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Encyclopedia > Lorenz curve

The Lorenz curve is a graphical representation of the cumulative distribution function of a probability distribution; it is a graph showing the proportion of the distribution assumed by the bottom y% of the values. 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 uses, some political doctrines (e.g. socialism) consider it to represent social inequality. It was developed by Max O. Lorenz in 1905 for representing income distribution. In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... Income, generally defined, is the money that is received as a result of the normal business activities of an individual or a business. ... A percentage is a way of expressing a proportion, a ratio or a fraction as a whole number, by using 100 as the denominator. ... In business and accounting an asset is anything owned which can produce future economic benefit, whether in possession or by right to take possession, by a person or a group acting together, e. ... Socialism refers to a broad array of doctrines, and may also refer to political movements that aspire to put these doctrines into practice. ... Social inequality refers to disparities in the distribution of material wealth in a society. ... Max Otto Lorenz (1880 – 1962) was an American economist who developed the Lorenz curve in 1905 to describe income inequalities. ... 1905 (MCMV) was a common year starting on Sunday (see link for calendar). ...

A typical Lorenz curve

Contents

Image File history File links Lorenz-curve1. ...


Explanation

Every point on the Lorenz curve represents a statement 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 or the 45° line.


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.


If the variable being measured cannot take negative values, it is impossible for the Lorenz curve to rise above the line of perfect equality, or sink below the line of perfect inequality; it is increasing and convex to the y-axis. In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...


The Gini coefficient is the area between the line of perfect equality and the observed Lorenz curve, as a percentage of the area between the line of perfect equality and the line of perfect inequality. Graphical representation of the Gini coefficient The Gini coefficient is a measure of inequality of a distribution, defined as the ratio of area between the Lorenz curve of the distribution and the curve of the uniform distribution, to the area under the uniform distribution. ...


For any distribution, the Lorenz curve L(F)  is written in terms of the probability density function (f(x)) or the cumulative distribution function (F(x)) as: In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

L(F)=frac{int_{-infty}^{x(F)} xf(x),dx}{int_{-infty}^infty xf(x),dx} =frac{int_0^F x(F'),dF'}{int_0^1 x(F'),dF'}

where x(F) is the inverse of the cumulative distribution function F(x) (for example, see the Pareto distribution). The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ...


References

Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American Statistical Association. 9: 209-219.


[also Will Dawson's contributions]


See also

To meet Wikipedias quality standards, this article or section may require cleanup. ... Welfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine the allocational efficiency of a macroeconomy and the income distribution consequences associated with it. ... Income inequality metrics or income distribution metrics are techniques used by economists to measure the distribution of income among members of a society. ... Graphical representation of the Gini coefficient The Gini coefficient is a measure of inequality of a distribution, defined as the ratio of area between the Lorenz curve of the distribution and the curve of the uniform distribution, to the area under the uniform distribution. ... The Robin Hood index is a measure of income inequality. ... Receiver operating characteristic analysis (ROC analysis) provides tools to select possibly optimal models and to discard suboptimal ones independently from (and prior to specifying) the cost context or the class distribution. ... Social welfare can be taken to mean the welfare or well-being of a society. ... A poster printed by the Industrial Workers of the World, dramatising economic inequality under capitalism and aiming to gain support for Industrial unionism. ... Originally, Zipfs law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. ... The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ...

External links

  • Measuring income inequality: a new database, with link to dataset
  • Free Online Software (Calculator) computes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset

  Results from FactBites:
 
Definitions Country Codes geographic.org Courty Profiles - Flags, Maps, Economy, Geography, Climate, Natural ... (12538 words)
The index is calculated from the Lorenz curve, in which cumulative family income is plotted against the number of families arranged from the poorest to the richest.
The index is the ratio of (a) the area between a country's Lorenz curve and the 45 degree helping line to (b) the entire triangular area under the 45 degree line.
If income were distributed with perfect equality, the Lorenz curve would coincide with the 45 degree line and the index would be zero; if income were distributed with perfect inequality, the Lorenz curve would coincide with the horizontal axis and the right vertical axis and the index would be 100.
LORENZ CURVE (420 words)
Lorenz curve was developed by Max O. Lorenz in order to describe the extent of inequality in a society.
The Lorenz curve is a graphical representation of the proportionality of a distribution (the cumulative percentage of the values).
The Gini coefficient is defined graphically as a ratio of two surfaces involving the summation of all vertical deviations between the Lorenz curve and the perfect equality line (A) divided by the difference between the perfect equality and perfect inequality lines (A+B).
  More results at FactBites »

 
 

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