Comparing proportions inside groups...?

Question

Let's say I wanted to compare the wealth of people in 2 groups to see in which one the wealth is distributed in a more progressively escalated manner:

In group A we have 3 people:

One with 12,000 $

One with 6,000 $

And the last one with 1,000 $

In group B we have four people:

One with 9,000 $

One with 6,000 $

One with 2,000 $

And the last one with 1,000 $

For group A, we have a total wealth of 19,000 $ and the contributed percentages from the people are:

63.16%

31.58%

5.26%

For group B, we have a total wealth of 18,000 $ and the contributed percentages from the people are:

50%

33.33%

11.11%

5.56 %

When comparing the proportions, here is what I've done:

Group A:

63.16/31.58 = x2

31.58/5.26 = x6

6/2 = 3

Group B:

50/33.33 = x1.5

33.33/11.11 = x3

11.11/5.56 = x2

2/1.5 = 1.33

3/1.5 = 2

3/2 = 1.5

Mean: (1.5+2+1.33)/3 = 4.83/3 = 1.61

Since 1.61 is less than 3, I conclude that group B has less differences in wealth among the members of the group and therefore the money is distributed in a more progressively escalating manner.

Is this correct? Did I made any mistakes?

Would it be better to just compare the standard deviations of each group to see in which of them the money escalates with fewer differences among the members of the group? And in the case where I have a group with a smaller total proportion in the way that I calculated it but with a bigger standard deviation, what would have a higher "priority"? The standard deviation or the calculated total proportion?

Or perhaps computing a linear regression over each frequency distribution and compare the average residuals

Answer 1

One way of looking at inequality is to draw a Lorenz curve: you sort the values and draw the cumulative distribution, rescaling the axes so different sets of data can be compared. With your data you would get

• the blue line (o) for Group A
• the red line (x) for Group B

Perfect equality would give the green diagonal.

You could then calculate the Gini coefficient as the area between the lines and the green diagonal, divided by the area of the green triangle, so a larger Gini coefficient would suggest more inequality. $0$ would result from total equality while $1$ would be approached as the number of people increased if a single person had everything.

For Group A this is about $0.386$ while for Group B this is about $0.389$, so the two groups are, on this measure, very similar.