The state exists primarily for the people. Therefore, the main task of management should be to solve the issue of improving the standard of living of the population and the well-being of each individual. And for this, you first need to analyze the distribution of material wealth between different social strata. One of the indicators that make it possible to do this is the decile coefficient of income differentiation. It is on its basis that the structure of the well-being of the population is often formed.
Economists calculate the difference in the total incomes of various segments of the population using a number of indicators. Among them, the main ones are the Gini index and the decile coefficient of differentiation. Graphically, social inequality is displayed using the Lorentz curve. The degree of inequality characterizes the angle of deviation of the broken line from the bisector. If the incomes are equal, then the Lorenz curve completely coincides with it. This situation means that any 10% of the population has the same percentage of the total national material resources. If someone alone has appropriated all the income, then the Lorenz curve will first go along the x-axis, and then rise vertically. From the resulting graph, you can calculate the Gini index.
The distribution of material resources among members of different social classes within a country is measured without plotting a graph in sociology using a variety of tools. The decile coefficient is one of them. It is the ratio of the average incomes of the richest 10% of the inhabitants of the state to the same percentage of the poorest. The smaller the result, the more stable the situation in society. It is believed that a coefficient value greater than 10 means the hypothetical possibility of civil unrest and the start of a coup d'état. This is due to the fact that the difference in the assets belonging to different "castes" is so great that the injustice of the structure of society immediately catches the eye.
The analysis of social stratification begins with the division of the population into groups. This can be done both with the help of statistical programs and manually. To do this, you need to collect information about the income level of the population, and then arrange it in descending order. There should be ten groups. The first will include the richest residents, the tenth - low-income citizens. The decile coefficient will be equal to the ratio of the average income of the last group to the income of the richest. To facilitate memorization, you can make a mathematical formula, denoting all the components with letters. Let d 1 and d 10 be the incomes of the 10% richest and poorest residents, and K d be the decile coefficient. In this case, the formula will look like this: K d \u003d K 10 / K 1.
The decile coefficient of income differentiation shows the differences between the incomes of the richest and the poorest segments of the population. It indicates the concentration of income in one hand. If the value of this coefficient is ten, then this means that the richest group receives 10 times more profit than the poorest group of similar size. Thus, the decile coefficient shows the difference between different segments of the population and is a tool economic analysis. It is on the basis of the calculation of such indicators that competent public administration should be formed.
Consider the countries of the world in terms of income inequality in them between rich and poor based on UN and CIA data. In Europe, the lowest value is in Denmark, Sweden and Finland. In these countries, the incomes of the rich and the poor differ by 3 times. In countries such as Germany and France, this figure is three times higher on average. At the same time, these countries are implementing programs aimed at reducing social tension, so income inequality tends to decrease. The highest rate is typical for Namibia, Bolivia, Sierra Leone, Honduras, Haiti, Botswana and Brazil (75 on average).
Generalized decile ratio in countries European Union is 6 (the smallest - in the Scandinavian countries - about 4), in the USA - 15, in Japan - 6, in North Africa - 6.
The situation with inequity in the distribution of income is a notorious situation in our society. Since the 90s of the last century, the decile coefficient in the Russian Federation has only been growing. The gap between social groups has reached such proportions that the problem is visible to the naked eye. And what kind of justice can be said in this case? The richest 10% have 20 times more tangible assets than a similarly sized group of the poor. However, the situation was not always so sad.
At the beginning of the last century, the decile coefficient did not exceed 6, while in the USA it exceeded 18. By the standards of Western countries, even the royal family lived in poverty. This once again proves that the Bolshevik revolution of 1917 was caused by political reasons, not economic ones. It is believed that any unrest can occur if the indicator exceeds 10.
According to the study of Professor B. Mironov, to the poorest segments of the population in 1901-1904. belonged to the following categories of people:
The income of the poorest 10% of the population was 6.5 rubles per month, or about 78 per year.
