The Gini Coefficient

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Authors: Dr. Jean-Paul Rodrigue

1. The Lorenz Curve

The Gini coefficient was developed to measure the degree of concentration (inequality) of a variable in a distribution of its elements. It compares the Lorenz curve of a ranked empirical distribution with the line of perfect equality. This line assumes that each element has the same contribution to the total summation of the values of a variable. The Gini coefficient ranges between 0, where there is no concentration (perfect equality), and 1 where there is total concentration (perfect inequality).

Geographers and many others have used the Gini coefficient in numerous instances, such assessing income distribution among a set of contiguous regions (or countries) or to measure other spatial phenomena such industrial location. Its major purpose as a method in transport geography has been related to measuring the concentration of traffic, mainly at terminals, such as assessing changes in port system concentration. Economies of scale in transportation can favor the concentration of traffic at transport terminals, while other considerations such as accessibility to regional markets can be perceived as a countervailing force to concentration. So, the temporal variations of the Gini coefficient reflect changes in the comparative advantages of a location within the transport system.

Three different measures of inequality linked to the Gini Coefficient are presented below. There are all linked to the concept of comparing the Lorenz curve with the lines of perfect equality and inequality.

2. Index of Dissimilarity (ID)

The dissimilarity index is the summation of vertical deviations between the Lorenz curve and the line of perfect equality, also known as the summation of Lorenz differences. The closer the ID is to 1 (or 100 if percentages are used instead of fractions), the more dissimilar the distribution is to the line of perfect equality.

Where X and Y are percentages (or fractions) of the total number of elements and their respective values (traffic being the most common). N is the number of elements (observations). For instance, the following considers the distribution of traffic among 5 terminals:

Terminal	Traffic	X	Y	\|X-Y\|
A	25,000	0.2	0.438	0.238
B	18,000	0.2	0.316	0.116
C	9,000	0.2	0.158	0.042
D	3,000	0.2	0.053	0.147
E	2,000	0.2	0.035	0.165
Total	57,000	1.0	1.0	0.708

Terminal B, with a traffic of 18,000 accounts for 0.2 (or 20%; X) of all terminals and 0.316 (or 31.6%; Y) of all traffic. The index of dissimilarity of this distribution is 0.354 (0.708 * 0.5), which indicates an average level of concentration. A more complex example is provided here.

3. Gini's Coefficient (G)

The Gini Coefficient represents the area of concentration between the Lorenz curve and the line of perfect equality as it expresses a proportion of the area enclosed by the triangle defined by the line of perfect equality and the line of perfect inequality. The closer the coefficient is to 1, the more unequal the distribution.

Where σX and σY are cumulative percentages of Xs and Ys (in fractions) and N is the number of elements (observations). Using the same example as above, the following table demonstrates the calculation of the Gini coefficient:

Terminal	Traffic	X	Y	σX	σY	σX_i-1 – σX_i (B)	σY_i-1 + σY_i (A)	A*B
A	25,000	0.2	0.438	0.2	0.438	0.2	0.438	0.088
B	18,000	0.2	0.316	0.4	0.754	0.2	1.192	0.238
C	9,000	0.2	0.158	0.6	0.912	0.2	1.666	0.333
D	3,000	0.2	0.053	0.8	0.965	0.2	1.877	0.375
E	2,000	0.2	0.035	1.0	1.000	0.2	1.965	0.393
Total	57,000	1.0	1.000					1.427

The Gini coefficient for this distribution is 0.427 (|1-1.427|). A more complex example is provided here.

4. Gini's Means Difference (GMD)

The mean of the difference between each observation and every other observation.

Where X is the cumulative percentage (or fractions) and N is the number of elements (observations).