Determine the variance and standard deviation of crop areas by using the method of moments to calculate the arithmetic mean and variance.

The distribution of commercial banks by assets is characterized by the following data:

Determine the total variance in two ways:

1. according to the method of moments.

Data on labor productivity of the three shops of the textile industry are characterized by the following data:

Compare the variation in labor productivity in these shops, draw conclusions.

Company turnover Catering per employee per quarter is characterized by the following data:

Determine for each enterprise: the coefficient of variation and compare the variation in the turnover of public catering in these enterprises. Draw your own conclusions.

The average value of the trait in the population is 20, and the mean square of the individual values ​​of this trait is 400. Determine the coefficient of variation.

The variance of a feature is 10, the mean square of its individual values ​​is 140. What is the mean?

The average value in the population is 16, the standard deviation is 8. Determine the average square of the individual values ​​of this feature.

The average square of the deviations of the individual values ​​of a trait from their average value is 100, and the average is 15. Determine what is the average square of the deviations of the individual values ​​of a trait from the value equal to 10 and 25.

The average value of the feature is 14, and the variance is 60. Determine the mean square of the deviations of the feature options from 19.

The average square of deviations of the trait options from an arbitrary value is 300, and the arbitrary value itself is 70 units. Determine the variance of a feature if it is known that the mean value of its variant is 80.

The average square of deviations of the trait options from an arbitrary value is 61. The average value of the trait is greater than the arbitrary value by 6 units and is equal to 10. Find the coefficient of variation.

The following data are available on the balance sheet profit of enterprises for two quarters:

Define:

the average of the intra-group, inter-group and total dispersion of the balance sheet profit of the enterprise;

coefficient of determination and empirical correlation.

Draw your own conclusions.

The distribution of families of employees of a financial corporation by the number of children is characterized by the following data:

Define:

intragroup dispersions;

intergroup dispersion;

overall variance.

Check the correctness of the calculations made using the rule of addition of variances and calculate the empirical correlation ratio.

The distribution of the cost of products intended for export deliveries at enterprise prices is characterized by the following data:

Define:

intrashop dispersions of a share;

the average of intrashop dispersions;

intergroup dispersion;

overall variance.

Check the correctness of the calculations made using the rule for adding variances of the share.

Below are data on individual dairy farms of the economy on the total number of cows and the number of dairy cows as of July 1, 2009:

Define:

dispersion of the share of dairy cows in the total number of cows by individual dairy companies;

middle of intra-group variances;

intergroup dispersion;

total variance in the proportion of dairy cows by farming generally.

Check the correctness of the calculations using the rule of addition of variances.

Distribution construction firms in terms of investment is characterized by the following data:

Define distribution characteristics:

1. coefficient of variation and asymmetry.

Draw conclusions about the nature of the distribution of construction firms.

The distribution of city families by the number of children is characterized by the following data:

Determine the coefficients of skewness and kurtosis using the central moments of the first four orders. Draw conclusions about the nature of the distribution of families.

According to task 6, determine the characteristics of the distribution:

2. standard deviation;

   coefficient of variation and Pearson's skewness.

Draw conclusions about the nature of the distribution of trade.

According to task 17, determine the indicators of asymmetry and kurtosis of the distribution of commercial banks by asset size. Draw your own conclusions.

In the study of the labor activity of employees of the organization (man-days worked out per year), average values ​​and central moments were obtained:

Using indicators of asymmetry and kurtosis, compare the nature of the distribution of men and women by labor activity. Draw your own conclusions.

According to a sample study of households on the number of cohabitation of their members, the following data were obtained:

Determine Pearson's skewness and normalized moments of the 3rd and 4th order. Draw your own conclusions.

Based on the data of Problem 14, determine Pearson's goodness of fit criterion (χ2) and check the closeness of the empirical and theoretical distributions of the number of unemployed for 2009.

Using the data of problem 14, check the closeness of the empirical and theoretical distributions of the number of unemployed for 2009 using Romanovsky and Kolmogorov's goodness-of-fit tests.

Topic 7. Statistical study of the relationship of socio-economic phenomena

7.1. Causality, regression, correlation

The study of objectively existing links between socio-economic phenomena and processes is the most important task of the theory of statistics. In the process of statistical study of dependencies, causal relationships between phenomena are revealed, which makes it possible to identify factors (signs) that have a major impact on the variation of the studied phenomena and processes. A causal relationship is such a connection between phenomena and processes, when a change in one of them - the cause - leads to a change in the other - the effect.

