 # Gini Coefficient

The Gini coefficient represents the income or wealth distribution of an area’s residents, and is the most commonly used measure of inequality. It was developed by the Italian statistician and sociologist Corrado Gini and published in his paper Variability and Mutability (Gini, 1912).

A Gini coefficient of zero expresses perfect equality, where all values are the same (for example, where everyone has the same income). A Gini coefficient of 1 (or 100%) expresses maximal inequality among values (e.g., for a large number of people, where only one person has all the income or consumption, and all others have none, the Gini coefficient will be very nearly one). However, a value greater than one may occur if some persons represent a negative contribution to the total (for example, having negative income or wealth). For larger groups, values close to or above 1 are very unlikely in practice.

Similar to the Theil Index, the Gini Coefficient provides a measure of the amount of inequality there is in the distribution of your variable across space. If all of the individual areas across the study region that you’re looking at have the same, or similar proportions of a certain variable – say, unemployed people – then there is no inequality (perfect equality) with respect to the distribution of that variable. If there are large differences in the distribution of the variable, then the inequality across your study region is large.

The Gini Coefficient compares the area under a Lorenz curve to the area under a perfect distribution line. It is defined as the ratio of the area between the Lorenz curve and the diagonal of perfect equality, to the area of the triangle below this diagonal as shown in the diagram below. The Lorenz curve is a graphical representation of the cumulative distribution function of some variable, often income. The formula for the Gini Coefficient from Sen (1973) as cited in Anand (1983):

$\large{G=\frac{1}{n} \left ( n+1-2\frac{\sum_{i=1}^n(n+1-i)y_i}{\sum_{i=1}^{n}y_i} \right )}$

where the values $y_i,i=1$ to $n$ are the income levels in indexed in non-decreasing order and $n$ is the population size.

For a random sample, the numerator is the sample size less one. Hence, for a measure of inequality of unemployment across regions, the Gini Coefficient becomes:

$\large{G(S)=\frac{1}{r-1}\left(r+1-2\frac{\sum_{i=1}^r(r+1-i)u_i}{\sum_{i=1}^ru_i} \right )}$

where $u_i,i=1$ to $r$ are the unemployment rates indexed in non-decreasing order and $r$ is the number of regions. This is a consistent estimator of the Gini Coefficient, though not an unbiased one.

The Gini Coefficient can range from a value of 0 to 1. Where there is perfect equality, the Gini Coefficient is zero and it would imply a Lorenz curve that follows the perfect distribution line. A Gini Coefficient of 1 implies perfect inequality, for example where one region has all the unemployment within your study area.

### SET UP

To show the Gini coefficient tool in use, we will run it to calculate the coefficients for the distribution of male youth unemployment across NSW:

Select New South Wales as your area.
Select NATSEM – Social and Economic Indicators – Unemployment Rate SA2 2016 with the following attributes:

• SA2 Code
• SA2 Name
• Number of males in the labour force aged 15 – 24
• Number of unemployed males aged 15 – 24 in the area

### Inputs

Once you have set up your data, open the Gini Coefficient tool (Tools → Indices → Gini Coefficient). The input fields are as follows:

• Dataset Input: This is the dataset that contains the values you would like to include in the Gini coefficient calculation. In this instance we select NATSEM – Social and Economic Indicators – Unemployment Rate SA2 2016.
• Numerator: This is the column that contains the different counts for the specific variable that you would like to calculate the inequality of distribution across the study region. In this instance, we select Number of unemployed males aged 15 – 24 in the area.
• Denominator: This is the column that contains the total counts of the sample population that you are taking the numerator from. In this instance, we select Number of males in the labour force aged 15 – 24.

The input parameters are summarised in the image below, once complete click Run Tool. ### Outputs

Once you have run the tool, click the Display output button that appears on the pop-up dialogue box. This should open up a text box like the one shown below, which has the Gini coefficient value for your variable. In this instance, we have a coefficient of 0.189, which suggests a low inequality in the distribution of youth male unemployment in NSW SA2 regions. Anand, S. (1983). Inequality and poverty in Malaysia: Measurement and decomposition. The World Bank.
Centre of Full Employment and Equity. (2015). AURIN spatial statistics and econometrics e-tools help file. University of Newcastle.
Gini, C. (1912). Variabilità e mutabilità (Variability and Mutability). Tipografia Di Paolo Cuppini, Bologna, Italy, 156.
Sen, A. (1973). Poverty, inequality and unemployment: Some conceptual issues in measurement. Economic and Political Weekly, 1457–1464.

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