The assessment of the wealth of the richest people in the country was carried out by the Commission for the Introduction income tax, established under the Ministry of Finance in 1905. She calculated that on average their income was 2,130 rubles a year, or 178 rubles a month. But this is only 1% of the tenth group. The remaining 9% of the rich received only 320.5 rubles a year. In this way, average income tenth decile group - 493 rubles, and the differentiation coefficient is 6.3.
Low-income families are the most vulnerable segment of the population. They don't have enough savings to survive the cyclical economic crises. The presence of such groups of the population not only provokes conflicts and unrest, but also forces the state to spend significant funds to maintain them. At the same time, subsidizing is not a solution to the problem, but only delays the escalation of the situation. A much more rational way is to create new jobs and introduce progressive taxation.
It is expedient to supplement the analysis of variation in the distribution series indicators of differentiation.
To assess the differentiation of the values of a characteristic of a series, we use decile coefficient of differentiation and coefficient of funds.
Decile coefficient equals the ratio of the ninth decile to the first decile. The decile coefficient is widely used when measuring the ratio of income levels of 10% of the most wealthy and 10% of the least wealthy population (in times).
Funds ratio equals the ratio of the average level of the 10th decile to the average level of the 1st decile. It gives a more precise level of differentiation.
State statistics regularly publishes the coefficient of funds to characterize the differentiation of income. However, in research work, the decile coefficient of differentiation is more often used. Its application is especially effective if, for example, in the distribution of income at the beginning of the first decile there are extremely low incomes, and the tenth decile ends with abnormally high incomes, which significantly affect the amount of income in these deciles. In such a situation, it is more correct to use the decile coefficient of differentiation, and not the coefficient of funds.
Close to differentiation indicators concentration indicators: Gini coefficient and Herfindahl coefficient.
Gini concentration coefficient calculated by the formula:
, (6.27)
where pi is the accumulated share (frequency) of the number of units in the series
qi- the accumulated share of the attribute values, attributable to all units of the series with the attribute values not more than x i .
The Gini coefficient can take values from 0 to 1, so the result should be divided by either 100 if pi or qi expressed as a percentage, or per 10,000 if both are expressed as a percentage. The greater the concentration of a trait, the closer the Gini coefficient is to 1. The Gini coefficient is used to characterize the degree of unevenness in the distribution of a population (for example, population) by the level of a trait (for example, income).
Herfindahl coefficient is calculated on the basis of data on the share of the studied trait in i-that group in the total volume of the trait:
or 
(6.28)
where is the share of revenue i-that group in the total volume of all attribute values;
The amount of characteristic values in i- to that group.
Index H depends on the number of units in the groups.
Example 6.6. There is data on the received balance sheet profit of the 50 largest banks in Russia (as of 01/01/1998), (in million rubles)
| - | 974,2 | - | 188,8 | - | 143,9 | - | 85,4 | - | 69,3 | |||||
| - | 609,2 | - | 187,3 | - | 134,6 | - | 84,5 | - | 66,4 | |||||
| - | 588,3 | - | 186,8 | - | 120,9 | - | 82,4 | - | 66,2 | |||||
| - | 562,9 | - | 171,1 | - | 112,2 | - | 79,6 | - | 59,7 | |||||
| - | 436,3 | - | 167,9 | - | 108,5 | - | 74,3 | - | 59,1 | |||||
| - | 432,5 | - | 164,3 | - | 101,6 | - | 74,0 | - | 58,3 | |||||
| - | 283,6 | - | 160,3 | - | 101,3 | - | 73,5 | - | 57,4 | |||||
| - | 265,8 | - | 159,9 | - | 97,4 | - | 73,2 | - | 53,8 | |||||
| - | 231,5 | - | 157,5 | - | 97,4 | - | 73,0 | - | 51,4 | |||||
| - | 211,7 | - | 147,6 | - | 92,0 | - | 71,5 | - | 51,2 |
The balance sheet profit of Sberbank of Russia as of July 1, 1997 was 4,353.283 million rubles.
