationship
The methods of correlation and variance analysis can be applied when all the characteristics under study are quantitative. Meanwhile, in statistical practice one has to face the problems of measuring the relationship between qualitative features.
To determine the tightness of the connection between two qualitative features, each of which consists of only two groups, association and contingency coefficients are used. When studying the connection, the numerical material is placed in the form of contingency tables:
Table I
Table for calculating association coefficients and contingent
| A | V | a+b |
| With | d | c+d |
| a+c | c+d | a+b+c+d |
The coefficients are determined by the formulas:
associations 
contingents 
The contingency coefficient is always less than the association coefficient. The connection is considered confirmed if K a 0.5 or K k 0.3.
When each of the qualitative features consists of more than two groups, then to determine the tightness of the connection, it is possible to use the Pearson's (C) and Chuprov's (K) mutual contingency coefficient:
where 2 - the indicator of root-mean-square contingency, determined by subtracting one from the sum of the ratios of the squares of the frequencies of each cell of the table to the product of the frequencies of the corresponding column and row;
K is the number of groups for each of the signs.
The value of the C and K coefficients ranges from 0 to 1. The Chuprov coefficient usually gives a more cautious estimate of the connection.
^ TOPIC 8. STATISTICAL INDICATORS OF PRODUCTS,
WORKFORCE AND EFFICIENCY
PRODUCTION
I. Statistical accounting industrial products
^ Under production industry understand the direct useful result of the industrial and production activities of enterprises, expressed either in the form of products or in the form of works and services of an industrial nature.
In order to correctly reflect the composition and volume of industrial products produced in each period, it is necessary to distinguish between the stages of its readiness. After the object of labor has entered the first stage of its processing and living labor has been applied to it, the initial degree of product readiness is formed - unfinished production. An object of labor that has passed all the necessary operations in the process of processing within a given workshop, but is subject to subsequent processing in other workshops, is called semi-finished product. A product that is completely finished by processing within a given enterprise - ready product.
The result of the activity of the enterprise can take the form of a new consumer value, be the result of the transformation of the object of labor into new form product and the result of the activity may be the restoration of a completely or partially lost use value due to wear and tear of a previously created thing (repair). This form of performance industrial enterprise is called industrial works.
To ensure the correct accounting of products, it is necessary to have a firmly established nomenclature and units of measurement. Accounting can be carried out in natural, conditionally natural and cost meters.
In the theory and practice of planning, accounting and statistics at various stages of the production process, a number of interrelated indicators of the volume of industrial output in monetary terms are used.
The cost of the total volume of products produced for a certain period by all industrial production departments of the enterprise is called gross production turnover. Part of the gross turnover is the so-called internal turnover- is the value of products produced by some and consumed by other shops of the enterprise during the same period.
An indicator characterizing overall result industrial and production activities of the enterprise for a given period in monetary terms, is called gross output according to the factory method.
The value of the gross output of an industrial enterprise can be determined in two ways. First, by excluding the cost of intra-factory turnover from the value of gross turnover. Secondly, by direct summation of the cost of finished products produced (minus those spent in the same period for industrial production needs), semi-finished products released to the side and industrial work performed on orders from outside, as well as the cost of changing the balance of semi-finished products and work in progress.
The final result of industrial and production activities, fully prepared in the reporting period for the release to the side, characterizes the volume indicator marketable products. The value of marketable output can be determined by summing up its constituent elements or by subtracting from the value of gross output the value of its internal elements.
^ Sold products represents the shipped products paid for in this period. At the same time, paid products can be shipped both in this and in previous periods.
2. Classification work force by economic activity
And status in employment
^ Economically active population (labor force) is the part of the population that provides the supply of labor for the production of goods and services. The economically active population rate is the share of the economically active population in the total population.
TO busy includes persons of both sexes aged 16 years and over, as well as persons younger ages who during the period under review were employed, were temporarily absent from work for reasons permitted by labor law, or performed work without pay at a family enterprise.
The unemployed include persons aged 16 and over who during the period under review did not have a job (profitable occupation), were looking for work or were ready to start working. When referring to the unemployed, all three of these criteria must be met simultaneously.
^ Unemployment rate is the proportion of the unemployed in the economically active population.
Economically inactive population population that is not part of the labor force. This part of the population is represented by the following categories:
A) pupils and students, listeners and cadets attending daytime educational institutions;
b) persons receiving pensions;
c) persons engaged in housekeeping, caring for children, the sick, etc.;
D) people who are desperate to find a job;
E) other persons who do not need to work, regardless of the source of income.
