VIDEOS 1 TO 50

Wealth/Income Inequality: The Gini coefficient, The Lorenz Curve and The Pareto Distribution

Published: 2015/10/26

Channel: Quant Channel

Fun with gnuplot and the Lorenz attractor.

Published: 2013/05/20

Channel: Jip Hendrix

Lorenz system in OpenGL

Published: 2016/03/29

Channel: Samrat Man Singh

The L-Curve: Income Distribution of the U.S.

Published: 2007/02/05

Channel: David Chandler

Gini coefficient - Wiki Videos

Published: 2015/10/19

Channel: Wiki Videos

Diverging trajectories in the Lorenz attractor

Published: 2012/08/09

Channel: zh84

The Butterfly Curve

Published: 2010/02/17

Channel: einsteinino

Lorenz: Hitler's "Unbreakable" Cipher Machine

Published: 2014/09/07

Channel: singingbanana

Lorenz Attractor: Increasing Rayleigh Number

Published: 2007/04/22

Channel: Grantisu

Pattern from Tweaking Lorenz System

Published: 2016/07/07

Channel: Ambdellsh

Lorenz System representation with SciLab

Published: 2015/04/15

Channel: John Smith

The Gini Coefficient

Published: 2014/03/24

Channel: Bernard Chapin

Fun with gnuplot and the Harter--Heighway dragon curve

Published: 2013/05/11

Channel: Jip Hendrix

Lorenz 84 Strange Attractor driven by Bach

Published: 2011/06/23

Channel: Timothy Jones

Simple Model of the Lorenz Attractor

Published: 2010/02/06

Channel: HumaticS

Income Inequality (US and the world)

Published: 2011/04/19

Channel: axydlbaaxr

Mehr Erfolg mit der 80:20 Regel - Arm und Reich: Die Paretoverteilung

Published: 2015/03/15

Channel: Sonntagssoziologe

Phillips Curve

Published: 2008/04/05

Channel: pajholden

Rössler Attractor

Published: 2016/02/13

Channel: Physics Graphics

The Long-run Phillips Curve

Published: 2012/01/11

Channel: Jason Welker

Mathematical Butterfly

Published: 2011/10/13

Channel: Bill

Closed timelike curve

Published: 2015/02/12

Channel: Wikivoicemedia

Closed timelike curve

Published: 2016/01/22

Channel: WikiAudio

Gini-Koeffizient

Published: 2016/06/26

Channel: WikiTubia

How Wages Can Reduce Income Inequality & Increase Income Mobility

Published: 2014/02/28

Channel: John De Montfort

An Introduction to Concentration Inequalities and Statistical Learning Theory

Published: 2016/06/21

Channel: Microsoft Research

WEALTH GAP IN AMERICA (Wealth Inequality) - Economics

Published: 2010/08/29

Channel: Different Tnereffid

Economie filmpje sociale zekerheid

Published: 2014/10/12

Channel: dave duivenvoorden

Proof of The Pythagorean Theorem: Proof by Rearrangement

Published: 2009/07/31

Channel: CurvesOfFractals

Some Freedom for All *or* All Freedom for Some?

Published: 2010/01/30

Channel: churchofstfu

VCE HHD - The Human Development Index

Published: 2015/07/15

Channel: Engage Wiki

Line 22 7b272 Chirikov Standard Circle Map Rotor Hamiltonian Chaotic Wormholes WOW SETI

Published: 2014/04/04

Channel: theideagirlsays

No Agenda #250 (complete) Multidimensional Poverty Index

Published: 2011/06/09

Channel: Derek Sypniewski

Stereographic projection of Riemann sphere

Published: 2007/11/16

Channel: AMS Public Awareness Office

Two component reaction-diffusion model (Gray-Scott)

Published: 2016/06/02

Channel: coleraby

Line 22 7b208 Timelike Congruence Euclid Elements Vector Field Spacetime WOW SETI

Published: 2014/02/10

Channel: theideagirlsays

Statistik: Die Lorenzkurve Übung (Video 25) | Naturwissenschaften und Mathematik | Statistik

Published: 2014/03/25

Channel: sofa Mathe Wirtschaft

Rössler attractor

Published: 2016/01/22

Channel: WikiAudio

The Impact of Productivity on the Economy: Alan Greenspan on Finance, Economics (2002)

Published: 2016/01/14

Channel: Remember This

What does Gini coefficient mean?

Published: 2015/01/17

Channel: What Does That Mean?

Comment réparer un écran plat ( Tuto )

Published: 2015/09/22

Channel: Systeme D

What is GINI COEFFICIENT? What does GINI COEFFICIENT mean? GINI COEFFICIENT meaning & explanation

Published: 2016/08/10

Channel: Audiopedia

Line 22 7b209 Pascal's Triangle Spacetime Quantum Dance Logic Gate Formula WOW SETI

Published: 2014/02/11

Channel: theideagirlsays

Guns and Butter | Smash It & Grab It | Guns vs Butter Video

Published: 2012/10/10

Channel: Ron Holland

What does loxodromic mean?

Published: 2015/05/18

Channel: What Does That Mean?

Freedom

Published: 2008/08/03

Channel: JXjizo

Di nuovo in gioco - Trailer Italiano

Published: 2012/10/12

Channel: Fox Movie Channel.

Line 22 7b109 Twistor Spinors Minkowski Space Hyperplane Lorentz Formula WOW SETI

Published: 2013/12/23

Channel: theideagirlsays

Gini Index

Published: 2012/10/10

Channel: Christopher Hunt

11 THE HUMAN DEVELOPMENT INDEX

Published: 2015/04/28

Channel: Shafiq Fakir

From Wikipedia, the free encyclopedia

Not to be confused with Lorenz equation or Lorentz distribution.