Financial and economic processes are the result of the simultaneous impact of a large number of causes. Therefore, when studying these processes, it is necessary to identify the main, main causes, abstracting from secondary ones.

At the heart of the first stage of the statistical study of communication is qualitative analysis associated with the analysis of the nature of a social or economic phenomenon by methods economic theory, sociology, concrete economics. The second stage is the construction of a communication model, based on statistical methods: groupings, averages, and so on. The third and last stage, the interpretation of the results, is again associated with the qualitative features of the phenomenon under study. Statistics has developed many methods for studying relationships. The choice of the method of studying the connection depends on the cognitive purpose and objectives of the study.

According to their essence and significance for the study of the relationship, signs are divided into two classes. Signs that cause changes in other related signs are called factorial, or simply factors. Signs that change under the influence of factor signs are called effective.

In statistics, functional and stochastic dependencies are distinguished. A functional relationship is such a relationship in which a certain value of a factor attribute corresponds to one and only one value of the effective 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 called stochastic. A special case of a stochastic connection is a correlation, in which the change in the average value of the effective attribute is due to a change in factor signs.

Relationships between phenomena and their features are classified according to the degree of tightness, direction and analytical expression.

According to the degree of closeness of communication, they distinguish (Table 31):

Table 31

Quantitative criteria for assessing the closeness of the relationship

In direction, there is a direct and reverse relationship. A direct link is a relationship in which with an increase or decrease in the values ​​of the factor attribute, an increase or decrease in the values ​​of the effective attribute occurs. Thus, the growth of production volumes contributes to an increase in the profit of the enterprise. When feedback the values ​​of the effective attribute change under the influence of the factor attribute, but in the opposite direction compared to the change in the factor attribute, that is, the inverse is the relationship in which, with an increase or decrease in the values ​​of one attribute, there is a decrease or increase in the values ​​of another attribute. Thus, a decrease in the unit cost of production results in an increase in profitability.

According to the analytical expression, rectilinear (or simply linear) and non-linear connections are distinguished. If a statistical relationship between phenomena can be approximately expressed by a straight line equation, then it is called a linear relationship of the form:

If the connection can be expressed by the equation of any curved line, then such a connection is called nonlinear or curvilinear, for example:

parabolas -

hyperbole -

To identify the presence of a connection, its nature and direction in statistics, the following methods are used: bringing parallel data; graphic; analytical groupings; correlations, regressions.

The method of bringing parallel data is based on the comparison of two or more series of statistical values. Such a comparison allows you to establish the presence of a connection and get an idea of ​​its nature.

Graphically, the relationship of two features is depicted using the correlation field. In the coordinate system, the values ​​of the factor attribute are plotted on the abscissa axis, and the resulting attribute is plotted on the ordinate axis. Each intersection of lines drawn through these axes is indicated by a dot. In the absence of close connections, there is a random arrangement of points on the graph. The stronger the connection between the features, the more closely the points will be grouped around a certain line expressing the form of the connection.

Rice. 25. Correlation field plot

In statistics, it is customary to distinguish between the following variants of dependencies:

Pair correlation - the relationship between two signs (effective and factorial, or two factorial).

Partial correlation - the relationship between the effective and one factor signs with a fixed value of other factor signs.

Multiple correlation - the dependence of the resultant and two or more factor characteristics included in the study.

Correlation analysis has as its task the quantitative determination of the tightness and direction of the connection between two signs (with a paired connection) and between the effective and the set of factor signs (with a multifactorial connection).

The tightness of the connection is quantitatively expressed by the value of the correlation coefficients, which, giving a quantitative characteristic of the tightness of the connection between the signs, allow us to determine the "usefulness" of the factor signs when constructing the multiple regression equation. The signs at the correlation coefficients characterize the direction of the connection between the features.

Regression is closely related to correlation and allows you to explore the analytical expression of the relationship between features.

Regression analysis consists in determining the analytical expression of the relationship, in which the change in one value (called the dependent or effective feature) is due to the influence of one or more independent values ​​(factorial features).