Classification by status in employment involves the division of the economically active population into employees; self-employed persons and employers. Employees, in turn, are divided into two subgroups - civilian population and military personnel, as well as by the duration of employment for permanent, temporary, seasonal workers, as well as workers hired for casual work.
3. Indicators of employment and employment of the population
With the birth of the labor market in statistical reporting, information about the unemployed appeared, the number of which can be characterized both by absolute and relative indicators.
The absolute number of unemployed is given as a momentary indicator at the beginning of each month. Dynamics are noted within the monthly cycle: how many unemployed are deregistered, employed, issued for early retirement, directed to professional education, employed after completion of vocational training.
The qualitative composition of the unemployed is characterized by gender, level of education, and place of residence.
Relative indicators include the percentage of unemployed in the total number of unemployed able-bodied citizens registered with the employment service, and the percentage of those receiving unemployment benefits.
In world practice, the unemployment rate is calculated by the formula:
To quantify employment of the population, statistics uses special indicators, absolute and relative. The absolute indicators include the number of people employed in the national economy; distribution of employees by spheres and sectors of the economy, gender, age, level of education; the number of people of working age employed in the national economy, etc.
Relative indicators include: population employment rate:
- 
Employment rate of labor resources 
Employment rate of the working age population
Employment rate of the able-bodied population of working age 
Where S z.n.- the number of employed people;
S - total population;
TR- the number of labor resources;
S TV - population of working age;
S TNTV - the number of able-bodied population of working age.
4. Balance of labor resources
The system of balances of labor resources is a series of interrelated tables that characterize the processes of reproduction and use of the labor resources of the country and its individual territories in specific conditions of social development.
The balance of labor resources for the year is compiled in average annual employees and is detailed. It contains the most important groupings of labor resources by spheres of production and sectors of the economy.
The main indicator of the resource part of the balance sheet is the population of working age. Working age limits are regulated by labor legislation. In Russia, the working-age population includes women aged 16-54 and men aged 16-59. But since only the able-bodied population is included in the labor force, the population of working age should be reduced by the number of non-working disabled people of I and II groups of working age and the number of non-working pensioners of working age who receive an old-age pension on preferential terms. The labor force includes persons of retirement age who continue to work.
Taking into account the fact that when determining the number of unemployed, pensioners who are looking for work and ready to start work are also included in the composition of the unemployed, this category of persons is also included in the composition of the labor force. The composition of the labor force also includes persons under 16 years of age employed in the economy.
The expenditure part of the balances provides for the distribution of labor resources by types of employment and sectors of the economy. The analytical possibilities of the balance of labor resources are expanding as a result of the distribution of employees among enterprises of various forms of ownership and those employed in the field of private entrepreneurship.
5. Indicators of the use of working time,
Funds of working time
Working time is a part of the calendar time spent on the production of products or on the performance of a certain type of work. In statistical practice, the man-day and man-hour serve as the unit for the use of working time.
Spent A man-day is considered for an employee such a day when he appeared and started work, regardless of its duration, incl. days spent on business trips.
Accounting for working time in man-days does not allow revealing the loss of working time within a working day, therefore, it is also recorded in man-hours. Worked man-hour count the hour of actual work of one person.
According to the accounting of working time in man-days, the working time funds are determined. There are calendar, personnel and maximum possible time funds. calendar fund consists of the number of person-days of appearances and absences. If we subtract from it the number of person-days of absenteeism on holidays and weekends, we get personnel fund, and excluding the number of man-days of paid annual leave - maximum possible fund working time.
The degree of use of one or another working time fund is determined using coefficients determined by the ratio of the number of man-days worked to the corresponding fund.
According to the accounting of working time in man-days and man-hours, the following indicators of the use of working time are calculated: - average actual length of the working day:
The average number of days of work per one listed worker;
the average number of hours worked per listed worker.
6. Main indicators and methods of calculation
labor productivity
Labor productivity means the fruitfulness, productivity of people's activities. In economic practice, the level of labor productivity is characterized through indicators of output and labor intensity. output (W) products per unit of time is measured by the ratio of the volume of output (q) and the cost (T) of working time: W \u003d q: T. The inverse indicator is laboriousness: t=T:q.
System statistical indicators labor productivity is determined by the unit of measurement of the volume of products produced. Accordingly, natural, conditionally natural, labor and cost methods are used to measure the level and dynamics of labor productivity.
