# Is it possible to make quantitative comparisons using income inequality metrics?

I've calculated income inequality metrics (Gini/Hoover/Theil coefficients) for several populations. I know I can make claims like "this population has a higher Gini coefficient, so its distribution in this variable is more unequal than this other population."

But is it possible to make more quantifiable claims? For example, to say that "since the Gini coefficient for this population is twice as large, the variable is distributed eight times more unequally?" Or something like that?

UPDATE

The data isn't economic data - this has to do with relative numbers of contributions to a volutneer project. There are 5 datasets, with ~10,000 rows each. Each row corresponds to a single Internet user who did one or more tasks to contribute to that project. Some did just one, some did thousands.

The question is: of the five projects, which ones are such that a small core group has done a lot of tasks each, and which ones are such that many people have done a few tasks each to add up to the total?

FURTHER UPDATE

Thinking about this more, maybe the Gini coefficient doesn't have enough information to make this kind of comparison. Maybe I need to compare the actual distributions of the variable among population members, using something like a K-S test?

• Could you update your question text and question with what you are trying to do? The question suggests you're interested in income, but you are looking at something different given your comment below. – Michelle Jan 31 '12 at 21:07
• You should have a look at the methods in the book: Handcock, Morris: "Relative Distribution Methods in the Social Sciences" (Springer). Plots such as the relative distribution might be more interpretable than the better known QQplot. – kjetil b halvorsen Aug 14 '15 at 11:18

The issue with this type of inter-country comparison for income is that it appears to ignore government transfers, e.g. tax rebates for lower income couples with children, socialised healthcare, etc. This means that it is very difficult to get consistent measures of income between countries, and that any relatively simplistic measure of income (e.g. earnings plus interest/dividends) won't reflect "in practice" income.

So while it looks like a coefficient of .5 shows twice as high inequality as a coefficient of 0.25, that may not be an accurate portrayal of the real levels of inequality.

Update questions: the extra information you have given is valuable and it looks like something other than an income inequality measure will be more appropriate given your data. For the task information, are the tasks the same duration, or could some tasks take longer? One would expect that longer tasks would equate to fewer tasks completed, so comparing number of tasks performed may not be a good comparator for "volunteer effort". A volunteer who completed a few long duration tasks could have contributed the same amount of effort as a volunteer who completed many short tasks. I'm not sure if this is relevant in your case.

If tasks are the same length, you could look at boxplots of number of tasks per subject (use project as the group for the plot). That will clearly show which projects are associated with more tasks.

The boxplot information should also suggest what type of test is best to use to compare the projects. It could be that an ANOVA (Analysis of Variance) is the best way forward.

• Actually, this isn't economic data, it's data about the numbers of contributions to volunteer projects. The research question is: which projects come from a few volunteers who do many tasks vs. many who do only a few tasks each? The situation is mathematically like income distribution, where "income" is actually volunteer contributions. (No judgment implied here - just a mathematical model. Also, let me know if this analogy is not accurate.) I know I'm safe treating this as ordinal - projects with higher Ginis are more unequal - but can I say anything about relative amounts of inequality? – Jordan Jan 31 '12 at 14:03

I talked to a friend of mine here. As I was starting to suspect in my edited question, you can't compare Gini coefficients in this way. The Gini coefficient is a summary statistic that erases details of the distributions. To make quantitative statements about the different levels of inequality in distributions, you have to compare the distributions themselves.

I think I can do this with a K-S test. If I run into trouble, I'll post a new question.

• The K-S test has exactly the same problem you describe with the Gini coefficient: it "is a summary statistic that erases details of the distributions." If you want to get beyond that, draw a picture of the distributions. A good one would be a q-q plot. (The K-S merely tells you the maximum deviation between that plot and a diagonal line.) – whuber Feb 1 '12 at 17:22