Absolute and relative indicators of changes in structure. Absolute and relative indicators of changes in structures Initial general, do not have an image

Calculate:

1) the level and dynamics of labor productivity for each enterprise separately;

2) for two enterprises together:

a) average labor productivity index of variable composition;

b) index of average labor productivity of permanent (fixed) composition;

c) index of the impact of structural changes due to changes in the number of employees;

d) absolute change in the volume of production in the second quarter compared to the first quarter as a result of changes in each of the factors.

Show the relationship between the calculated indicators. Analyze the results and draw conclusions.

Solution.

1. Determine the level and dynamics of labor productivity for each enterprise

a) for enterprise No. 1

performance index

Labor productivity at enterprise No. 1 increased by 25.9%.

b) for enterprise No. 2

in the 1st quarter, million rubles. for one person

in the 2nd quarter, million rubles. for one person

performance index

Labor productivity at enterprise No. 2 increased by 24.4%.

2. Let us determine for two enterprises together:

a) average labor productivity index of variable composition:

b) index of average labor productivity of permanent (fixed) composition:

c) index of the impact of structural changes due to changes in the number of employees

Index Relationship

d) absolute change in the volume of production in the second quarter compared to the first quarter as a result of changes in each of the factors

Thousand rub.

Thousand rub.

Thousand rub.

Average labor productivity at two enterprises in the second quarter increased by 22.8% (or 1.13 thousand rubles) compared to the first quarter, including due to an increase in productivity at individual enterprises by an average of 25.1% ( or by 1.22 thousand rubles) and a change in structure by - 1.8% (or a decrease by 0.09 thousand rubles).

Example 2. The following data on the export of metal products from the Russian Federation is known.

Table 39

Export of metal products from the Russian Federation

According to the given data:

a) calculate the price indices and physical volume of exported metal products;

b) determine by what amount (millions of US dollars) export revenue changed under the influence of changes in contract prices.

Analyze the obtained indicators and draw conclusions.

Solution.

a) Transform the aggregate form of the price index

=> => or 91%

Index of physical volume of exported products

=> or 104%

b) absolute change in export revenue due to the influence of changes in contract prices, million US dollars

Metal prices decreased by an average of 9%. The growth in the physical volume of exported metal products amounted to 4%. Changes in contract prices for metal products led to a decrease in export revenue by 434.2 million US dollars.

Example 3. The following data is available on the structure of income (Table 40).

Table 40

Income structure in groups with different average per capita monetary income in some regions of the Russian Federation in 2002.

Determine the significance of structural differences in the incomes of various groups using the Salai and Gatev indices.

Solution. 1. Let's define the Szalai index.

Szalai index I s = ,

Where d 1– income structure in the second group

d 0- income structure in the first group

n– number of groups

We present the calculated data in Table 41.

Table 41

Data for calculating the Salai index

Continuation of table 41

	Property income	-0,74	3,34	-0,2216	0,0491
	Other income	4,6	48,9	0,0941	0,0089
	Total:	-	-	-	0,2075

Thus, the Salai index shows quite significant differences in the distribution of per capita incomes of different groups.

2. Calculate the integral coefficient of K. Gatev:

The calculated data are given in table 42.

Table 42

Data for calculating the integral coefficient of K. Gatev

Thus, K. Gatev’s coefficient shows the differences in the distribution by type of income between the group with low and high per capita income.

Control questions

1. The concept of indices.

2. Individual indices and their types.

3. Main types of economic indices. Aggregate index as the main form of economic index.

5. Relationship between chain and basic indices.

The structure of a particular set does not remain constant either in time or in space. The need to analyze changes in structures arises either when comparing the structures of different periods of time, or the structures of different territorial objects. In the first case they talk about structural shifts, in the second - about structural differences.

The difference in the structures of the compared populations can be expressed in the difference in the specific weights of individual parts of these populations. All indicators characterizing changes in structures are divided into absolute and relative. Absolute indicators of changes in structures are based on the difference between the specific gravities of the corresponding parts of different structures. They are measured in percentage points, can be positive or negative, and their sum is zero. They show by how many percentage points the share of the analyzed part in one structure increased or decreased (positive or negative value, respectively) compared to its value in another structure. Relative indicators are calculated by the ratio of the corresponding specific weights: if the result is greater than one, then the share of this element in the compared structure is greater than in the basic structure; if it is less than one, then the share of the analyzed element of the compared structure is the corresponding part of the share of this element in the basic structure. It should be noted that when analyzing changes in two structures, in order to obtain an objective picture of these changes, it is necessary to use both absolute and relative indicators. Let's consider official statistical data on the structure of cash income of the population of the Russian Federation by source of income for 2000 and 2011. (Table 6.5).