Depending on how labor costs are measured, there are average hourly (W r), average daily (W g), and average monthly output (W). They are obtained by dividing the volume of manufactured products, respectively, by the number of man-hours worked during a given period of time; the number of man-days worked by all the workers of the enterprise; the average number of workers (employees).
There is a relationship between the indicators of the average hourly output of workers and the indicators of their use of working time:
To get an idea of the average monthly (quarterly, annual) output of one worker of industrial and production personnel, one more factor must be introduced - the share of workers in the average headcount of the PPP (d p) . Then:
W=W r TDd p .
Based on this dependence, factor analysis labor productivity index method.
Labor productivity is studied on different levels- from individual labor productivity to the productivity of social labor in the national economy of the whole country as a whole:
The dynamics of labor productivity, depending on the method of measuring its level, is analyzed using statistical indices: natural (I), labor (2, 3) and cost
(4):
To analyze the change in average output under the influence of a number of factors, a system of indices of average values is used, in which output is the indexed value, and the share in total labor costs is the weight.
7. Product cost indicators
^ Production cost -expressed in monetary terms, the costs of enterprises for the production and sale of products. Thisone of the most generalizing indicators characterizing the efficiency of the enterprise.
In the practice of planning, accounting and statistics, two main types of production costs are distinguished:production , covering only the costs associated with the production process andcomplete , including the cost of production and the costs associated with the storage and marketing of products.
According to the economic content, production costs are divided into those associated with the use of living labor, means of labor and objects of labor, and they are taken into account separately for these economic elements.
According to the nature of the connection with the production process, they distinguishmain costs directly related to the production process, andinvoices associated with the process of organizing and managing production. The main expenses are usually calledvariables , i.e. changing in proportion to the growth of output, invoices -conditionally permanent .
To study the cost of production, the main statistical methods are used: groupings, average and relative values, graphic, index, and also the method of comparison.
The most important groupings in the study of cost are:
I) grouping the costs of production by economic elements (What is spent on the production of products?);
2) grouping of production costs according to costing items (Where is it spent?);
3) grouping by costs that occupy the largest share in total costs (labor-intensive, material-intensive, energy-intensive, capital-intensive);
4) by types of production costs (technological, production, workshop, complete);
5) grouping depending on the method of attributing costs to the cost price (indirect and direct);
6) grouping according to the volume of production (proportional, non-proportional).
The method of average and relative values is used in calculating average cost levels for homogeneous products, in studying the structure and dynamics of cost.
The graphical method allows you to visualize the cost structure, the changes taking place in it, as well as the dynamics of its components.
The index method is necessary for a summary description of the dynamics of the cost of comparable and all commercial products, to study the dynamics and identify the influence of individual factors on it.
8. Analysis of the structure and dynamics of production costs
The analysis of production costs is carried out by comparing the proportion of actual costs by elements with planned data or with data for the previous (reporting) period. When analyzing production costs by elements, it must be borne in mind that indicators for the previous period are taken without recalculation for the volume and range of products actually produced in the reporting period at current prices.
Having data on the unit cost of the product for the previous period (Z 0), according to planned calculations (Z pl) and for the reporting period (Z 1), we can give general characteristics the degree of fulfillment of the planned task to reduce the cost and its dynamics, as well as to determine the absolute amount of savings or overspending as a result of changes in the cost.
In this case, individual cost indices will be determined:
Scheduled Job Index 
Cost dynamics index 
The given indices are interconnected:
The total amount of savings (overspending) from a change in the cost of the product is determined by the formula 
.
Subtracting the planned savings from the actual savings, we get the above-planned savings (overspending):
When studying the dynamics of the cost price for a group of enterprises manufacturing products of the same type, indices of variable composition, constant composition and structural changes are used.
At those enterprises where different types of products are manufactured and comparable products predominate in the total output, indicators of reducing the cost of comparable commercial products are calculated. Comparable products include products that were produced in the reporting and previous periods. In this case, the following three indices are used:
Scheduled Job Index 
Plan task execution index 
Index of actual change in the cost of comparable commercial products 
The difference between the numerator and denominator of these indices characterizes the corresponding changes in the cost of comparable marketable products in absolute terms.
9. Statistics of the financial activity of the enterprise.
Profit and profitability indicators
The subject of study of enterprise finance statistics is a quantitative description of their financial and monetary relations, taking into account their qualitative features, due to the formation, distribution and use of financial resources, the fulfillment of obligations of economic entities to each other, to the financial and banking system and the state.