One of the problems in constructing regression equations is their dimension, that is, determining the number of factor features included in the model. Their number should be optimal. Reducing the dimension by eliminating secondary, unimportant factors makes it possible to obtain a model that is faster and better implemented. At the same time, the construction of a low-dimensional model may lead to the fact that it will not fully describe the phenomenon or process under study.

When building regression models, the following requirements must be met:

The totality of the initial data under study must be homogeneous and mathematically described by continuous functions.

Possibility of describing the simulated phenomenon by one or more equations of cause-and-effect relationships.

All factor signs must have a quantitative (numerical) expression.

The presence of a sufficiently large volume of the sample under study.

Workshop onstatistics ...

Task 3.1 - section Mathematics, Statistics ( general theory statistics) workshop The distribution of construction companies by investment is characterized by the following ...

Define:

a) the average volume of investments;

c) standard deviation;

d) coefficient of variation.

Draw your own conclusions.

End of work -

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All topics in this section:

Typical task 1
To analyze the fulfillment of the production norms of the enterprise, a 10% mechanical repeated sample survey was carried out, the results of which showed the following distribution of workers according to the fulfillment of

Solution
1) Let us determine the possible limits of the average fulfillment of production standards for the enterprise, i.e. confidence interval

Solution
The sample size is determined by the formula: , where t = 3 (at p = 0.997

Task 4.1
The data of a random repeated sample survey on the distribution of depositors by the size of deposits in the bank of the city are known:

Task 4.2
With a 30% mechanical non-repetitive sample of workers, the following initial data were obtained: Labor productivity, thousand rubles / person. Number of workers

Task 4.3
With a 25% mechanical non-repetitive sampling, enterprises were studied in terms of profitability: Enterprise Profit before taxation, thousand rubles.

Task 4.4
In order to study the performance of four types of machines performing the same operations, a 10% typical sample was made (within the groups, a random non-repetitive sampling method was used).

Task 4.5
The following data on the underweight of completed products in boxes weighing 20 kg were obtained during selective random selection: Underweight of 1 box, kg

Typical task 1
The output of the enterprise is characterized by the following data: Indicator January February March

Solution
1. a) Absolute growth: chain basic

Typical task 2
The following data on the sale of confectionery products are known trading company cities: Months

Solution
Let's apply the three-term moving average method to the initial data. The results of the calculations will be entered in the table. Months

Solution
First, we calculate the average value of revenue for two years for February:

Task 5.1
Known data on the sale of canned meat in one of the regions for 2003-2007:

Task 5.2
Using the relationship of dynamics indicators, determine the levels of a series of dynamics and the missing chain indicators: Year Production, million rubles.

Task 5.3
Known data on the volume of product sales: Years Sales volume, t

Task 5.6
Known data on the volume of sales of products for 2 years: Month Sales volume, tons base year

Task 5.7
Known data on the volume of sales of product "A" in the retail trade of the city for 3 years (t): Quarter

Task 5.8
Realization of confectionery products in city stores by months 2003-2006 are characterized by the following data: Month

Basic formulas for calculating individual and composite indices
Index name Index calculation formulas Individual index Aggregate index Average index

Typical task 1
Known data on the sale of goods in one of the markets of the city: Goods Sold goods Price, rub. April

Solution
1) Determine the change in prices in the reporting period compared to the base for each type of product, i.e. individual price indices.

Typical task 2
Known production data construction company: Type of products Production output in the I quarter, mln. rub. Volume change pro

Solution
Let us determine the total change in the physical volume of production, i.e. composite index of physical volume. It follows from the condition that individual indices of physical volume by types of products have

Typical task 3
The following data on the wages of workers in 3 sectors of the city's economy are known: Branch of the economy Wage, rub. p number

Solution
1. To determine the index wages of variable composition, we first determine the average wage for three industries for January and June.