Using the presented data, we will calculate indicators characterizing structural changes in 2011 compared to 2000.

Table 6.5

Statistical data on the structure of cash income of the population of the Russian Federation by source of income for 2000 and 2011.

It is obvious that there were changes in the structure of cash income of the population of the Russian Federation in 2011 compared to 2000: the share of income from business activities and income from property decreased, and the share of other income items increased. This is confirmed by the signs of absolute change (pluses and minuses). Based on the results obtained, we can say that in terms of absolute changes, the largest changes occurred in the shares of income from business activities, social benefits and wages, and in relative terms, the most significant changes are observed for the shares of other income and income from property. The relative change is more clearly visible by the relative increase (decrease). Relative increase (decrease) is calculated from relative change (multiplying by 100 and subtracting 100%). This means that the share of income from business activities decreased by 6.3 percentage points in 2011 compared to 2000, or amounted to 41% in 2011 of its value in 2000; the share of wages in 2011 compared to 2000 increased by 4.3 percentage points, or 1.07 times, or 7%. Similarly, conclusions can be drawn about other sources of income. The different degrees of changes in absolute and relative indicators are explained by differences in the size of the share of individual elements. An increase in the share of other income by 0.8 percentage points gave the maximum increase in relative change, since the very value of the share of this source of income generation is the smallest. At the same time, the increase in the share of wages by 4.3 percentage points amounted to the smallest relative change of 1.07, or an increase of 7%. It is worth paying attention to the content of the changes that have occurred over the past 10 years, reflected in this example. In the structure of income of the population of the Russian Federation, the shares of wages and social payments have increased and the shares of income from business activities, income from property and other income have decreased.

Absolute and relative indicators of change in individual parts of the whole are disproportionate to each other: smaller absolute changes may correspond to larger relative changes, and larger absolute changes may correspond to smaller relative ones. That is why, when analyzing changes in the structure of any population, both absolute and relative indicators of changes in structures should be calculated in order to obtain a more accurate idea of ​​the structural changes in the compared structures.

Moving on to general indicators, let us pay attention to the following point. If the total volume of the population under study grows, then the relative indicators of change for individual elements of the population may be greater or less than unity, i.e. they can grow and contract. Moreover, if the relative indicator of change in an individual element is greater than the relative change in the entire aggregate, this means that the specific weight of this element in the aggregate is growing. Accordingly, if the relative indicator of change for any element or part of the population is less than the same indicator for the entire population as a whole, this means that the share of this part in the total volume is decreasing. Thus, a change in the structure of the whole is a consequence of the uneven intensity of change in its individual parts, i.e. differences in relative changes in specific gravity.

When analyzing changes in structures, a generalized description of these changes is often required. The following indicators can be used for this.

1. Sum of absolute changes in specific gravity

where are the proportions of individual elements of the two populations being compared; n- the number of elements (groups) in total.

The sum of absolute changes in specific weights is expressed in percentage points. This value characterizes the total volume of deviations of one structure from another.

.

The difference index, calculated through specific gravities expressed as percentages, can take values ​​from 0 to 100%; approaching zero means no change; approaching maximum indicates a significant change in the structure.

3. Integral coefficient of structural shifts K. Gateva. The above indicators do not provide an idea of ​​changes in the shares of individual elements of the population. This indicator takes into account the intensity of changes in individual groups in the compared structures:

.

The number of groups into which the population under study is divided affects the final assessment of structural changes.

4. Salai Structural Difference Index. This indicator also takes into account the number of groups or elements in the compared structures:

.

Both last presented coefficients (or indexes) can take values ​​from zero to one. The closer the resulting value is to unity, the more significant the structural changes that have occurred. The Szalai coefficient takes values ​​close to one when the total number of units is large.

5. Ryabtsev index. The values ​​of this indicator do not depend on the number of gradations of structures. The assessment is made on the basis of the maximum possible value of discrepancies between the components of the structure; the actual discrepancies of individual components of the structures are compared with the maximum possible values:

.