The final financial result of the enterprise is the balance sheet profit (loss). Balance sheet profit is the sum of profit from the sale of products (works, services), profit (or loss) from other sales, income and expenses from non-sales operations.
^ Profit from sales products is defined as the difference between the proceeds from the sale of products at the wholesale prices of the enterprise (excluding VAT) and its full cost.
^ Net income is the profit remaining at the disposal of the enterprise. It is defined as the difference between the taxable balance sheet profit and the value of taxes, taking into account benefits.
Profit indicators characterize the absolute efficiency of the economic activity of the enterprise. Along with this absolute assessment, relative indicators of economic efficiency are also calculated - indicators of profitability. Depending on what indicators are used in the calculations, there are several indicators of profitability. Their numerator is usually one of three values: profit from sales (PR), balance sheet profit (PB) or net profit (NP). In the denominator - one of the following indicators: the cost of production of sold products (Z etc ), production assets 
, gross income, equity, etc.
Distinguish:
Profitability of production balance sheet (total) 
Profitability of sold products 
Product profitability 
And other types of it.
In the process of analyzing the influence of various factors on the profit from sales of products, which has the largest share in the structure of balance sheet profit, calculations are made according to the following formulas.
1) Impact of price changes (tariffs):
2) The impact of changes in the cost of goods sold:
3) The impact of changes in the volume of sales of products:
4) The impact of changes in the structure of products sold:
where PR' - actual profit of the reporting period at prices and cost of the previous period.
The influence of various factors on the change in the profitability of production according to the method of factorial index analysis is carried out according to the following model:
Where a \u003d IIB: ПР - coefficient of change in balance sheet profit;
b \u003d PR: Z pr - profitability of sold products;
in \u003d W pr : 
- the turnover ratio, calculated on the basis of the total cost of goods sold;
r = 
- the share of working capital in the total cost of production assets.
^ TOPIC 9. STATISTICAL ESTIMATION OF THE ECONOMIC
COUNTRY DEVELOPMENT
1. Statistics of national wealth and national property
national wealth- this is a set of material resources, accumulated products of past labor and natural resources taken into account and involved in the economic turnover, which society has at a certain point in time.
National wealth statistics solves problems related to the development of a system of indicators and the justification of the methodology for their calculation, as well as the tasks of the practical organization of statistical observation and processing of the information received.
The indicator system of national wealth statistics used in the analysis includes the following main characteristics:
1) the presence (volume) and structure of wealth;
2) reproduction of its most important parts;
3) the dynamics of all wealth and its constituent elements;
4) distribution of wealth on the territory of the country;
5) protection natural resources and their replenishment.
Using this system, it is possible to characterize changes in the volume and composition of all wealth from various angles by constructing appropriate groupings, series of dynamics, calculating indices and compiling a balance of national wealth and its individual parts.
National wealth statistics as a whole is constructed as a statistic accumulated wealth and natural resource statistics. The accumulated wealth is in the form of a set of material goods for various purposes and uses.
The grouping of the elements of wealth according to the characteristics of their circulation (fixed production assets, circulating production assets, etc.) and according to the natural material composition, depending on the role they play or can play in the reproduction process, is widely used. Of particular interest is the distribution of wealth by forms of ownership and social groups of the population, by economic regions and territories, as well as by branches of material production and the non-productive sphere.
With the transition to the system of national accounts, it becomes of particular importance perpetual inventory method. The advantage of this method is that it is designed to estimate the value of real wealth.
2. Indicators of statistics of fixed production assets
^ Fixed assets participate repeatedly in the production process and transfer their value to the finished product in parts in the form of depreciation.
The most important tasks of the statistical study of fixed assets are:
1) establishing the availability and study of the composition of fixed assets;
2) study of the state, movement and use of fixed production assets;
3) study of the armament of labor by the main production assets.
The presence of both fixed assets as a whole and their individual types can be characterized by momentary and average indicators. When studying the composition of fixed assets, various types of their groupings are used. This is primarily their division into production and non-production, by industry National economy, as well as according to their common species classification.
To analyze the dynamics and structure of fixed assets, develop their balance sheets and determine the effectiveness, it is necessary to distinguish between the types of valuation of fixed assets (full initial, residual value, full replacement, restoration, taking into account depreciation).
The most complete picture of the availability and dynamics of fixed assets is given by balance of fixed assets. Such a balance, along with data on the availability of fixed assets at the beginning and end of the reporting period, contains data on their receipt from various sources and on their disposal for various reasons. It can be compiled both for all fixed assets, and for their individual types, either at the full initial cost, or at the residual. Balance sheets are compiled for enterprises, industries and the national economy as a whole.
