Gini Coefficient Formula

Gini Coefficient Formula (Table of Contents)

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What is the Gini Coefficient Formula?

The term “Gini coefficient” of a nation refers to the measurement of income distribution across the population of the nation. In other words, it can be seen as the measure of inequality of income distribution in the society and its value lies in the range of 0 and 1 where the value of 1 indicates a higher degree of income inequality. The formula for Gini coefficient involves computation of an aggregate score which is a complex function of the income contribution (fraction of income) by a different segment of the population (fraction of population). Then, the Gini coefficient is calculated by deducting the aggregate score from 1. Mathematically, Gini Coefficient Formula is represented as,

Examples of Gini Coefficient Formula (With Excel Template)

Let’s take an example to understand the calculation of the Gini Coefficient in a better manner.

Gini Coefficient Formula – Example #1

Let us take a simple example of a population of 20 people to understand the concept of the Gini coefficient. As per the given information, first 5 people made $50 per month per person, the next 10 people made $100 per month per person and the last 5 people made $300 per month per person. Calculate the Gini coefficient for the population.

Solution:

Total Income is calculated as

• Total Income = (5 * $50) + (10 * $100) + (5 * $300)
• Total Income = $2,750

Fraction of Income is calculated as

First 5 People

• Fraction of Income = (5 * $50) / $2,750
• Fraction of Income = 0.09

Next 10 People

• Fraction of Income= (10 * $100) / $2,750
• Fraction of Income = 0.36

Last 5 People

• Fraction of Income = (5 * $300) / $2,750
• Fraction of Income = 0.55

Fraction of Population is calculated as

First 5 People

• Fraction of Population = 5 / 20
• Fraction of Population = 0.25

Next 10 People

• Fraction of Population = 10 / 20
• Fraction of Population= 0.50

Last 5 People

• Fraction of Population = 5 / 20
• Fraction of Population = 0.25

Fraction of Richer Population is calculated as

First 5 People

• Fraction of Richer Population = 1 – 0.25
• Fraction of Richer Population = 0.75

Next 10 People

• Fraction of Richer Population = 1 – 0.25 – 0.50
• Fraction of Richer Population = 0.25

Last 5 People

• Fraction of Richer Population = 1 – 0.25 – 0.50 – 0.25
• Fraction of Richer Population = 0.00

Score is calculated using the formula given below

Score = Fraction of Income * (Fraction of Population + 2 * Fraction of Richer Population)

First 5 People

• Score = 0.09 * (0.25 + 2 * 0.75)
• Score = 0.159

Next 10 People

• Score = 0.36 * (0.50 + 2 * 0.25)
• Score = 0.364

Last 5 People

• Score = 0.55 * (0.25 + 2 * 0.00)
• Score = 0.136

Aggregate Score is calculated as

• Aggregate Score = 0.159 + 0.364 + 0.136
• Aggregate Score = 0.659

Gini Coefficient is calculated using the formula given below

Gini Coefficient = 1 – Aggregate Score

• Gini Coefficient = 1 – 0.659
• Gini Coefficient = 0.341

Therefore, the Gini coefficient of the population is 0.341.

Gini Coefficient Formula – Example #2

Let us take the example of two countries (Country X and Country Y) and check which country exhibits higher income inequality. Based on the following information, calculate their Gini coefficient.

Solution:

Fraction of Richer Population is calculated as

Group 1

• Fraction of Richer Population = 1 – 0.30
• Fraction of Richer Population = 0.70

Group 2

• Fraction of Richer Population = 1 – 0.30 – 0.60
• Fraction of Richer Population = 0.10

Group 3

• Fraction of Richer Population = 1 – 0.30 – 0.60 – 0.10
• Fraction of Richer Population = 0.00

For Country X

Score is calculated using the formula given below

Score = Fraction of Income * (Fraction of Population + 2 * Fraction of Richer Population)

Group 1

• Score = 0.03 * (0.30 + 2 * 0.70)
• Score = 0.051

Group 2

• Score = 0.45 * (0.60 + 2 * 0.10)
• Score = 0.360

Group 3

• Score = 0.52 * (0.10 + 2 * 0.00)
• Score = 0.052

Aggregate Score is calculated as

• Aggregate Score = 0.051 + 0.360 + 0.052
• Aggregate Score = 0.463

Gini Coefficient is calculated using the formula given below

Gini Coefficient = 1 – Aggregate Score

• Gini Coefficient = 1 – 0.463
• Gini Coefficient = 0.537

For Country Y

Score is calculated using the formula given below

Score = Fraction of Income * (Fraction of Population + 2 * Fraction of Richer Population)

Group 1

• Score = 0.08 * (0.30 + 2 * 0.70)
• Score = 0.136

Group 2

• Score = 0.37 * (0.60 + 2 * 0.10)
• Score = 0.296

Group 3

• Score = 0.55 * (0.10 + 2 * 0.00)
• Score = 0.055

Aggregate Score is calculated as

• Aggregate Score = 0.136 + 0.296 + 0.055
• Aggregate Score = 0.487

Gini Coefficient is calculated using the formula given below

Gini Coefficient = 1 – Aggregate Score

• Gini Coefficient = 1 – 0.487
• Gini Coefficient = 0.513

Therefore, it can be seen that Country Y has a lower income inequality than Country X.

Explanation

The formula for the Gini coefficient can be derived by using the following steps:

Step 1: Firstly, collect the income information for the entire population and arrange the data set in ascending order of income.

Step 2: Next, group the population into different segments based on the level of income.

Step 3: Next, calculate the contribution of each group to the overall income, i.e. total income of the group divided by the income of the entire population, and it is known as the Fraction of Income.

Step 4: Next, calculate the contribution of each group to the overall population, i.e. the population of the group divided by the entire population, and it is known as the Fraction of Population.

Step 5: Next, calculate the Fraction of Richer Population for each group which is one minus the cumulative fraction of population (lower income group to higher income group).

Step 6: Next, the score for each group can be calculated by using a fraction of income, a fraction of population and fraction of richer population as shown below.

Score = Fraction of Income * (Fraction of Population + 2 * Fraction of Richer Population)

Step 7: Next, sum up the score of each group to arrive at the aggregate score.

Step 8: Finally, the formula for the Gini coefficient can be derived 1 minus the aggregate score as shown below.

Gini Coefficient = 1 – Aggregate Score

Relevance and Use of Gini Coefficient Formula

It is quintessential to understand the concept of the Gini coefficient as it is one of the most important economic tool used for analyzing the wealth or income distribution of a country. A lower with a lower Gini coefficient indicates the economic well-being and prosperity is adequately distributed among the population, while a higher value indicates the concentration of wealth to a selected few which is not seen as a sign of national prosperity.

Gini Coefficient Formula Calculator

You can use the following Gini Coefficient Calculator

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