This coefficient (index) also takes values ​​from zero to one. An advantage of this indicator can be considered the presence of a scale for assessing the obtained indicator values ​​(Table 6.6).

Table 6.6

Scale for assessing the significance of structural differences using the Ryabtsev index

Thus, the listed indicators represent a generalized characteristic of structural changes, but do not give an idea of ​​the magnitude of these changes.

The following indicators give this idea.

6. Average linear change in shares

.

7. Mean square change

.

The average estimate of the measure of change (per one group, population unit) is represented by the average linear change in shares or the root mean square of these changes. The obtained values ​​show how many percentage points on average the specific weights of the compared structures deviate from each other. The analytical content of these two indicators is the same. However, the square mean is always greater than the arithmetic mean, so the value of the root mean square change will be greater than the linear mean. Two indicators will be equal if the absolute changes in the specific weights of all parts of the whole are equal in absolute value. In the absence of changes in structures, these indicators are equal to zero. Since the degree of the average linear change corresponds to the degree of the indicator itself, this estimate should be considered more accurate, however, the mean square change is more often used, since it reacts more sensitively to weak fluctuations in the structure.

When using the listed indicators, the analysis of changes in structures occurs without taking into account the size of the base from which this change occurred. A more accurate assessment can be made by using relative rather than absolute changes. In particular, the average relative linear change can be calculated as the average of the relative linear deviations (i.e., growth rates) taken modulo:

.

The result multiplied by 100 can be expressed as a percentage and easily evaluated.

Graphical comparative analysis of structure

In socio-economic research, situations often arise in which it is necessary to analyze the structures of phenomena or processes over a number of periods. One of the methods of analysis in this case is to consider structural diagrams.

The most common structural diagram is the pie or pie

Figure - Composition and structure of the unemployed by education in 2003, %

This type of diagram is most convenient to use when illustrating the structure of a phenomenon for one, two or three periods, but in practice a situation may arise when it is necessary to compare the structure for 5 or more periods. In this case, you need to use a donut chart.

Figure - Composition and structure of the unemployed by education in 1992. and 2003, %

Figure - Composition and structure of the unemployed by education in 1992, 1998, 2002-2003, %

To assess changes in the structure of the population over time and determine the structures of individual groups, indicators of structural differences and shifts are used. The simplest indicators of structural differences are [page 37, Timofeeva]:

Linear coefficient of structural differences (shifts) or Re index:

Where d1, do- structure of the reporting and base periods, %

P - number of lines.

Shows how much, on average, the structure of the reporting period does not correspond to the structure of the base period. A disadvantage of the indicator is the fact that its value depends on n. If n is small, then the index takes small values ​​and vice versa.

Quadratic coefficient of structural changes:

0 £ d£100 or £0 s£100 (if data measured in %).

The closer the value of the indicators is to 0, the smaller the differences in the structures of the populations being studied; or the smaller the changes that have occurred in the structure of the population in dynamics.

Linear and quadratic coefficients are used mainly to study the dynamics of structure indicators, because clearly allow one to draw conclusions about the intensity of changes in structures in certain periods of time.

Gateva Index(Gatev index) distinguishes structures with equal sums of squared deviations.

Ryabtsev index(Ryabtsev index) differs slightly from the Gatev index and takes lower values:

Salai Index(Szalai index) was introduced when studying differences in the structure of time budget use among different population groups:

The Salai index differs from all the indices of this group discussed above. It takes values ​​close to one when the total is a large number of units.

The given indices take values ​​in the range from 0 to 1. If one or another index is equal to zero, then complete similarity of structures is observed, if one is a complete difference. If more than 0.5, then the differences in the structure of the reporting and current periods are considered significant.

The following conditional data are available on the structure of cash incomes of the population of the region, in percentage:

It is necessary to draw a conclusion about changes in the structure of cash incomes of the population.

Solution.

Based on the above indicators, we can conclude that in the composition of the population’s cash income, the share of wages decreased (from 60% in the base period to 42% in the reporting period) with an increase in the share of income from property and business activities (from 24% to 44%, respectively) .

A generalizing characteristic of the measure of structural changes is given by integral indicators of structural differences, the calculation of which is illustrated in the table:

The magnitude of the calculated indicators of structural differences indicates significant changes in the structure of monetary incomes of the population of the region.

Problems 5-6 involve studying the dynamics of indicators, i.e. the intensity of changes in phenomena over time, which are carried out using the following indicators: absolute growth, growth rates, growth rates, the absolute value of one percent of growth, as well as average generalizing indicators.