To characterize the intensity of movement of fixed assets, the following indicators are calculated:
1) The general receipt coefficient shows the share of all received (P) in the reporting period of fixed assets in their total volume at the end of this period:
2) The coefficient of retirement of fixed assets, equal to the ratio of the value of all fixed assets retired for a given period (B) to the value of fixed assets at the beginning of this period 
3) The depreciation coefficient of fixed assets is calculated on a certain date as a percentage of the ratio of the amount of depreciation of fixed assets (I) to their total cost 
4) The coefficient of usefulness of fixed assets, defined as the difference between 100% and the depreciation coefficient.
The general indicator of the use of fixed production assets is the return on assets - the ratio of the volume of products produced in a given period (O) to the average value of fixed production assets over this period: FD = 0 / F.
To quantify output at the level of individual enterprises and industries, its volume is used, and for the national economy as a whole, the national income or the total social product.
Along with capital productivity, its inverse indicator is used - capital intensity: FE \u003d F / 0.
The capital-labor ratio has a great influence on the value of capital productivity and capital intensity: FV \u003d F / T
Where T is the number of workers or employees.
The stock-labor ratio can be defined as a momentary indicator (as of a certain date) or as an interval indicator (for a certain period).
The capital-labor ratio and capital productivity are interconnected through the indicator of labor productivity, determined by the formula PT \u003d 0 / T. This dependence has the form: PT = FO FV.
The effect of improving the use of fixed assets can be determined by various statistical methods, primarily index.
When analyzing the dynamics of the average indicators of the use of fixed assets for a set of enterprises, their values depend not only on the corresponding indicators at each enterprise, but also on changes in the structure. The system of indices for determining the impact of structural shifts on capital productivity for a group of enterprises is as follows:
Capital productivity index of variable composition 
permanent staff 
structural changes 
Where dФ is the share of the value of fixed assets of the i-th enterprise in their total value for the group of enterprises.
Determination of the impact of changes in capital productivity and the value of fixed assets on the change in the volume of production by the index method is carried out according to the following structural model: 0= FD F, i.e.
As a result As a result
Change in output = change + change in quantity
Return on assets of fixed assets
In relative terms: 
In absolute terms:
By company
By group of enterprises
Similarly, the index method establishes the influence of changes in other indicators of the use of fixed assets, for example, the influence of the degree of use of fixed assets on their total need is established according to the following structural dependence: F = FU 0.
3. Indicators of the volume, structure and use of reserves
material values
In the statistical literatureresources most often, material values are implied, including raw materials, materials, fuel, semi-finished products used to meet production and operational needs and capital construction.
Stocks of material assets are measured both in absolute terms and in days of average daily consumption. The amount of reserves is calculated in monetary terms or in kind in accordance with the accepted classification. The presence of reserves in monetary terms is characterized by momentary and average indicators.
^ Average stocks can be determined by the formulas of the arithmetic mean (simple or weighted) or chronological mean.Supply of the enterprise with reserves in days is calculated by dividing the size of stocks of material assets by the average daily consumption of this type of stock.
The structure of material resources is characterized by the relative values of the share of each type of reserves in accordance with the established classification.
To characterize the efficiency of resource use at the level of the national economy, the generalizing indicator is material intensity of the national income, reflecting the amount of material costs spent on the production of one ruble of national income (gross national product), and for individual sectors of the production sector - for one ruble of gross or marketable output.
Specific consumption indices allow us to conclude what changes have occurred in specific consumption over the reporting period compared to the baseline or norm.
To characterize the use various kinds materials for the production of several types of products, a composite index of unit costs is used:
Where m 0 and m 1 are the specific costs of a given type of material for the production of each type of product in the base and reporting periods.
The difference between the numerator and denominator of this index shows savings (overruns) in material costs (in monetary terms) only due to changes in unit costs.
To characterize the use of inventory, the following indicators are used:
Turnover ratio (turnover rate)
K about \u003d R: Z
average turnaround time in days
fixing factor K closed = 3: Р
R - sales of products or services; 3 - volume of stocks.
Turnover indicators for a set of enterprises represent the average value of similar indicators for individual enterprises. At the same time, K about and K closed calculate
8.1. Basic concepts of correlation and regression analysis
Exploring nature, society, economy, it is necessary to take into account the relationship of observed processes and phenomena. At the same time, the completeness of the description is somehow determined by the quantitative characteristics of the cause-and-effect relationships between them. Evaluation of the most significant of them, as well as the impact of some factors on others, is one of the main tasks of statistics.