Task 6.1
The following data on the sale of goods in the markets of the city are known: Goods Sold, t Average price of goods, rub. Basis

Task 6.2
The following data on the sale of goods in the retail trade of the city is known: Goods Sold, t Average sale price of 1 kg of goods in the base unit

Task 6.3
Production costs at the enterprise are characterized by the following data: Product type Production costs in the reporting month

Task 6.4
Known data on the sale of goods in the area: Product groups Goods sold in 2006, million rubles Growth in the number of sales in 2007 K

Task 6.5
Known data on the sale of products in the city: Commodity Turnover, thousand rubles. Price index in the reporting period to the base, %

Task 6.6
Information about one of the workshops of the enterprise is known: Product March April Manufactured, pcs. Persons spent -

Task 6.7
The following data are known on the output of products of the same name "A" and its cost for two enterprises: Enterprises Production, thousand tons

Task 6.8
The following data on the wages of employees of enterprises by sectors of the economy are known: Industry Average wages, thousand rubles. At

Task 6.9
The dynamics of production indicators of two enterprises of JSC, producing the same name products "A", are characterized by the following data:

Task 6.10
Costs in the reporting period increased by 8% compared to the base period, the total costs of production and sale of products in the reporting period amount to 659 million rubles, the total cost index is equal to

Typical task
The following data on average life expectancy and meat consumption per capita for 20 countries of the world are known: No. Country Average

Solution
Let us take x ¾ meat consumption per capita per year (kg) as a factor indicator, and y ¾ average life expectancy (years) as a resultant sign. For

Task 7.1
Data are available for ten enterprises for the reporting period: Enterprises Average annual cost of fixed assets production assets, billion rubles

Task 7.2
Known data on non-bank credit institutions: Organizations Equity, billion rubles Raised capital, billion rubles

Baseline data by country for 2002
Country Per capita income, USD (U) Human development index HDI (X1) Human poverty index, HPI, (X2)

Task 7.4
To identify the dependence of labor productivity of workers on their length of service, a linear correlation coefficient was found equal to 0.8. In addition, the following data are known: § average length of service

Task 7.5
For 20 enterprises, a model was obtained that reflects the dependence of output per month on the amount of fixed capital: y = 12.0 + 0.5x. In addition, the following data are known: § average s

Guidelines for the implementation of control work

When starting the test, you should familiarize yourself with the relevant sections of the course program and guidelines, study the recommended educational literature. In this case, special attention should be paid to the methods of construction, calculation technique and economic sense statistical indicators.

Tasks for the test work are made in six versions, the number of the option is selected in accordance with the initial letter of the surname.

When performing control work, you must be guided by the following requirements:

1. Tasks should be completed in the order in which they are given in the individual task.

2. Give the condition of the problem in full, and separate its solution by some interval.

3. It is necessary to follow the sequence in the calculations, give formulas with symbols give brief written explanations.

4. In the process of solving problems, it is necessary to check the calculations made, using the relationship between the calculated indicators, and pay attention to the economic content of the latter. Tasks presented by the student, to which answers are given without detailed calculations, explanations and brief conclusions, will be considered unresolved.

5. If possible, problem solving should be presented in the form of tables. All calculations of relative indicators must be made with the accuracy accepted in statistics: coefficients - up to 0.001, percentages - up to 0.1.

6. The control work must be neatly designed, written legibly, without blots and strikethroughs. It is not recommended to arbitrarily abbreviate words (only generally accepted abbreviations are allowed). All given tables must be drawn up in accordance with the rules adopted in statistics.

7. The pages of the work should be numbered and leave enough wide margins for the reviewer's comments and corrections (additions) made by the student after the review.

8. At the end of the work, a list of used literature should be given (author, title of the textbook, publisher, year of publication). The work must be signed by the student indicating the date of its completion.

9. The work must be registered with the laboratory assistant. If comments are made in the credited work, the student must make the necessary corrections and additions without rewriting the work again. The student must provide the work with the corrected comments to the examiner when passing the exam (test).

10. Students who submitted unsatisfactory papers for verification, perform them again or correct them in accordance with the reviewer's comments.

If a student cannot independently complete the test or some part of it, then you should seek advice from the Department of Economics and Management.

Methodological instructions for completing tasks

AT task 1 it is necessary to perform an analytical grouping of the presented statistical data. To solve this problem, it is important to understand the essence of the analytical grouping, with the help of which the relationships of the studied features are studied.

Analytical grouping allows you to establish the presence and direction of the relationship between factorial and effective features. In each variant, these signs are different and it is very important to determine which of the signs is factorial (influencing), and which is effective (accepting the influence of the factor).

Grouping is carried out according to a factor criterion, and the selected groups must be characterized by the given indicators in the condition of the problem.

To compile an analytical grouping, the value of the interval is first determined by the formula:

where x tt and x tah - minimum and maximum value of the factor attribute;

k- the number of groups (indicated in the condition of the problem).

