Gini Coefficient Equation, Derivation & Rearrangements

Core idea

Overview

The Gini coefficient is a statistical measure of economic inequality that quantifies the dispersion of income or wealth among a population. It is derived from the Lorenz curve, where 'A' represents the area between the line of perfect equality and the Lorenz curve, and 'B' represents the area beneath the curve.

When to use: Use this formula when evaluating income distribution across different countries or analyzing wealth gaps within a single region over time. It is particularly useful for comparing the effectiveness of tax systems and social welfare programs in reducing inequality.

Why it matters: The Gini coefficient provides a single, easy-to-understand metric for social equity, helping international organizations like the World Bank identify regions at risk of social instability. A higher coefficient suggests significant disparity, which can impact economic growth and political stability.

Symbols

Variables

Gini = Gini Coeff, A = Area A, B = Area B

G ini

Gini Coeff

V a r iab l e

A

Area A

V a r iab l e

B

Area B

V a r iab l e

Walkthrough

Derivation

Geometric Definition: Gini Coefficient

The Gini coefficient is derived from the area between the Lorenz curve and the line of perfect equality.

1

Ratio of areas:

A is the area of inequality; A+B is the total area under the diagonal of perfect equality.

G ini = \frac{A}{A + B}

Result

G ini = \frac{A}{A + B}

Source: Lorenz Curve Analysis

Free formulas

Rearrangements

Solve for $G ini$

Make Gini the subject

G ini = \frac{A}{A + B}

Gini is already the subject of the formula.

Difficulty: 1/5

Solve for $A$

Make A the subject

A = \frac{G ini \times B}{1 - G ini}

To make A the subject of the Gini Coefficient formula, first clear the denominator by multiplying both sides by (A + B). Then, expand the expression, gather all terms containing A on one side, factor out A, and finally divide to fully isola...

Difficulty: 2/5

Solve for $B$

Make B the subject

B = A (\frac{1 - G ini}{G ini})

Start from the Gini Coefficient formula. First, clear the denominator and isolate the term `(A + B)`. Then, subtract `A` from the isolated `(A + B)` term to make `B` the subject.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a linear function passing through the origin with a slope of 1 divided by the sum of A and B. Because A is directly proportional to the Gini coefficient, the line rises steadily as A increases, with the domain restricted from zero to the total area of A plus B. For an economics student, this means that larger values of A represent higher levels of income inequality, while smaller values indicate a more equal distribution. The most important feature is that the linear relationship means doubling the are

Graph type: linear

Why it behaves this way

Intuition

The Gini coefficient presents a statistical picture of income distribution as a ratio of areas within a square plot, comparing the actual cumulative wealth distribution (Lorenz curve)

Gini

A quantitative measure of the statistical dispersion of income or wealth within a population.

A value of 0 indicates perfect equality (everyone has the same income), while a value of 1 indicates perfect inequality (one person has all the income).

A

The area between the line of perfect equality (the 45-degree line) and the Lorenz curve on a cumulative distribution plot.

This area visually represents the total 'gap' or shortfall from perfect income equality; a larger 'A' signifies greater inequality.

B

The area under the Lorenz curve, which plots the cumulative proportion of total income against the cumulative proportion of the population.

This area represents the actual distribution of income or wealth; its size, when combined with 'A', forms the total area under the line of perfect equality.

Free study cues

Insight

Canonical usage

The Gini coefficient is a dimensionless quantity, typically reported as a decimal between 0 and 1, or sometimes as a percentage.

Common confusion

Students may incorrectly attempt to assign physical units to the Gini coefficient or to the areas A and B, when they are inherently dimensionless proportions on a normalized graph.

Dimension note

The Gini coefficient is a ratio of two areas (A and A+B) derived from the Lorenz curve. On the Lorenz curve graph, both axes represent dimensionless proportions (cumulative population share and cumulative income share).

Unit systems

$A$ dimensionless · Represents the area between the line of perfect equality and the Lorenz curve on a graph where both axes are dimensionless proportions (cumulative population share and cumulative income share).

$B$ dimensionless · Represents the area beneath the Lorenz curve on a graph where both axes are dimensionless proportions (cumulative population share and cumulative income share).

Ballpark figures

Quantity:
Quantity:

One free problem

Practice Problem

A researcher studying a small town finds that the area between the line of equality and the Lorenz curve (A) is 0.12, while the area under the Lorenz curve (B) is 0.38. Calculate the Gini coefficient.

Area A0.12

Area B0.38

Solve for: $G ini$

Hint: Divide the area A by the total area of the triangle (A + B).

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

If area A is 10 and area B is 30, Gini = 10 / (10+30) = 0.25.

Study smarter

Tips

A result of 0 indicates absolute equality where everyone has the same income.
A result of 1 indicates absolute inequality where one person has all the income.
In a unit square model, the sum of A and B always equals 0.5.
The Gini coefficient is often expressed as a percentage by multiplying the result by 100.

Avoid these traps

Common Mistakes

Swapping A and B.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The Gini coefficient is derived from the area between the Lorenz curve and the line of perfect equality.

Use this formula when evaluating income distribution across different countries or analyzing wealth gaps within a single region over time. It is particularly useful for comparing the effectiveness of tax systems and social welfare programs in reducing inequality.

The Gini coefficient provides a single, easy-to-understand metric for social equity, helping international organizations like the World Bank identify regions at risk of social instability. A higher coefficient suggests significant disparity, which can impact economic growth and political stability.

Swapping A and B.