The forms of manifestation of interrelations are very diverse. As the two most common types allocate functional(complete) and correlation(incomplete) connection. In the first case, the value of the factor attribute strictly corresponds to one or more values of the function. Quite often, the functional connection is manifested in physics, chemistry. In economics, an example is the directly proportional relationship between labor productivity and an increase in production.
Correlation (which is also called incomplete, or statistical) appears on average, for mass observations, when the given values of the dependent variable correspond to a certain number of probable values of the independent variable. The explanation for this is the complexity of the relationships between the analyzed factors, the interaction of which is influenced by unaccounted random variables. Therefore, the relationship between the signs is manifested only on average, in the mass of cases. With a correlation, each value of the argument corresponds to randomly distributed values of the function in a certain interval.
For example, some increase in the argument will entail only an average increase or decrease (depending on the direction) of the function, while specific values for individual units of observation will differ from the average. These dependencies are ubiquitous. For example, in agriculture, this may be the relationship between yield and the amount of fertilizer applied. Obviously, the latter are involved in the formation of the crop. But for each specific field, plot, the same amount of applied fertilizers will cause a different increase in yield, since there are a number of other factors (weather, soil conditions, etc.) in interaction that form the final result. However, on average, such a relationship is observed - an increase in the mass of applied fertilizers leads to an increase in yield.
In the direction of communication, there are straight, when the dependent variable increases with the increase in the factor trait, and reverse, at which the growth of the latter is accompanied by a decrease in the function. Such relationships can also be called positive and negative, respectively.
Regarding their analytical form of communication, there are linear And non-linear. In the first case, on average, linear relationships appear between the signs. A non-linear relationship is expressed by a non-linear function, and the variables are interconnected on average non-linearly.
There is one more rather important characteristic of connections from the point of view of interacting factors. If a relationship between two characteristics is characterized, then it is called steam room. If more than two variables are being studied − multiple.
The above classification features are most often found in statistical analysis. But in addition to the above, there are also direct, indirect And false connections. Actually, the essence of each of them is obvious from the name. In the first case, the factors interact directly with each other. An indirect relationship is characterized by the participation of some third variable, which mediates the relationship between the studied traits. A false connection is a connection established formally and, as a rule, confirmed only by quantitative estimates. It does not have a qualitative basis or is meaningless.
They differ in strength weak And strong connections. This formal characteristic is expressed by specific values and is interpreted in accordance with generally accepted criteria for the strength of the connection for specific indicators.
In the most general form, the task of statistics in the field of studying relationships is to quantify their presence and direction, as well as to characterize the strength and form of influence of some factors on others. To solve it, two groups of methods are used, one of which includes the methods of correlation analysis, and the other - regression analysis. At the same time, a number of researchers combine these methods into a correlation-regression analysis, which has some grounds: the presence of a number of common computational procedures, complementarity in interpreting the results, etc.
Therefore, in this context, we can talk about correlation analysis in the broad sense - when the relationship is comprehensively characterized. At the same time, there are correlation analysis in the narrow sense - when the strength of the connection is studied - and regression analysis, during which its form and the impact of some factors on others are evaluated.
Tasks proper correlation analysis are reduced to measuring the closeness of the relationship between varying traits, identifying unknown causal relationships and assessing the factors that have the greatest impact on the resulting trait.
Tasks regression analysis lie in the field of establishing the form of dependence, determining the regression function, using an equation to estimate unknown values of the dependent variable.
The solution of these problems is based on appropriate techniques, algorithms, indicators, the use of which gives reason to talk about the statistical study of relationships.
It should be noted that traditional methods correlations and regressions are widely represented in various kinds of statistical software packages for computers. The only thing left for the researcher is to properly prepare the information, choose a software package that satisfies the requirements of the analysis, and be ready to interpret the results obtained. There are many algorithms for calculating communication parameters, and at present it is hardly advisable to carry out such a complex type of analysis manually. Computational procedures are of independent interest, but knowledge of the principles of studying the relationships, possibilities and limitations of certain methods of interpreting the results is a prerequisite for research.
Methods for assessing the tightness of the connection are divided into correlation (parametric) and non-parametric. Parametric methods are based on the use, as a rule, of normal distribution estimates and are used in cases where the population under study consists of quantities that obey the normal distribution law. In practice, this position is most often taken a priori. Actually, these methods are parametric and are commonly called correlation methods.