Then the numerical values ​​of the groups according to the factor attribute are determined and a working table is compiled. The worksheet data is formatted as a group analytical table.

The analytical table should have a title, the name of the subject and predicate, units of measurement, calculated and total indicators. In conclusion, it is necessary to give economic analysis indicators of the group table and draw conclusions about the presence and direction of communication.

In these tasks, interval variation distribution series are presented, for which it is necessary to calculate the mean value of the feature, mode and median, standard deviation and coefficient of variation (Appendix).

To calculate these indicators, it is necessary to move from the intervals to their midpoints, while the open intervals are closed by the intervals adjacent to them (the first - by the next, the last - by the previous one).

Task 3 compiled on the use of sample observation in statistical practice. Attention should be paid to the calculation of the mean and marginal sampling errors for various kinds sampling (Appendix).

If the selection is actually random or mechanical, then the total variance is used to calculate the sampling error, with typical selection, the average of intragroup variances, and with serial selection, intergroup variance. You should also pay attention to the features of calculating the sampling error for the share of an alternative attribute.

Task 4 compiled for the calculation and analysis of analytical indicators of dynamic series, which are determined by the formulas set forth in the Appendix. The calculated indicators of dynamics are presented in the table.

To obtain general indicators of dynamics, average indicators of dynamics are calculated: the average level of the series, the average absolute increase, the average growth rate and the growth rate, using the formulas given in the Appendix on this topic.

Task 5 compiled on the topic "Indexes". In the first variant, it is necessary to calculate the individual and aggregate indices, show the relationship of the relevant indices and, on their basis, determine the relative and absolute change in the effective indicator by factors.

It is necessary to understand the rule for choosing weights for qualitative (cost, price, profit, etc.) and quantitative (product output, sales volume of goods, etc.) features when constructing the aggregate form of general indices.

Formulas for calculating typical general indices in aggregate form are presented in the Appendix.

In the sixth variant, the general indices are calculated not in the aggregate, but in the average form. The choice of arithmetic mean or mean harmonic form is carried out independently.

The task in the second version is designed to calculate the indices of variable, constant composition and the index of structural shifts. This system of indices is used to study the dynamics of the average value of a qualitative indicator for a set of homogeneous units (markets, shops, firms, etc.), including due to the action of individual factors: changes in the value of the indexed indicator for individual units of the studied population and changes in the structure of the population .

In the tasks of the third, fourth and fifth options, in the first part it is necessary to calculate individual and general indices, in the second - variable, constant composition and structural changes.

After calculating each index system, show the relationship between them.

Options for control tasks

Option 1

Task 1

The following data are available for 25 enterprises of one of the branches of the economy of the region:

In order to study the relationship between the average annual cost of fixed assets and the average annual number of employees, make a grouping of enterprises, forming three groups at equal intervals.

For each group and set of enterprises as a whole, calculate:

1) number of enterprises;

3) average annual number workers - total and on average per enterprise.

Present the results of the calculations in the form of a group table and draw conclusions.

Task 2

The distribution of enterprises by the number of employees is characterized by the following data:

Define:

1) the average number of employees per enterprise;

2) mode and median;

3) dispersion;

5) coefficient of variation.

Build a histogram and a polygon for the distribution of enterprises by the number of employees.

Task 3

According to a sample survey (sample of 5% typical proportional with mechanical selection) of workers in three industries, the following data were obtained on their savings:

Define:

1) the average percentage of employees with savings;

2) limits on the share of all workers in three industries with savings, with a probability of 0.954.

Task 4

The dynamics of passenger turnover in the region is characterized by the following data, million pas.km:

To analyze the dynamics of passenger turnover for 2001-2009. compute:

1) absolute increments, rates of growth and gain in the basic and chain way; the absolute content of one percent increase. Present the obtained indicators in the table, build a graph of the dynamics of passenger turnover for 2001-2009;

2) average annual passenger turnover;

3) the average annual growth rate and increase in the size of passenger traffic;

4) the expected size of passenger traffic for three years in advance, provided that the average annual growth rate is maintained.

Draw your own conclusions.

Task 5

The sale of products in the retail network of the city is characterized by indicators:

Define:

1) general aggregate price indices and physical volume of trade;

2) the general index of trade in actual prices;

3) an absolute change in turnover due to changes in prices and sales volume.