1. Build a variation series, forming 7-8 intervals arbitrarily.
2. Calculate the average size book profit per bank based on arithmetic mean, mode and median.
3. Calculate the variation rates.
4. Measure bank differentiation based on decile ratio and fund ratio.
5. Calculate the concentration coefficient of Gini and Herfindahl.
Solution:
1. Distribution of 50 banks of the Russian Federation by the amount of balance sheet profit (BP) as of 01/01/1998
| BP, million rubles xk-1-xk | Number of banks | Middle of the interval x i | x i f i | On-accumulation hour-thots Vi, % | On-accumulation hour-tos-ty pi | Share of BP of groups of banks in total BP | ||||
| fi | in % of total | growing as a result, qi | ||||||||
| BUT | ||||||||||
| 50-60 | 0,042 | 0,042 | 0,02 | |||||||
| 60-80 | 0,076 | 0,118 | 0,006 | |||||||
| 80-100 | 0,059 | 0,177 | 0,003 | |||||||
| 100-150 | 0,109 | 0,286 | 0,012 | |||||||
| 150-300 | 0,318 | 0,604 | 0,101 | |||||||
| 300-500 | 0,087 | 0,691 | 0,008 | |||||||
| 500-800 | 0,212 | 0,902 | 0,045 | |||||||
| 800-1000 | 0,098 | 1,0 | 0,010 | |||||||
| Total | - | - | - | - | 0,187 |
2. Averages:
a) the average balance sheet profit per bank is calculated by the arithmetic weighted average
b) we calculate the mode according to the formula (5.6)
The modal interval is 150-300, because the frequency of this interval, equal to 13, is the maximum.
c) we calculate the median using the formula (5.5)
The median interval is 100-150, because the cumulative frequency of this interval, equal to 31, is the first cumulative frequency that exceeds half the sum of the frequencies of the series.
3. Indicators of variation:
a) dispersion (according to formula 6.6)
= 
b) standard deviation (according to formula 6.7)
c) coefficient of variation (according to formula 6.11)
V>35%, which indicates the heterogeneity of the population.
4. Indicators of differentiation:
a) to find the decile coefficient, we first determine the first and ninth deciles using the formula 5.4
The interval corresponding to the first decile is 50-60, because the cumulative frequency of this interval, equal to 7, the first cumulative frequency greater than 0.1 of the sum of frequencies.
The interval corresponding to the ninth decile is 300-500, because the cumulative frequency of this interval, equal to 14, the first cumulative frequency greater than 0.9 of the sum of frequencies.
Then the decile coefficient will be: 
b) because 10% of the largest and 10% of the smallest banks are the same value (in our example), then the stock ratio will be (according to the original table):
5. Concentration indicators:
a) we calculate the Gini coefficient according to the formula 6.27, having made preliminary calculations
| 1,652 | 1,428 |
| 6,018 | 5,428 |
| 13,156 | 10,974 |
| 37,448 | 25,168 |
| 60,808 | 55,568 |
| 82,984 | 67,718 |
| 90,02 | |
b) the Herfindahl coefficient is determined by the formula 6.28 (see the result of column 9):
Example 6.7. To illustrate the principle of calculating the Gini and Herfindahl coefficients, we will use data from a sample survey of the daily revenue of 20 grocery stores (thousand rubles):
| Store numbers i | Characteristic values (store revenue) x i | Cumulative characteristic values | Accumulated share of feature values q i | The accumulated share of the number of units in the series: p i | |||
| 0,022 | 0,05 | 0,002 | - | 0,0005 | |||
| 0,044 | 0,1 | 0,007 | 0,002 | 0,0005 | |||
| 0,071 | 0,15 | 0,014 | 0,007 | 0,0007 | |||
| 0,1 | 0,2 | 0,025 | 0,015 | 0,0009 | |||
| 0,137 | 0,25 | 0,041 | 0,027 | 0,0013 | |||
| 0,176 | 0,3 | 0,062 | 0,044 | 0,0015 | |||
| 0,218 | 0,35 | 0,087 | 0,065 | 0,0017 | |||
| 0,262 | 0,4 | 0,118 | 0,092 | 0,0019 | |||
| 0,308 | 0,45 | 0,154 | 0,123 | 0,0021 | |||
| 0,359 | 0,5 | 0,198 | 0,162 | 0,0026 | |||
| 0,411 | 0,55 | 0,246 | 0,205 | 0,0026 | |||
| 0,472 | 0,6 | 0,307 | 0,296 | 0,0037 | |||
| 0,533 | 0,65 | 0,373 | 0,320 | 0,0037 | |||
| 0,597 | 0,7 | 0,447 | 0,388 | 0,0040 | |||
| 0,66 | 0,75 | 0,528 | 0,462 | 0,0040 | |||
| 0,724 | 0,8 | 0,615 | 0,543 | 0,0040 | |||
| 0,787 | 0,85 | 0,709 | 0,630 | 0,0040 | |||
| 0,853 | 0,9 | 0,811 | 0,725 | 0,0044 | |||
| 0,927 | 0,95 | 0,927 | 0,834 | 0,0054 | |||
| 1,0 | 1,0 | - | 0,95 | 0,0054 | |||
| å | 5,670 | 5,584 | 0,05528 |
The Gini coefficient is 0.086, which indicates a low level of store revenue concentration. The Herfindahl coefficient value of 0.05528 confirms this conclusion.