Nonparametric methods do not impose restrictions on the law of distribution of the studied quantities. Their advantage is also the simplicity of calculations.
8.2. Pair Correlation and Pair Linear Regression
The simplest technique for identifying a relationship between two features is to build correlation table:
| \ Y \ X\ |
Y 1 | Y2 | ... | Yz | Total | Y i |
| x1 | f 11 | 12 | ... | f 1z | ||
| x1 | f 21 | 22 | ... | f2z | ||
| ... | ... | ... | ... | ... | ... | ... |
| X r | f k1 | k2 | ... | fkz | ||
| Total | ... | n | ||||
| ... | - |
The grouping is based on two traits studied in the relationship - X and Y. Frequencies f ij show the number of corresponding combinations of X and Y. If f ij are arranged randomly in the table, we can talk about the absence of a relationship between the variables. In the case of the formation of any characteristic combination f ij, it is permissible to assert a connection between X and Y. In this case, if f ij is concentrated near one of the two diagonals, there is a direct or reverse linear relationship.
A visual representation of the correlation table is correlation field. It is a graph where X values are plotted on the abscissa axis, Y values are plotted along the ordinate axis, and the combination of X and Y is shown by dots. By the location of the points, their concentration in a certain direction, one can judge the presence of a connection.
In the results of the correlation table for rows and columns, two distributions are given - one for X, the other for Y. Let's calculate for each X i the average value of Y, i.e. , How
The sequence of points (X i , ) gives a graph that illustrates the dependence of the average value of the effective feature Y on the factor X, - empirical regression line, showing how Y changes as X changes.
In essence, both the correlation table, and the correlation field, and the empirical regression line previously characterize the relationship when the factor and resultant features are selected and it is required to formulate assumptions about the form and direction of the relationship. At the same time, a quantitative assessment of the closeness of the connection requires additional calculations.
In practice, to quantify the tightness of the connection, the linear correlation coefficient. It is sometimes referred to simply as the correlation coefficient. If the values of the variables X and Y are given, then it is calculated by the formula
You can use other formulas, but the result should be the same for all calculation options.
The correlation coefficient takes values in the range from -1 to + 1. It is generally accepted that if |r| < 0,30, то связь слабая; при |r| = (0.3÷0.7) – average; at |r| > 0.70 - strong, or close. When |r| = 1 – functional connection. If r takes a value near 0, then this gives grounds to speak about the absence of a linear relationship between Y and X. However, in this case, a nonlinear interaction is possible. which requires additional verification and other meters discussed below.
To characterize the influence of changes in X on the variation in Y, regression analysis methods are used. In the case of a paired linear dependence, a regression model is built
where n –
number of observations;
a 0 , a 1 – unknown parameters of the equation;
e i is the error of the random variable Y.
The regression equation is written as
where Y itheor is the calculated equalized value of the effective feature after substitution into the equation X.
The parameters a 0 and a 1 are estimated using procedures, the most widely used of which is least square method. Its essence lies in the fact that the best estimates for ag and a are obtained when
those. the sum of the squared deviations of the empirical values of the dependent variable from those calculated using the regression equation should be minimal. The sum of squared deviations is a function of the parameters a 0 and a 1 . Its minimization is carried out by solving the system of equations
You can use other formulas that follow from the least squares method, for example:
The linear regression apparatus is quite well developed and, as a rule, is available in a set of standard programs for evaluating the relationship for a computer. The meaning of the parameters is important: and 1 is a regression coefficient that characterizes the effect that a change in X has on Y. It shows how many units Y will change on average when X changes by one unit. If a is greater than 0, then a positive relationship is observed. If a has a negative value, then an increase in X by one entails a decrease in Y on average by a 1 . The parameter a 1 has the dimension of the ratio Y to X.
Parameter a 0 is a constant value in the regression equation. In our opinion, it has no economic meaning, but in some cases it is interpreted as the initial value of W.
For example, according to the data on the cost of equipment X and labor productivity Y, the least squares method obtained the equation
Y \u003d -12.14 + 2.08X.
Coefficient a means that an increase in the cost of equipment by 1 million rubles. leads on average to an increase in labor productivity by 2.08 thousand rubles.
The value of the function Y \u003d a 0 + a 1 X is called the calculated value and forms on the graph theoretical regression line.
The meaning of theoretical regression is that it is an estimate of the mean value of the variable Y for a given value of X.