Option 2

Task 1

Data are available for 25 manufacturing firms in the region:

In order to study the relationship between the amount of profit from sales and the average annual cost of fixed assets, group manufacturing firms by the average annual cost of fixed assets, forming three groups at equal intervals. For each group and set of firms, calculate:

1) the number of firms;

2) the average annual cost of fixed assets - total and on average per firm;

3) profit from sales - total and on average per firm.

Present the results of the calculations in the form of a group table and draw brief conclusions.

Task 2

The distribution of students by the time spent on travel to the place of study is as follows:

Define:

1) average time spent on travel to the place of study;

2) mode and median;

3) dispersion;

4) standard deviation;

5) coefficient of variation.

Build a histogram and a polygon for the distribution of students by the time spent on travel to the place of study.

Task 3

To study staff turnover at the enterprises of the region during the year, 200 people (10%) who quit their jobs were interviewed. own will(selection is actually random). The survey results are characterized by the following data:

Of those who quit, 50 people were dissatisfied with the mode of work and working conditions.

Determine with a probability of 0.954:

1) the limits within which the average length of service of those who quit of their own free will will be;

2) the limits of the proportion of workers who quit due to dissatisfaction with the mode of work and working conditions. Draw your own conclusions.

Task 4

Turnover dynamics retail per capita in the region is characterized by the following data, thousand rubles:

To analyze the dynamics of retail trade turnover per capita for 2001-2009. compute:

Present the obtained indicators in the table; build a graph of the dynamics of retail trade turnover per capita for 2001-2009;

2) the average annual turnover of retail trade per capita in the region;

3) the average annual growth rate and increase in retail trade turnover per capita;

4) the expected turnover of retail trade per capita in the region for three years in advance, provided that the average annual growth rate is maintained.

Draw your own conclusions.

Task 5

We have the following data on the production of fabric at different enterprises:

Define:

1) the average cost of 1 m of fabric for all enterprises;

2) change in the average cost, including due to changes in the cost at each individual enterprise and due to changes in the structure of production.

3) the amount of savings (overspending) of costs from changes in the average cost, including due to individual factors (cost and structural changes).

Option 3

Task 1

For the reporting period, the following data on the work of industrial enterprises in the region are available:

To study the relationship between the proceeds from the sale of products and the average annual number of employees, make a grouping of enterprises, forming three groups at equal intervals.

For each group and for the totality of enterprises, calculate:

1) number of enterprises;

2) the average annual number of employees in total and on average per enterprise;

Present the results in the form of a group table and draw conclusions.

Task 2

There is data on the distribution of the region's population in terms of average per capita income:

Define:

1) average per capita income per inhabitant of the region;

2) mode and median;

3) dispersion;

4) standard deviation;

5) coefficient of variation.

Construct a histogram and a polygon for the distribution of the region's population by the level of average per capita income.

Draw conclusions based on the results of the calculations.

Task 3

There are the following sample data (5% mechanical sample) on the loss of working time in one of the industries for the year:

Define:

1) the sampling error for the amount of lost working time per enterprise and the boundaries within which there will be lost working hours in the industry in the general population, with a probability of 0.954;

2) sampling error for the share of enterprises with loss of working time over 72 thousand man-days and the boundaries in which the general share will be, with a probability of 0.997.

Task 4

The number of employees in the region is characterized by the following data, thousand people:

To analyze the dynamics of employment in the region for 2001-2009. compute:

1) absolute growth, growth rates and growth rates in the basic and chain ways, the absolute content of one percent of the growth. Present the obtained indicators in the table; build a graph of the dynamics of the employed for 2001-2009;

2) the average annual number of employees for 2001-2009;

3) the average annual growth rate and increase in the number of people employed in agricultural production;

4) the expected number of employed for three years in advance, provided that the average annual absolute growth is maintained .

Make a short summary.

Task 5

The sale of fruits in the two markets of the city is characterized by the following data:

Define:

1. For market 1 (for two types of fruit together):

b) general price index;

c) the general index of the physical volume of trade;

d) an absolute increase in trade turnover, including due to changes in prices and the volume of fruit sales.

2. For two markets together (for apples):

c) the index of the influence of changes in the structure of apple sales on the dynamics of the average price.

Explain the difference between the values ​​of constant and variable composition indices.

