Are Lorenz curves and QQ-plots the same? If not, where are the differences? I read about both of them and they appear to be two terms for the same type of plot / statistical technique to compare distributions. I was not able to find any confirmatory source for this. Perhaps you know?
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The Lorenz curve is just a cumulative distribution function for a random variable bounded between 0 and 1, e.g., a proportion. In economics, the Lorenz curve asks, "what fraction of income is earned by the lowest x% of earners?" Typically, it is compared to the uniform distribution over [0,1], a distribution that would arise under perfect equality in income. The Gini coefficient is the area under the perfect equality curve less the area under the Lorenz curve, normalized by the area under the perfect equality curve; note that the area under the perfect equality curve is equal to 0.5.
So, to be clear, while a QQ plot compares two distributions to one another, the Lorenz curve considers only one distribution that has a range over [0,1].
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$\begingroup$ So, this would mean when having two distributions to compare, I could plot two Lorenz curves and calculate two Gini coefficients for them. This should then be comparable with creating one QQ plot with the advantage that the Lorenz/Gini solution would allow me to compare the mentioned distributions numerically, whereas the QQ-plot only allows me to compare visually. Correct? $\endgroup$ Commented Sep 21, 2010 at 3:56
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$\begingroup$ What question are you trying to answer? If you are asking, "are these two distributions the same?", then you should use the QQ plot for a visual and the Kolmogorov-Smirnov test for the similarity of the distributions. This test basically uses the absolute maximum distance from the 45-degree line in the QQ plot as its statistic. The KS test can also be thought of as the maximum distance between two cumulative distribution functions plotted in the same figure. A CDF is a general concept; a Lorenz curve is a special case of the CDF when the domain of the random variable is from 0 to 1. $\endgroup$– CharlieCommented Sep 21, 2010 at 5:28
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$\begingroup$ I have a distribution ranging from 1 to 1024 (screen coordinates) with frequencies of user activities at these coordinates. I produce a density plot of this distribution and, since I have a number of activities I would like to compare, I have several of such distributions. So, to answer your question, my rage is naturally not [0,1]. However, if I normalize the screen between 0% and 100% I can fit it that way (just like in economic problems where the income level is scaled between 0% and 100% in 20% steps). Does that sound reasonable? $\endgroup$ Commented Sep 21, 2010 at 5:47
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$\begingroup$ I'm sorry, but I'm still not understanding the hypothesis that you want to test. Is it, "clicks are uniformly distributed across the screen"? If so, you can use the KS test to answer whether your distribution of clicks is statistically different from a normal distribution. You wouldn't need to normalize or think of this as a Lorenz curve; just think of it as a CDF and the KS test statistic is the maximum absolute distance between your CDF (with a domain from 0 to 1024) to the CDF of the uniform "reference" distribution (the 45-degree line). $\endgroup$– CharlieCommented Sep 21, 2010 at 6:01
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$\begingroup$ The KS test can compare one empirical distribution to a known theoretical distribution or one empirical distribution to another empirical distribution. It is a nonparametric test for equality of distributions. See the Wikipedia page: en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test $\endgroup$– CharlieCommented Sep 21, 2010 at 6:02