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# Gini

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### Gini

1. 1. Dynamics of libre software communities Analysis of the concentration of parameters in libre software communities Master on Free Software
2. 2. 1. The problem Measure the concentration (or level of ● inequality) of the distribution of a certain parameter among the members of a population. Usually depends on 'subjective' opinion. – Can we represent this graphically? – Can we quantify it? – Master on Free Software
3. 3. 2. Lorenz curve Graphical representation of inequality. ● Steps: ● 1. Sort the values of the parameter for each member from lowest to highest. 2. Cumulative sum of parameter's values --> calculate cumulative %. 3. Represent those results against the cumulative % of individuals in the population. Master on Free Software
4. 4. 3. Example: Lorenz curve Distribution of the hectares of terrain ● cultivated by 10 farmers. 1 7,5 6,5 7,5 7,5 1 6,5 7,5 1 First, we have to sort the values, from lowest – to highest: 1; 1; 1; 4; 6,5; 6,5; 7,5; 7,5; 7,5; 7,5 ● Master on Free Software
5. 5. 3. Example: Lorenz curve Work out values for graphic: (p(i)~q(i)). ● x(i) N(i) u(i) p(i) q(i) 1 1 1 10 2 1 2 2 20 4 1 3 3 30 6 4 4 7 40 14 6,5 5 13,5 50 27 6,5 6 20 60 40 7,5 7 27,5 70 55 7,5 8 35 80 70 7,5 9 42,5 90 85 7,5 10 50 100 100 Master on Free Software
6. 6. 3 Example: Lorenz Curve Plot the graph. ● 100 90 80 70 60 Equitative distribu- 50 tion Lorenz curve 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 Master on Free Software
7. 7. 3. Example: Lorenz Curve Problem: Comparing inequality levels of ● different distributions. 100 100 90 90 80 80 70 70 60 60 Equitative Equitative 50 50 distribution distribution Case 1 Case 3 40 40 Case 2 Case 4 30 30 20 20 10 10 0 0 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Master on Free Software
8. 8. 4. Conclusions about the Lorenz curve Graphical resume of the concentration of ● a certain distribution. Very useful to offer a general picture of the – whole case under study. It is invariant with respect to scale changes – (uses %). Some drawbacks. – We cannot compare crossing Lorenz curves. ● It is difficult to quickly compare different case ● studies. Master on Free Software
9. 9. 5. The Gini coefficient Introduced by Corrado Gini in 1936. ● Original intention: Measure inequalities in the – wealth distribution within a population . Corrado Gini. (May 23, 1884 - March 13, 1965) Master on Free Software
10. 10. 5. The Gini Coefficient It allows us to quantify the inequality ● level of a distribution, with a single value. More condensed resume than the Lorenz – curve. But we loose more information about the precise ● details of the distribution. We can quickly compare the inequality of – different distributions. Comparing the Gini coefficients for each one. ● Master on Free Software
11. 11. 5. The Gini Coefficient The Gini coefficient is given by: ● n−1 ∑ [p i−qi] i=1 Notice that the G= summatory goes from 1 to n-1 n−1 ∑ p i i=1 Master on Free Software
12. 12. 5. The Gini coefficient Geometrically: Area between the line of ● perfect equality and the Lorenz curve. 100 90 80 Gini coefficient 70 60 Equitative distribu- 50 tion Lorenz curve 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 Master on Free Software
13. 13. 5. The Gini coefficient Some properties: ● It can take values in the interval [0, 1]. – It allows us to compare distributions with – crossing Lorenz curves. It is not affected by scale changes. – It is independent from the absolute level of – the values of the distribution. We can compare distribution of the same ● parameter in different population, or in the same population at different points in time. Master on Free Software