Option 4

Task 1

The following data are available for 25 banks in one of the regions, million rubles:

In order to study the relationship between the size of assets and authorized capital, group banks by size authorized capital, forming three groups of banks at equal intervals.

For each group of banks and for all banks, calculate:

1) the number of banks;

2) the size of the authorized capital in total and on average per bank;

3) total and average assets per bank. Present the results of the calculations in the table and draw brief conclusions.

Task 2

There is data on the distribution of workers by wage level:

Define:

1) the average wage of one worker;

2) mode and median;

3) dispersion;

4) standard deviation;

5) coefficient of variation.

Build a histogram and a polygon for the distribution of workers by wage level.

Draw conclusions based on the results of the calculations.

Task 3

To study the duration of the job search for young people under the age of 25 in the region, a 2% typical proportional sample with mechanical selection in population groups by sex was carried out, as a result of which the following generalizing indicators were obtained:

Determine with a probability of 0.997 the possible time limits for the youth of the region to search for work.

Task 4

The number of students in higher educational institutions characterized by the following data, thousand people:

To analyze the dynamics of the number of students in the region for 2001-2009. compute:

1) absolute growth, growth rates and growth rates in the basic and chain ways; the absolute content of one percent increase. Present the obtained indicators in the table, build a graph of the dynamics of the number of students in the region for 2001-2009;

2) the average annual number of students for 2001-2009;

3) the average annual growth rate and increase in the number of students;

4) the expected number of students in the region for three years in advance, provided that the average annual growth rate is maintained.

Draw your own conclusions.

Task 5

The dynamics of output and its cost for two firms is characterized by the following data:

Define:

1. For firm 1 (for two types of products together):

a) general indexes of production costs; production cost; the physical volume of production;

b) change total amount production costs, including due to changes in the cost and volume of manufactured products. Show the relationship between the calculated indices.

2. For two firms together for product A:

a) average cost index;

b) average change in cost;

c) the index of the cost of structural changes.

Option 5

Task 1

For the reporting period, there are data on the sale of goods and distribution costs for grocery stores in the city, million rubles:

To identify the relationship between distribution costs and the volume of retail turnover, group the stores according to the volume of retail turnover, forming three groups at equal intervals.

For each group and set of stores, calculate:

1) the number of stores;

2) the volume of turnover in total and on average per store;

3) the sum of distribution costs in total and on average per store.

Task 2

The distribution of construction firms by investment volume is characterized by the following data:

Define:

1) the average annual volume of investments;

2) mode and median;

3) dispersion;

4) standard deviation;

5) coefficient of variation.

Build a histogram and a polygon for the distribution of construction firms by investment volume. Draw conclusions based on the results of the calculations.

Task 3

From 400 boxes of 100 parts each, received at the warehouse finished products, in the order of random non-repetitive serial sampling, 5 boxes were selected, all parts of which were checked for weight. The sampling results are as follows:

Define:

a) possible limits on the average weight of the part for the entire batch received at the warehouse (with a probability of 0.954);

b) the volume of random non-repetitive serial sampling, so that with a probability of 0.683 the marginal sampling error in determining the average weight of one part for the entire batch does not exceed 0.7 g.

Task 4

The dynamics of the commissioning of residential buildings is characterized by the following data, thousand square meters. m:

To analyze the dynamics of the commissioning of residential buildings for 2001-2009. compute:

1) absolute growth, growth rates and growth rates in the basic and chain ways; the absolute content of one percent increase.

Present the obtained indicators in the table; build a graph of the dynamics of the commissioning of residential buildings for 2001-2009;

2) average annual level commissioning of residential buildings in the region;

3) the average annual growth and growth rate;

4) the expected commissioning of residential buildings in the region for three years in advance, provided that the average annual growth rate is maintained.

Draw your own conclusions.

Task 5

The dynamics of average prices and sales volume in the food markets of the city is characterized by the following data:

Define:

1. For market 1 (for two types of goods together):

a) general turnover index;

b) general price index;

c) general index of the physical volume of sales;

d) an absolute increase in trade turnover due to a change
prices and sales of goods.

Show the relationship between the calculated indices.

2. For two markets together (for carrots):

a) price index of variable composition;

b) fixed composition price index;

c) price index of structural shifts.

Explain the difference between the values ​​of the calculated indexes.

