In

_{1},...,''y''_{''n''} in non-decreasing order ( ''y''_{''i''} ≤ ''y''_{''i''+1}) and their probabilities $f(y\_j):=\backslash Pr(Y=y\_j)$ the Lorenz curve is the _{''i''}, ''L''_{''i''} ), ''i'' = 0 to ''n'', where ''F''_{0} = 0, ''L''_{0} = 0, and for ''i'' = 1 to ''n'':
:$F\_i\; :=\; \backslash Sigma\_^i\; \backslash ;\; f(y\_j)\backslash ,$
:$S\_i\; :=\; \backslash Sigma\_^i\; \backslash ;\; f(y\_j)\backslash ,y\_j\backslash ,$
:$L\_i\; :=\; S\_i\; /\; S\_n\; \backslash ,$
When all ''y''_{''i''} are equally probable with probabilities 1/''n'' this simplifies to
:$F\_i\; =\; i/n\backslash ,$
:$S\_i\; =\; 1/n\backslash Sigma\_^i\; \backslash ;\; y\_j\backslash ,$
:$L\_i\; =\; S\_i\; /\; S\_n\; \backslash ,$
For a _{1}) = inf
For an example of a Lorenz curve, see

_{X}(''F''), then −''X'' has the Lorenz curve:
: ''L''_{ − X } = 1 − ''L''_{ X }(1 − ''F'')
The Lorenz curve is changed by translations so that the equality gap ''F'' − ''L''(''F'') changes in proportion to the ratio of the original and translated means. If ''X'' is a random variable with a Lorenz curve ''L''_{ X }(''F'') and mean ''μ''_{ X }, then for any constant ''c'' ≠ −''μ''_{ X }, ''X'' + ''c'' has a Lorenz curve defined by:
:$F\; -\; L\_(F)\; =\; \backslash frac\; (\; F\; -\; L\_X(F))\backslash ,$
For a cumulative distribution function ''F''(''x'') with mean ''μ'' and (generalized) inverse ''x''(''F''), then for any ''F'' with 0 < ''F'' < 1 :
*If the Lorenz curve is differentiable:
::$\backslash frac\; =\; \backslash frac$
*If the Lorenz curve is twice differentiable, then the probability density function ''f''(''x'') exists at that point and:
::$\backslash frac\; =\; \backslash frac\backslash ,$
*If ''L''(''F'') is continuously differentiable, then the tangent of ''L''(''F'') is parallel to the line of perfect equality at the point ''F''(''μ''). This is also the point at which the equality gap ''F'' − ''L''(''F''), the vertical distance between the Lorenz curve and the line of perfect equality, is greatest. The size of the gap is equal to half of the relative

WIID

World Income Inequality Database, a source of information on inequality, collected by WIDER (World Institute for Development Economics Research, part of United Nations University)

glcurve

Free add-on to STATA to compute inequality and poverty measures

Free Online Software (Calculator)

computes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset * Free Calculator

an

downloadable scripts

(

R

data analysis software can install the "ineq" package which allows for computation of a variety of inequality indices including Gini, Atkinson, Theil. *

MATLAB Inequality Package

including code for computing Gini, Atkinson, Theil indexes and for plotting the Lorenz Curve. Many examples are available. *

complete handout

about the Lorenz curve including various applications, including a

Excel spreadsheet

graphing Lorenz curves and calculating Gini coefficients as well as coefficients of variation.

LORENZ 3.0

is aGini coefficient
In economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science ...

s and

economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interact ...

, the Lorenz curve is a graphical representation of the distribution of income In economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour ...

or of wealth
Wealth is the abundance of valuable financial asset
A financial asset is a non-physical asset whose value is derived from a contractual claim, such as deposit (finance), bank deposits, bond (finance), bonds, and participations in companies' sh ...

. It was developed by Max O. Lorenz in 1905 for representing inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...

of the wealth distribution
The distribution of wealth is a comparison of the wealth of various members or groups in a society
A society is a Social group, group of individuals involved in persistent Social relation, social interaction, or a large social group shar ...

.
The curve is a graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

showing the proportion of overall income or wealth assumed by the bottom ''x''% of the people, although this is not rigorously true for a finite population (see below). It is often used to represent income distribution In economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour ...

, where it shows for the bottom ''x''% of households, what percentage (''y''%) of the total income they have. The percentage
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of households is plotted on the ''x''-axis, the percentage of income on the ''y''-axis. It can also be used to show distribution of asset
In financial accounting
Financial accounting is the field of accounting
Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic entities such a ...

s. In such use, many economists consider it to be a measure of social inequality
Social inequality occurs when resources in a given society are distributed unevenly, typically through Norm (social), norms of allocation, that engender specific patterns along lines of socially defined categories of persons. It is the differ ...