In economics, the **Lorenz curve** is a graphical representation of the distribution of income or of wealth. It was developed by Max O. Lorenz in 1905 for representing inequality of the wealth distribution.

The curve is a graph 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, where it shows for the bottom *x*% of households, what percentage (*y*%) of the total income they have. The percentage 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 assets. In such use, many economists consider it to be a measure of social inequality.

The concept is useful in describing inequality among the size of individuals in ecology^{[1]} and in studies of biodiversity, where the cumulative proportion of species is plotted against the cumulative proportion of individuals.^{[2]} It is also useful in business modeling: e.g., in consumer finance, to measure the actual percentage *y*% of delinquencies attributable to the *x*% of people with worst risk scores.

Points on the Lorenz curve represent statements like "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."

The Gini coefficient 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.

The Lorenz curve 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 population of size *n*, with a sequence of values *y*_{i}, *i* = 1 to *n*, that are indexed in non-decreasing order ( *y*_{i} ≤ *y*_{i+1}), the Lorenz curve is the continuous piecewise linear function connecting the points ( *F*_{i}, *L*_{i} ), *i* = 0 to *n*, where *F*_{0} = 0, *L*_{0} = 0, and for *i* = 1 to *n*:

Note that the statement that the Lorenz curve gives the portion of the wealth or income held by a given portion of the population is only strictly true at the points defined above, but not at the points on the line segments between these points. For instance, in a population of 10 households, it doesn't make sense to say that 45% of them earn a certain portion of the total. If the population is modeled as a continuum then this subtlety disappears.

For a discrete probability function *f*(*y*), let *y*_{i}, *i* = 1 to *n*, be the points with non-zero probabilities indexed in increasing order ( *y*_{i} < *y*_{i+1}). The Lorenz curve is the continuous piecewise linear function connecting the points ( *F*_{i}, *L*_{i} ), *i* = 0 to *n*, where *F*_{0} = 0, *L*_{0} = 0, and for *i* = 1 to *n*:

For a probability density function *f*(*x*) with the cumulative distribution function *F*(*x*), the Lorenz curve *L*(*x*) is given by:

where 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 *F*(*x*) with inverse *x*(*F*), the Lorenz curve *L*(*F*) is directly given by:

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*_{1}) = inf {*y*:*F*(*y*) ≥*F*_{1}}

For an example of a Lorenz curve, see Pareto distribution.

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 a continuous function. 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 and the Lorenz asymmetry coefficient.^{[1]}

The Lorenz curve cannot rise above the line of perfect equality. If the variable being measured cannot take negative values, the Lorenz curve:

- cannot sink below the line of perfect inequality,
- is increasing.

Note however that a Lorenz curve for net worth 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

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*_{− 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

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:

- If the Lorenz curve is twice differentiable, then the probability density function
*f*(*x*) exists at that point and:

- 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 mean absolute deviation:

Wikimedia Commons has media related to .Lorenz curve |

- Distribution (economics)
- Distribution of wealth
- Welfare economics
- Income inequality metrics
- Gini coefficient
- Hoover index (a.k.a. Robin Hood index)
- ROC analysis
- Social welfare (political science)
- Economic inequality
- Zipf's law
- Pareto distribution
- Mean deviation

- ^
^{a}^{b}Damgaard, Christian; Jacob Weiner (2000). "Describing inequality in plant size or fecundity".*Ecology*.**81**(4): 1139–1142. doi:10.1890/0012-9658(2000)081[1139:DIIPSO]2.0.CO;2. **^**Wittebolle, Lieven; et al. (2009). "Initial community evenness favours functionality under selective stress".*Nature*.**458**(7238): 623–626. Bibcode:2009Natur.458..623W. doi:10.1038/nature07840. PMID 19270679.

- Lorenz, M. O. (1905). "Methods of measuring the concentration of wealth".
*Publications of the American Statistical Association*. Publications of the American Statistical Association, Vol. 9, No. 70.**9**(70): 209–219. Bibcode:1905PAmSA...9..209L. doi:10.2307/2276207. JSTOR 2276207. - Gastwirth, Joseph L. (1972). "The Estimation of the Lorenz Curve and Gini Index".
*The Review of Economics and Statistics*. The Review of Economics and Statistics, Vol. 54, No. 3.**54**(3): 306–316. doi:10.2307/1937992. JSTOR 1937992. - Chakravarty, S. R. (1990).
*Ethical Social Index Numbers*. New York: Springer-Verlag. ISBN 0-387-52274-3. - Anand, Sudhir (1983).
*Inequality and Poverty in Malaysia*. New York: Oxford University Press. ISBN 0-19-520153-1.

- WIID: World Income Inequality Database, the most comprehensive source of information on inequality, collected by WIDER (World Institute for Development Economics Research, part of United Nations University)
- glcurve: Stata 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: Online and downloadable scripts (Python and Lua) for Atkinson, Gini, and Hoover inequalities
- Users of the R data analysis software can install the "ineq" package which allows for computation of a variety of inequality indices including Gini, Atkinson, Theil.
- A MATLAB Inequality Package, including code for computing Gini, Atkinson, Theil indexes and for plotting the Lorenz Curve. Many examples are available.
- A complete handout about the Lorenz curve including various applications, including an Excel spreadsheet graphing Lorenz curves and calculating Gini coefficients as well as coefficients of variation.
- LORENZ 3.0 is a Mathematica notebook which draw sample Lorenz curves and calculates Gini coefficients and Lorenz asymmetry coefficients from data in an Excel sheet.

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