It should be noted that the above calculations are for illustrative purposes only, since economic sense Gini and Herfindahl coefficients are most fully manifested only when comparing the studied phenomena in time and space. For example, the Gini coefficient to characterize the differentiation of incomes of the population in different regions of the Russian Federation or countries, the Herfindahl coefficient to characterize the concentration of production, capital. The main advantage of the Herfindahl coefficient is its high sensitivity to changes in the total turnover of the shares of the largest participants, which allows you to track the concentration market turnover and responds to the number of market participants. The Herfindahl coefficient can be used as a measure of the diversification of a bank's loan portfolio. The smaller the value of the Herfindahl coefficient, i.e. the more diversified loan portfolio, the lower the capital requirements for the loan portfolio may be.
6.2. test questions to topic 6
1. Why is it necessary to study the variation of a trait?
2. Specify the main indicators of variation.
3. What methods do you know for calculating variance and standard deviation?
4. How is the variance of an alternative feature determined?
5. What is the coefficient of variation?
6. Rule for adding variances. What do private (intragroup), average of private, intergroup and total variances show?
7. How are the coefficient of determination and the empirical correlation ratio calculated and what do they characterize?
8. How do they calculate and what characterize the differentiation coefficient and the coefficient of funds?
9. Concentration indicators: Gini and Herfindahl coefficients.
6.3. Control tasks to topic 6
1. The variance of the feature is 600. The size of the population is 10. The sum of the squares of the individual values of the feature is 6250. Find the average value.
2. The average value in the population is 15, the standard deviation is 10. What is the average square of the individual values of this feature?
3. The average value in the aggregate is 13, and the mean square of the individual values of this trait is 174. Determine the coefficient of variation.
4. The variance of the trait is 360000, the coefficient of variation is 50%. What is the mean value of the feature?
5. The variance of the trait is 360, the mean square of the individual values is 585. What is the mean?
6. Determine the variance of the feature if the mean value of the feature is 2600 and the coefficient of variation of the feature is 30%.
7. The total variance is 8.4. The average value of the trait for the entire population is 13. The group averages are 10, 15, and 12, respectively. The number of units in each group is 32, 53, and 45. Determine the average intragroup variance.
8. Based on a population of 100 units, the following are known: arithmetic mean - 47.0; the sum of the squares of the individual values of the feature is 231592. Determine whether the study population is homogeneous.
9.
Determine the magnitude of the empirical correlation if the total variance is 15.2; group averages ; 
, and the number of groups is 75, 60, and 65, respectively.
10. To study the level wages workers at the enterprise examined 500 men and 300 women. The results of the study showed that the average salary for men was 1200 USD. with a standard deviation of 200 c.u., for women, respectively, 800 c.u. and 150 c.u.
Determine: 1) the average wage of employees;
2) dispersion of wages and coefficient of variation;
11. There are data on net profit(balance sheet net of taxes) enterprises of two districts:
Determine the variances of net profit: group (for each district), average of the group, intergroup and total.