Depending on the research objective, indicators can be calculated with a variable base of comparison (chain) and with a constant base of comparison (basic).

1. Absolute increase is the difference between the level being compared and the previous or baseline:

chain absolute increase:

base absolute increase: .

The sum of chain absolute increases is equal to the basic absolute increase for the corresponding period of time.

2. Growth rate– a relative indicator characterizing the intensity of development of the phenomenon; it is equal to the ratio of the level being studied to the previous or basic level and is expressed in coefficients or percentages.

chain growth rate: 100;

base growth rate: .

The product of the corresponding chain growth rates calculated in coefficients is equal to the base one.

3. Rate of increase determined in two ways:

a) as the ratio of absolute growth to the previous level (chain) or basic level (basic):

chain growth rate:

base growth rate: .

b) as the difference between the growth rate and 100%:

T pr = T r -100%.

4. Absolute value of one percent increase is defined as the ratio of the chain absolute increase to the chain growth rate (%) or for each subsequent level - as 0.01 of the previous level of the dynamics series:

5. Average absolute increase calculated using the simple arithmetic average, that is, dividing the sum of chain absolute increases by their number

Average growth rate found using the geometric mean formula:

Average growth rate found by subtracting 100% from the average growth rate:

Calculation methods mid-level Some dynamics depend on its type and completeness of information.

1) in interval series with equal time intervals, the average level is determined by the simple arithmetic mean formula:

2) in interval series with unequal time intervals - according to the weighted arithmetic mean formula (based on the size of the intervals):

3) in moment series with comprehensive data on changes in the moment indicator, the calculation is made using the arithmetic mean of the series levels that remained unchanged for certain periods of time, weighted by the value of the corresponding intervals;

4) in moment series of dynamics with equally spaced levels, the average chronological simple formula is used.

The development of a statistical population is manifested not only in the quantitative growth or reduction of elements of the system, but also in changes in its structure. Structure- this is the structure of the aggregate, consisting of individual elements and connections between them. For example, a country's exports (aggregate) consist of various types of goods (elements), the value of which varies by type and by country. In addition, there is a constant change in the dynamics of the export structure. Accordingly, the task of studying the structure of aggregates and their dynamics arises, for which special methods have been developed that will be discussed below.

In topic 2, the structure index was considered, calculated using formula (6), which characterizes the proportion of individual elements in the total absolute attribute of the population. Topic 3 discusses the system of indicators and the methodology for analyzing the distribution of a population according to the values ​​of any individual characteristic (variation series of distribution). Here are the indicators characterizing the change in the structure as a whole, i.e. "structural shift". We will consider the practical application of these indicators using two examples presented in tables 19 and 20 (the first 4 columns in bold are the original data, and the rest are auxiliary calculations).

Table 19. Distribution of the Russian population by average per capita cash income (ACI)

groups (j)	rub./person per month	Population shares		|d 1–d 0|			(d 1–d 0)2	(d 1+d 0)2
		2005 year (d 0)	2006 (d 1)
	up to 1500
	1500-2500
	2500-3500
	3500-4500
	4500-6000
	6000-8000
	8000-12000
	more than 12000
	Total

Table 20. Distribution of the number of unemployed in Russia by level of education in 2006

Group number (j)	Have an education	Men (d 0)	Women (d 1)	|d 1–d 0|			(d 1–d 0)2	(d 1+d 0)2
	Higher professional
	Incomplete higher professional
	Secondary professional
	Initial professional
	Average (full) general
	Basic general
	Initial general, do not have an image
	Total

A generalizing absolute indicator of changes in structure can be sum of absolute change modules of shares, determined by formula (50):

, (50)

Where d 1j– share of the j-th group of elements in the reporting period; d 0j– share of the j-th group of elements in the base period.

According to Table 19 in the 5th column, a calculation was made using the formula (50): =0.212, that is, the total change in shares in the distribution of Russians by income amounted to 21.2%. Similarly, according to the same formula according to Table 20: =0.276, that is, the difference in the structure of the unemployed among women and men by level of education is 27.6%.

Calculation of the average absolute change per share (group, element of the population) does not provide any additional information. But you can determine how strong the change in structure that has occurred is in comparison with the maximum possible value of the sum of the modules, which is equal to 2. For this, the indicator is used degree of absolute shift intensity(or Loosemore-Hanby index), which is determined by formula (51): the th object in the overall total of the indicator being studied; k– number of objects.