Pair correlation or pair regression can be considered as a special case of reflecting the relationship of some dependent variable, on the one hand, and one of the many independent variables, on the other. When it is required to characterize the relationship of the entire specified set of independent variables with the resultant attribute, one speaks of multiple correlation or multiple regression.
8.3. Assessing the significance of relationship parameters
Having obtained correlation and regression estimates, it is necessary to check them for compliance with the true parameters of the relationship.
Existing computer programs include, as a rule, several of the most common criteria. To assess the significance of the pair correlation coefficient, the standard error of the correlation coefficient is calculated:
As a first approximation, it is necessary that . The significance of r xy is checked by comparing it with , and one obtains
where t calc is the so-called calculated value of the t-criterion.
If t calc is greater than the theoretical (tabular) value of Student's t-test (t tabl) for a given level of probability and (n-2) degrees of freedom, then it can be argued that r xy is significant.
Similarly, based on the corresponding formulas, the standard errors of the parameters of the regression equation are calculated, and then the t-tests for each parameter. It is important to check again that the condition t calc > t tab. Otherwise, there is no reason to trust the obtained parameter estimate.
The conclusion about the correct choice of the type of relationship and the characteristic of the significance of the entire regression equation is obtained using the F-criterion, calculating its calculated value:
where n is the number of observations;
m is the number of parameters of the regression equation.
F calc should also be greater than F theor at v 1 = (m-1) and v 2 = (n-m) degrees of freedom. Otherwise, the form of the equation, the list of variables, etc., should be revised.
8.4. Nonparametric Methods for Estimating Relationships
The methods of correlation and variance analysis are not universal: they can be applied if all the characteristics under study are quantitative. When using these methods, one cannot do without calculating the main distribution parameters (averages, variances), so they are called parametric methods.
Meanwhile, in statistical practice, one has to deal with the problems of measuring the relationship between qualitative features, to which parametric methods of analysis in their usual form are not applicable. Statistical science has developed methods that can be used to measure the relationship between phenomena without using the quantitative values of the attribute, and hence the distribution parameters. Such methods are called nonparametric.
If the relationship of two qualitative features is studied, then the combinational distribution of population units is used in the form of the so-called cross-link tables.
Let's consider the method of analysis of tables of mutual contingency on specific example social mobility as a process of overcoming the isolation of certain social and professional groups of the population. Below is the data on the distribution of secondary school graduates by spheres of employment with the allocation of similar social groups of their parents.
The distribution of frequencies in the rows and columns of the cross-coupling table makes it possible to identify the main patterns of social mobility: 42.9% of the children of parents in group 1 (“Industry and construction”) are employed in the field of intellectual labor (39 out of 91); 38.9% of children. whose parents work in agriculture, work in industry (34 out of 88), etc.
One can also notice a clear heredity in the transfer of professions. Thus, out of those who came to agriculture, 29 people, or 64.4%, are children of agricultural workers; more than 50% in the field of intellectual labor have parents of the same social group etc.
However, it is important to obtain a generalizing indicator that characterizes the closeness of the relationship between the features and allows you to compare the manifestation of the relationship in different populations. For this purpose, for example, conjugacy coefficients Pearson (C) and Chuprov (C):
where f 2 is the root-mean-square contingency index, determined by subtracting one from the sum of the ratios of the squares of the frequencies of each cell of the correlation table to the product of the frequencies of the corresponding column and row:
K 1 and K 2 - the number of groups for each of the signs. The value of the coefficient of mutual contingency, reflecting the closeness of the relationship between qualitative features, fluctuates within the usual range for these indicators from 0 to 1.
In socio-economic studies, there are often situations when a feature is not expressed quantitatively, but the units of the population can be ordered. Such ordering of units of the population according to the value of the attribute is called ranking. Examples can be the ranking of students (pupils) according to their abilities, any set of people according to the level of education, profession, ability to be creative, etc.
When ranking, each unit of the population is assigned rank, those. serial number. If the value of the attribute is the same for different units, they are assigned a combined average serial number. For example, if the 5th and 6th units of the population have the same values of features, both will receive a rank equal to (5 + 6) / 2 = 5.5.
The relationship between ranked features is measured using rank correlation coefficients Spearman (r) and Kendall (t). These methods are applicable not only for qualitative, but also for quantitative indicators, especially with a small volume of the population, since non-parametric methods of rank correlation are not associated with any restrictions on the nature of the distribution of the trait.
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