Option 6

Task 1

The following data are available for 25 industrial enterprises region:

In order to study the relationship between output and the average annual cost of fixed assets, group enterprises according to the average annual cost of fixed assets, forming three groups of enterprises at equal intervals. For each group and set of enterprises, calculate:

1) number of enterprises;

2) the average annual cost of fixed assets - in total and on average per enterprise;

3) proceeds from the sale of products in total and on average per enterprise.

Present the calculation results in the form of a group table. Write short conclusions.

Task 2

The distribution of operating credit institutions by the amount of registered authorized capital is characterized by the following data:

Define:

1) average value authorized capital of one credit institution;

2) mode and median;

3) dispersion;

4) standard deviation;

5) coefficient of variation.

Build a histogram and a distribution polygon by the size of the authorized capital.

Draw conclusions based on the results of the calculations.

Task 3

Based on the materials of a sample survey of wages of workers public sector area (sample 1%, typical with mechanical selection), the following data were obtained.

1. Determine the average yield of grain crops for the collective farm.

Gross harvest = yield * crop area

Average values

For two enterprises producing the same type of product, the following data are known:

Distribution of juvenile delinquency in one of the regions of the Russian Federation for the 1st half of the year:

Determine the indicators of variation:

A) range b) average linear deviation c) standard deviation e) coefficient of variation

Variation indicators

The distribution of the mileage of a trading company van is characterized by the following data:

Define:

A. average mileage per flight

B. standard deviation

B. Coefficient of variation

Variation indicators

Determine: Mean, mode, standard deviation and coefficient of variation.

Dynamic indicators

Dynamic indicators

Calculate the relative rates of dynamics with a variable and constant base.

Relative values

In the 1st quarter, the retail trade turnover amounted to 250 million rubles; in the 2nd quarter, the retail turnover is planned to be 350 million rubles. determine the relative value of the planned task.

Relative values

The company, according to the plan, was to produce products during the quarter in the amount of 200 thousand rubles. In fact, it produced products worth 220 thousand rubles. determine the degree to which the company's production plan for the quarter is fulfilled.

Relative value.

According to the plan, labor productivity in the industry of the region was to increase by 2.9%. In fact, labor productivity increased by 3.6%. Determine the degree of implementation of the plan for labor productivity by the region.

Relative values

The number of telephone exchanges in Russia in 2006 amounted to 34.3 thousand, and in 2007 - 34.5 thousand. To determine the relative magnitude of the dynamics.

Series of dynamics

Task 3. The distribution of construction firms by investment volume is characterized by the following data:

It is necessary to calculate the indicators of the distribution center, the indicators of variation and the form of distribution. Formulate a conclusion.

Task 4. The following data are available on the results of a survey of the workers of the enterprise in terms of monthly wages:

The total dispersion of wages in the surveyed population of workers was 450. To determine the extent to which the variation in the wages of workers of the enterprise depends on age.

Task 5. Determine the group variance of the share, the average of the group variance of the share, the intergroup variance of the share and the total variance of the share according to the data characterizing the number of students of all forms of education and the proportion of full-time graduates who received diplomas with honors from the universities of the city.

Tasks for independent work

Task 1. The following information is known about the workers of one of the teams:

Determine from these data: intra-group dispersion in the production of parts by one worker with a given category; the average of the intragroup dispersions for the three groups of workers; intergroup dispersion; the total dispersion of the output of the workers of this brigade.

Task 2. The variance of the attribute is 600. The volume of the population is 10. The sum of the squares of the individual values ​​of the attribute is 6250. Find the average value.

Task 3*. There are data on the distribution of the population of Russia in terms of average per capita income for 2003 to 2007. Determine for each of the periods: 1) the size and intensity of variation in incomes of the population; 2) evaluate the degree of homogeneity of the distribution of the population of Russia in terms of average per capita cash income; 3) measure income differentiation based on the decile coefficient of differentiation; 4) calculate the coefficients of skewness and kurtosis of the distribution. Draw meaningful conclusions.

Tab. Distribution of the population by average per capita income (percentage).

Task 4. The share of the main workers in the three workshops of the enterprise was: 80%, 75% and 90% total strength workers. Determine the variance and standard deviation of the share of the main workers for the enterprise as a whole, if the number of workers in the three shops was 100, 200 and 150, respectively.