.
The concept is useful in describing inequality among the size of individuals in ecology
Ecology (from el, οἶκος, "house" and el, -λογία, label=none, "study of") is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms
In biol ...

and in studies of biodiversity
Biodiversity is the biological variety and Genetic variability, variability of life, life on Earth. Biodiversity is a measure of variation at the Genetics, genetic, species, and ecosystem level. Terrestrial biodiversity is usually greater near ...

, where the cumulative proportion of species is plotted against the cumulative proportion of individuals. It is also useful in business modeling
Business process modeling (BPM) in business process management and systems engineering is the activity of process modeling, representing processes of an enterprise, so that the current process may be analyzed, improved, and automated. BPM is typical ...

: e.g., in consumer finance
An alternative financial service (AFS) is a financial service
Financial services are the Service (economics), economic services provided by the finance industry, which encompasses a broad range of businesses that manage money, including credit ...

, to measure the actual percentage ''y''% of delinquencies attributable to the ''x''% of people with worst risk scores.
Explanation

Data from 2005. Points on the Lorenz curve represent statements such as, "the bottom 20% of all households have 10% of the total income." A perfectly equal income distribution would be one in which every person has the same income. In this case, the bottom ''N''% of society would always have ''N''% of the income. This can be depicted by the straight line ''y'' = ''x''; called the "line of perfect equality." By contrast, a perfectly unequal distribution would be one in which one person has all the income and everyone else has none. In that case, the curve would be at ''y'' = 0% for all ''x'' < 100%, and ''y'' = 100% when ''x'' = 100%. This curve is called the "line of perfect inequality." TheGini coefficient
In economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science ...

is the ratio of the area between the line of perfect equality and the observed Lorenz curve to the area between the line of perfect equality and the line of perfect inequality. The higher the coefficient, the more unequal the distribution is. In the diagram on the right, this is given by the ratio ''A''/(''A+B''), where ''A'' and ''B'' are the areas of regions as marked in the diagram.
Definition and calculation

The Lorenz curve is a probability plot (aP–P plot
In statistics, a P–P plot (probability–probability plot or percent–percent plot or P value plot) is a probability plot for assessing how closely two data set
A data set (or dataset) is a collection of data. In the case of tabular data, a d ...

) comparing the distribution of a variable against a hypothetical uniform distribution of that variable. It can usually be represented by a function ''L''(''F''), where ''F'', the cumulative portion of the population, is represented by the horizontal axis, and ''L'', the cumulative portion of the total wealth or income, is represented by the vertical axis.
For a discrete distribution of Y given by values ''y''continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

piecewise linear function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

connecting the points ( ''F''continuous distribution
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

with the probability density function
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

''f'' and the cumulative distribution function
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...

''F'', the Lorenz curve ''L'' is given by:
:$L(F(x))=\backslash frac\; =\backslash frac$
where $\backslash mu$ denotes the average. The Lorenz curve ''L(F)'' may then be plotted as a function parametric in x: ''L(x)'' vs. ''F(x)''. In other contexts, the quantity computed here is known as the length biased (or size biased) distribution; it also has an important role in renewal theory.
Alternatively, for a cumulative distribution function
In probability theory
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and space ...

''F''(''x'') with inverse ''x''(''F''), the Lorenz curve ''L''(''F'') is directly given by:
:$L(F)=\backslash frac$
The inverse ''x''(''F'') may not exist because the cumulative distribution function has intervals of constant values. However, the previous formula can still apply by generalizing the definition of ''x''(''F''):
:''x''(''F''Pareto distribution
The Pareto distribution, named after the Italian civil engineer
A civil engineer is a person who practices civil engineering
Civil engineering is a professional engineering
Regulation and licensure in engineering is established by variou ...

.
Properties

A Lorenz curve always starts at (0,0) and ends at (1,1). The Lorenz curve is not defined if the mean of the probability distribution is zero or infinite. The Lorenz curve for a probability distribution is acontinuous function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. However, Lorenz curves representing discontinuous functions can be constructed as the limit of Lorenz curves of probability distributions, the line of perfect inequality being an example.
The information in a Lorenz curve may be summarized by the Gini coefficient
In economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science ...

and the Lorenz asymmetry coefficient
The Lorenz asymmetry coefficient (LAC) is a summary statistic of the Lorenz curve
upright=1.2, A typical Lorenz curve
In economics
Economics () is the social science that studies how people interact with value; in particular, the Producti ...

