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# Gini

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• 1. Dynamics of libre software communities Analysis of the concentration of parameters in libre software communities Master on Free Software
• 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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