12. Wage data are available for two groups of workers:
Find all kinds of wage variances, coefficient of determination and empirical correlation.
13. The following data are available on the average daily employment time of married women in household:
Find the total variance of employment, the coefficient of determination and the empirical correlation ratio.
14. There are two groups of people with different monthly incomes (thousand rubles):
Group A: 3, 3, 3, 4.
Group B: 6, 6, 7.
Which group should include a person with an income of 5 thousand rubles. per month.
15. According to the results of a marketing study of travel companies organizing weekly tours to Turkey in various resort towns, the following data on the variation in the cost of tours were obtained (prices are given for hotels of the same class):
Find the total variance, coefficient of determination and empirical correlation.
16. According to a sample survey of workers' wages public sector the following indicators were obtained:
Determine: 1) the average wages of workers in two industries;
3) coefficient of determination and empirical correlation.
17. The following data (conditional) are available for three groups of workers:
2) dispersion of wages;
3) coefficient of determination and empirical correlation.
18. When studying the time budget of students, a survey of university students was conducted. During the survey, universities were divided into 7 groups according to their specialization. The following results of the average amount of time spent by students daily on independent work were obtained:
Using the rule of addition of variances, determine the relationship between the average number of hours for independent work on the student's specialization.
19. To determine the average amount of expenses for a semi-annual subscription to newspapers and magazines, the surveyed population of families was divided into income groups into three groups. The following results were obtained by groups:
Determine all types of cost dispersions, the coefficient of determination and the empirical correlation ratio.
20. There are 20 thousand families living in cities, urban-type settlements and countryside. As a result, the following data were obtained on the average number of children in families:
Using the rule for adding variances, determine the coefficient of determination and the empirical correlation ratio.
21. The following sample data on the contributions of the population of the district are available:
Determine the closeness of the relationship between the average size of the contribution and the type of population by calculating the coefficient of determination and the empirical correlation ratio.
22. The capital of commercial banks is characterized by the following data:
Determine the indicators of the tightness of the relationship between the size of the equity capital of banks and borrowed funds by calculating the coefficient of determination and the empirical correlation.
23. According to a survey of commercial banks in the city, 70% of the total number of customers were legal entities with an average loan size of 120 thousand rubles. and a coefficient of variation of 25%, and 30% - individuals with an average loan size of 20 thousand rubles. with a mean square deviation of 6 thousand rubles.
Using the rule of addition of variances, determine the closeness of the relationship between the size of the loan and the type of client by calculating the coefficient of determination and the empirical correlation ratio.
24. Trade turnover by enterprises Catering per employee per quarter is characterized by the following data:
Determine all types of variances in the turnover of public catering enterprises, the coefficient of determination and the empirical correlation.
25. There is data on the distribution of families of employees of a financial corporation by the number of children:
Calculate all kinds of variances using the rule of addition of variances.
26. The distribution of fixed assets among small enterprises in the industry is characterized by the following data:
Calculate the coefficient of determination and the empirical correlation ratio. Draw your own conclusions.
27. According to the 1926 census in Russia, the proportion of literate women was 46%, and among men - 77%. Determine the total (average) percentage of literacy of the entire population and the variance of this indicator if women accounted for 53% of the total population.
28. Determine the variance and standard deviation if, when checking a batch of products from 1000 pcs. 30 pcs. turned out to be married.
29. For determining specific gravity of women in the number of employees in the industry, all enterprises were divided according to the average number of employees into 3 groups:
1st gr. – with the number of employees up to 1000 people;
2nd gr. - with the number of employees from 1001 to 5000 people;
3rd gr. - employing more than 5000 people.
The total number of employees in the 1st group is 120 thousand people, in the 2nd group - 89 thousand, in the 3rd group - 50 thousand. The proportion of women in the first group was 47%, in the 2nd - 36%, in 3rd - 29%. Use the variance addition rule to determine the variance of the share of women in the industry.
30. The following data are available on the number of households in extreme poverty (average per capita income is two times lower than living wage):
Draw your own conclusions.
31. The following data are available on expenditures of households in the district on cultural and household goods:
Determine all types of dispersions, coefficient of determination and empirical correlation.