.
The Lorenz curve cannot rise above the line of perfect equality.
A Lorenz curve that never falls beneath a second Lorenz curve and at least once runs above it, has Lorenz dominance over the second one.
If the variable being measured cannot take negative values, the Lorenz curve:
*cannot sink below the line of perfect inequality,
*is increasing
Image:Monotonicity example3.png, Figure 3. A function that is not monotonic
In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that pres ...

.
Note however that a Lorenz curve for net worth
Net worth is the value of all the non-financial and financial asset
In financial accounting
Financial accounting is the field of accounting
Accounting or Accountancy is the measurement, processing, and communication of financial and no ...

would start out by going negative due to the fact that some people have a negative net worth because of debt.
The Lorenz curve is invariant under positive scaling. If ''X'' is a random variable, for any positive number ''c'' the random variable ''c'' X has the same Lorenz curve as ''X''.
The Lorenz curve is flipped twice, once about F = 0.5 and once about ''L'' = 0.5, by negation. If ''X'' is a random variable with Lorenz curve ''L''mean absolute deviation
The average absolute deviation (AAD) of a data set is the average
In colloquial
Colloquialism or colloquial language is the style (sociolinguistics), linguistic style used for casual (informal) communication. It is the most common functional sty ...

:
::$F(\backslash mu)\; -\; L(F(\backslash mu))\; =\; \backslash frac$
See also

*Distribution (economics)
In economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant ...

* Distribution of wealth
The distribution of wealth is a comparison of the wealth
Wealth is the abundance (economics), abundance of Value (economics), valuable financial assets or property, physical possessions which can be converted into a form that can be used fo ...

* Welfare economics
Welfare economics is a branch of economics that uses microeconomic
Microeconomics is a branch of mainstream economics
Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, tha ...

* Income inequality metrics
In microeconomics, income is the Consumption (economics), consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms.Smith's financial dictionary. Smith, Howard Irving. 190 ...

* Gini coefficient
In economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science ...

* Hoover index The Hoover index, also known as the Robin Hood index or the Schutz index, is a measure of income
In microeconomics
Microeconomics is a branch of mainstream economics
Mainstream economics is the body of knowledge, theories, and models of e ...

(a.k.a. Robin Hood index)
* ROC analysis
* Social welfare (political science)
* Economic inequality
There are wide varieties of economic inequality, most notably measured using the distribution of income In economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), di ...

* Zipf's law
Zipf's law (, not as in German) is an empirical law
Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage ...

* Pareto distribution
The Pareto distribution, named after the Italian civil engineer
A civil engineer is a person who practices civil engineering
Civil engineering is a professional engineering
Regulation and licensure in engineering is established by variou ...

* Mean deviation
References

Further reading

* * * *External links

WIID

World Income Inequality Database, a source of information on inequality, collected by WIDER (World Institute for Development Economics Research, part of United Nations University)

glcurve

Stata
Stata (, , alternatively , occasionally stylized as STATA) is a general-purpose statistical
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying stati ...

module to plot Lorenz curve (type "findit glcurve" or "ssc install glcurve" in Stata prompt to install)
Free add-on to STATA to compute inequality and poverty measures

Free Online Software (Calculator)

computes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset * Free Calculator

an

downloadable scripts

(

Python
Python may refer to:
* Pythonidae
The Pythonidae, commonly known as pythons, are a family of nonvenomous snakes found in Africa, Asia, and Australia. Among its members are some of the largest snakes in the world. Ten genera and 42 species
...

and Lua) for Atkinson, Gini, and Hoover inequalities
* Users of thR

data analysis software can install the "ineq" package which allows for computation of a variety of inequality indices including Gini, Atkinson, Theil. *

MATLAB Inequality Package

including code for computing Gini, Atkinson, Theil indexes and for plotting the Lorenz Curve. Many examples are available. *

complete handout

about the Lorenz curve including various applications, including a

Excel spreadsheet

graphing Lorenz curves and calculating Gini coefficients as well as coefficients of variation.

LORENZ 3.0

is a

Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning
Machine learning (ML) is the study of computer algorithms that can improve automatically through experi ...

notebook which draw sample Lorenz curves and calculates Lorenz asymmetry coefficient
The Lorenz asymmetry coefficient (LAC) is a summary statistic of the Lorenz curve
upright=1.2, A typical Lorenz curve
In economics
Economics () is the social science that studies how people interact with value; in particular, the Producti ...

s from data in an Excel sheet.
{{DEFAULTSORT:Lorenz Curve
Economics curves
Welfare economics
Statistical charts and diagrams
Income inequality metrics