The Gini CoefficientAuthors: Dr. Jean-Paul Rodrigue1. The Lorenz CurveThe 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).

Media The Lorenz Curve
Traffic Concentration and Lorenz Curves
World’s 50 Largest Container Ports, Passenger Airports and
Freight Airports
Lorenz and Perfect Inequality Differences
Calculation of the Index of Dissimilarity
Calculation of the Gini Coefficient