32. The following data are available on the costs of paid services district households:
Determine all types of dispersions, coefficient of determination and empirical correlation.
33. The following sample data on food expenditure by city households are available:
Determine all types of dispersions, coefficient of determination and empirical correlation.
34. There are data on the total revenue (billion rubles) for 1997. 50 largest audit and consulting firms
| - | 78,1 | - | 17,5 | - | 7,2 | - | 4,0 | - | 2,7 | |||||
| - | 44,8 | - | 15,8 | - | 7,0 | - | 3,8 | - | 2,7 | |||||
| - | 35,2 | - | 15,7 | - | 6,8 | - | 3,6 | - | 2,7 | |||||
| - | 34,6 | - | 14,5 | - | 6,6 | - | 3,6 | - | 2,7 | |||||
| - | 32,5 | - | 13,2 | - | 5,6 | - | 3,5 | - | 2,6 | |||||
| - | 31,8 | - | 12,0 | - | 5,1 | - | 3,1 | - | 2,6 | |||||
| - | 25,4 | - | 11,6 | - | 4,8 | - | 3,0 | - | 2,2 | |||||
| - | 23,0 | - | 9,4 | - | 4,5 | - | 3,0 | - | 2,1 | |||||
| - | 17,8 | - | 7,6 | - | 4,5 | - | 3,0 | - | 1,5 | |||||
| - | 17,7 | - | 7,3 | - | 4,4 | - | 2,9 | - | 1,5 |
1) Build a variation series, forming 7-8 intervals arbitrarily.
2) Calculate the average revenue per firm based on the arithmetic mean, fashion median.
3) Calculate the variation rates.
4) Measure revenue differentiation based on decile ratio and fund ratio.
5) Calculate the concentration coefficient of Gini and Herfindahl.
35. There is data on the distribution of the population of the Russian Federation by the size of the average per capita cash income in the first half of 2000:
Calculate: 1) per capita money income based on the arithmetic mean, mode and median;
2) indicators of concentration differentiation;
The use of the coefficient of variation makes sense when studying the variation of a trait that takes only positive values. It is absolutely wrong to use V in the case of measuring the fluctuation of a trait that takes both positive and negative values. It does not make sense, for example, V, calculated to study the fluctuations in the average annual air temperature, which is especially clear when the average annual temperature is close to zero.
There are many other analytical expressions for the Gini coefficient, but to save space, we will focus on one.
In the analysis of effectiveness government controlled important is the question of the standard of living of the population. The authorities of the country exercise control over its useful resources, the contents of the subsoil and the gross product of human activities. It is the state that organizes the distribution of material wealth among citizens. Why is it that in some countries rich people
Are you dissatisfied with your standard of living? And other states that do not have such resources do not know what popular indignation is. How to find a criterion for comparing measures of the distribution of material wealth?
For these purposes, sociology uses a number of tools. One of them is the population decile coefficient. It is calculated by comparing the income statistics of the richest 10% of the population and income data of the same number of poor citizens. In science, it is believed that the approximation of the value of the coefficient to 10 creates conditions for social unrest in the country. The people will begin to express their indignation at the current government, and riots may arise.
Sweden, Denmark and Finland have the lowest value of this coefficient. In these countries, the indicator fluctuates between 3-4. In France and Germany, it is at the level of 5-7. Insignificant dispersion in these states helps to maintain a favorable social climate. In Russia, the decile coefficient has shown continuous growth since the 1990s. The gap in the incomes of different segments of the population at the moment has reached impressive proportions.
The wealthiest citizens receive 15-20 times more than the poorest categories. And this applies to the working part of society. Persons who fell into the category of the poor from unwillingness to work are not included in the calculation. Consequently, a person who has worked all his life in Russia also runs the risk of remaining poor. The work of a simple locksmith, for example, has no value compared to the profitability of a Gazprom employee. The decile coefficient shows that conditions have formed in our country for the non-equilibrium distribution of material wealth.