According to Table 19 in the 6th and 7th columns, the Herfindahl coefficient was calculated using formula (52): H 2005=0.142 and H 2006=0.1687, that is, the level of concentration in the income distribution of Russians increased in 2006 compared to 2005. Similarly, using the same formula according to Table 20: H husband=0.2455 and H women = 0.2177, that is, the level of concentration in the distribution of the unemployed by level of education among men is higher than among women (the impact of education level on the status of the unemployed among men is higher than among women).

The reciprocal of the Herfindahl index is effective number of groups in the structure, which shows the number of groups without taking into account groups with negligible shares, is determined by formula (53):

E= 1/H. (53)

According to Table 19, the effective number of groups according to formula (53): E 2005=1/0.142=7.0 and E 2006=5.9, that is, the effective number of groups in the distribution of Russians by income decreased from 7 in 2005 to 6 in 2005, which indicates the need to revise the intervals of distribution of Russians by income next year. Similarly, using the same formula according to Table 20: E husband=1/0.2455=4.07 and E female = 1/0.2177 = 4.59, then the effective number of groups in the distribution of the unemployed by level of education among men is higher and among women – 4 for men and 5 for women.

Another option for assessing the degree of structuring of the phenomenon as a whole is Grofman index(54), which is the sum of the absolute change modules of the shares per one effective group:

. (54)

According to Table 19 in formula (54): =0.212*0.142=0.030, that is, the change in shares per effective group in the income distribution of Russians is insignificant (3.0%). Similarly, according to the same formula according to Table 20: =0.2455*0.276=0.068, that is, the difference in structure per effective group among unemployed women and men by level of education is weak (6.8%).

To assess changes in the two largest shares (dominant shares) Leaphart index (55):

. 55)

Where d 1m And d 0m– share m-th group of elements in the reporting period and base periods; m– maximum share in the aggregate.

According to Table 19 according to formula (55): =0.5*(0.083+0.023)=0.053, that is, the average change in shares in the two dominant groups of the income distribution of Russians was 5.3%. Similarly, according to the same formula according to Table 20: =0.5*(0.060+0.051)=0.056, that is, the difference in structure in the two dominant groups among unemployed women and men by level of education is 5.6%.

The considered indicators are based on the arithmetic mean in various variants, and due to their linearity in deviations, they take into account large and small deviations equally. Quadratic indices allow comparison of different structures that are indistinguishable in terms of the amount of change.

Quadratic index of structural changes Kazintsa (56):

. (56)

According to Table 19 according to formula (56): ==0.035, that is, the average change in shares in the group in the distribution of Russians by income was 3.5% (insignificant). Similarly, according to the same formula according to Table 20: ==0.049, that is, the difference in groups in the structure of the unemployed among women and men by level of education is 4.9% (insignificant).

Similar to the Kazinets index least squares index(or Gallagher index), in the calculation of which, in contrast to formula (51), small differences in shares have a weaker effect on the index than large ones, is determined by formula (57) = =0.117, that is, the difference in the structure of the unemployed among women and men by level of education according to the Monroe formula is 11.7%.

Integral coefficient of structural shifts Gatev(59), which distinguishes structures with equal sums of squared deviations (takes higher values ​​when groups have approximately equal shares):

. (59)

According to Table 19 according to formula (59): ==0.179, that is, the intensity of changes in shares in the distribution of Russians by income according to the Gatev method was 17.9% (insignificant). Similarly, according to the same formula according to Table 20: ==0.192, that is, the difference in the structure of the unemployed among women and men by level of education according to Gatev’s method is 19.2% (insignificant).

Index Ryabtseva, differing from (59) only in the denominator, usually takes lower values, calculated using formula (60):

. (60)

According to Table 19 using formula (60): = =0.127, that is, the intensity of changes in shares in the distribution of Russians by income according to Ryabtsev’s method was 12.7% (insignificant). Similarly, using the same formula according to Table 20: = =0.137, that is, the difference in the structure of the unemployed among women and men by level of education according to Ryabtsev’s method is 13.7% (quite significant).

Structural Difference Index Salai(61), the peculiarity of which is that the larger the fraction j Atkinson index, generalized entropy index, which will be discussed in the course of socio-economic statistics in the topic “Living Standards Statistics”.