The onset of most unfavorable economic factors affects, first of all, the unsecured segments of the population. Inflationary and crisis processes are reflected in the growth of housing and communal services tariffs, prices for services public transport and the cost of essential goods. At the same time, prices fluctuate in a smaller range. In addition, citizens with low incomes do not have the so-called financial "airbag". They do not have sufficient material savings to easily weather severe economic shocks. When such categories of the population form a social majority in a state, then the conditions for social unrest are created. The decile coefficient clearly shows this.
A generalized assessment of the degree of structurization of the phenomenon as a whole is usually carried out according to the formula concentration level(or Herfindahl coefficient), which is more sensitive to changes in the shares of groups with the largest specific gravity in the end, determined by the formula:
where is the share j-th group in the total result of the studied value of the attribute; m - the number of groups identified in the aggregate.
The maximum coefficient value is 10,000 (if the shares are in percentages) or 1 (if the shares are in numbers). The closer the value of the coefficient to the maximum, the higher the concentration of the trait under study in one or more groups. The minimum coefficient value is 100 (if the share is in percentage) or 0.01 (if the share is in numbers).
The reciprocal of the Herfindahl index is effective number of groups in the structure, which shows the number of groups, excluding groups with negligible shares, is determined by the formula:
E = 1/H.
|
Production per person |
number of people |
group mean (midpoint) |
production per group | ||
Herfindahl coefficient 
, which indicates a slight concentration of the studied trait (production) in one or more groups.
The number of effective groups E = 1/H = 1/0.179 = 5.6. This indicator shows that the trait under study is concentrated in 6 groups out of 7 available. Which again confirms the relatively even distribution of output among groups of workers.
On the basis of quantiles, various coefficients of differentiation of the studied trait are calculated: decile (quartile, percentile) coefficient, funds coefficient, Ginny coefficient. These coefficients characterize the uneven distribution of the trait in the studied population. For example, inequality in income distribution (how much the income of the richest group differs from the income of the least wealthy group).
Decile coefficient of differentiation,
This coefficient shows how many times the smallest value of a feature among the 10% of observation units with the highest values of the feature (the smallest income of the top 10%) is greater than the largest value of the feature among the 10% of observation units with the smallest values of the feature (the largest income of 10 % of the poorest).
Quartile coefficient of differentiation,
This coefficient shows how many times the smallest value of a feature among the 25% of observation units with the highest values of the feature (the smallest income of the top 25%) is greater than the largest value of the feature among the 25% of observation units with the smallest values of the feature (the largest income of 25 % of the poorest).
Funds ratio- the ratio of the average value of the studied trait in the tenth decile group to the average value of the studied trait in the first decile group
This coefficient shows how many times the average value of the feature among the 10% of observation units with the highest values of the feature (the average income of the top 10% of the wealth) is greater than the average value of the feature among the 10% of the observation units with the smallest values of the feature (the average income of the poorest 10%) .
The first decile group is the range of values from the lowest to the first decile. The tenth decile group is the range of values from the ninth decile to the maximum value. The mean of a range of values is the middle of the range.
Consequently,
;
The greater the value of these coefficients, the greater the inequality in the distribution of benefits (duties) between the tenth decile group and the first decile group.
Let's look at these coefficients with an example.
Information on the development of workers for October
|
Production of one worker |
number of people |
cumulative frequency |
||||||||
|
meaning |
||||||||||
|
16+(22-16)×((612/10)-5)/110= | ||||||||||
|
40+(46-40)×(9×(612/10)-567)/60= | ||||||||||
|
1 quartile |
22+(28-22)×((612/4)-115)/182= | |||||||||
|
3rd quartile |
34+(40-34)×(3×(612/4)-417)/90= | |||||||||
The values of the coefficients allow us to draw the following conclusions: the smallest output of the 10% of the best workers exceeds the best output of the 10% of the most unproductive workers by 2.3 times.
The lowest output of the 25% of the best workers exceeds the best output of the 25% of the most unproductive workers by 1.6 times.
To calculate the coefficient of funds, you need to find the average values of the 1st and 10th decile groups.
=
;
This coefficient allows us to conclude that the average output of the best workers is 3.3 times higher than the output of the lowest productive workers.
