In economics, the
Gini coefficient
Gini coefficient (/ˈdʒiːni/ JEE-nee), sometimes called Gini index, or Gini ratio, is a measure of statistical dispersion intended to represent the income or wealth distribution of a nation's residents, and is the most commonly used measurement of inequality. It was developed by the Italian statistician and sociologist Corrado Gini and published in his 1912 paper Variability
and Mutability (Italian: Variabilità e
mutabilità).[1][2]
The
Gini coefficient
Gini coefficient measures the inequality among values of a frequency distribution (for example, levels of income). A Gini coefficient of zero expresses perfect equality, where all values are the same (for example, where everyone has the same income). A Gini coefficient of 1 (or 100%) expresses maximal inequality among values (e.g., for a large number of people, where only one person has all the income or consumption, and all others have none, the Gini coefficient will be very nearly one).[3][4] However, a value greater than one may occur if some persons represent negative contribution to the total (for example, having negative income or wealth). For larger groups, values close to one are very unlikely in practice. Given the normalization of both the cumulative population and the cumulative share of income used to calculate the Gini coefficient, the measure is not overly sensitive to the specifics of the income distribution, but rather only on how incomes vary relative to the other members of a population. The exception to this is in the redistribution of income resulting in a minimum income for all people. When the population is sorted, if their income distribution were to approximate a well-known function, then some representative values could be calculated. The Gini coefficient
Gini coefficient was proposed by Gini as a measure of inequality of income or wealth.[5] For OECD countries, in the late 20th century, considering the effect of taxes and transfer payments, the income Gini coefficient
Gini coefficient ranged between 0.24 and 0.49, with Slovenia being the lowest and Mexico the highest.[6] African countries had the highest pre-tax Gini coefficients in 2008–2009, with South Africa the world's highest, variously estimated to be 0.63 to 0.7,[7][8] although this figure drops to 0.52 after social assistance is taken into account, and drops again to 0.47 after taxation.[9] The global income Gini coefficient
Gini coefficient in 2005 has been estimated to be between 0.61 and 0.68 by various sources.[10][11] There are some issues in interpreting a Gini coefficient. The same value may result from many different distribution curves. The demographic structure should be taken into account. Countries with an aging population, or with a baby boom, experience an increasing pre-tax Gini coefficient
Gini coefficient even if real income distribution for working adults remains constant. Scholars have devised over a dozen variants of the Gini coefficient.[12][13][14] Contents 1 Definition 2 Calculation 2.1 Example: two levels of income 2.2 Alternate expressions 2.3 Discrete probability distribution 2.4 Continuous probability distribution 2.5 Other approaches 3 Generalized inequality indices 4 Gini coefficients of income distributions 4.1 Regional income Gini indices 4.2 World income Gini index since 1800s 5 Gini coefficients of social development 5.1
6 Features of Gini coefficient 7 Countries by Gini Index 8 Limitations of Gini coefficient 9 Alternatives to Gini coefficient 10 Relation to other statistical measures 11 Other uses 12 See also 13 References 14 Further reading 15 External links Definition[edit]
Graphical representation of the Gini coefficientThe graph shows that
the
x ¯ displaystyle bar x , to normalize for scale. If xi is the wealth or income of person i,
and there are n persons, then the
G = ∑ i = 1 n ∑ j = 1 n
x i − x j
2 ∑ i = 1 n ∑ j = 1 n x j = ∑ i = 1 n ∑ j = 1 n
x i − x j
2 n ∑ i = 1 n x i = ∑ i = 1 n ∑ j = 1 n
x i − x j
2 n 2 x ¯ displaystyle G= frac displaystyle sum _ i=1 ^ n sum _ j=1 ^ n leftx_ i -x_ j right displaystyle 2sum _ i=1 ^ n sum _ j=1 ^ n x_ j = frac displaystyle sum _ i=1 ^ n sum _ j=1 ^ n leftx_ i -x_ j right displaystyle 2nsum _ i=1 ^ n x_ i = frac displaystyle sum _ i=1 ^ n sum _ j=1 ^ n leftx_ i -x_ j right displaystyle 2n^ 2 bar x When the income (or wealth) distribution is given as a continuous
probability distribution function p(x), the
G = 1 2 μ ∫ − ∞ ∞ ∫ − ∞ ∞ p ( x ) p ( y )
x − y
d x d y displaystyle G= frac 1 2mu int _ -infty ^ infty int _ -infty ^ infty p(x)p(y),x-y,dx,dy where μ = ∫ − ∞ ∞ x p ( x ) d x displaystyle textstyle mu =int _ -infty ^ infty xp(x),dx is the mean of the distribution, and the lower limits of integration may be replaced by zero when all incomes are positive. Calculation[edit]
Example: two levels of income[edit]
This section's tone or style may not reflect the encyclopedic tone
used on. See's guide to writing better articles
for suggestions. (February 2019) (Learn how and when to remove this
template message)
Richest u of population (red) equally share f of all income or
wealth; others (green) equally share remainder: G = f − u. A smooth
distribution (blue) with same u and f always has G > f − u.
The most equal society will be one in which every person receives the
same income (G = 0); the most unequal society will be one in which a
single person receives 100% of the total income and the remaining N
− 1 people receive none (G = 1 − 1/N).
While the income distribution of any particular country need not
follow simple functions, these functions give a qualitative
understanding of the income distribution in a nation given the Gini
coefficient.
An informative simplified case just distinguishes two levels of
income, low and high. If the high income group is a proportion u of
the population and earns a proportion f of all income, then the Gini
coefficient is f − u. An actual more graded distribution with these
same values u and f will always have a higher
Alternate expressions[edit] In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve. For example, (taking y to mean the income or wealth of a person or household): For a population uniform on the values yi, i = 1 to n, indexed in non-decreasing order (yi ≤ yi+1): G = 1 n ( n + 1 − 2 ( ∑ i = 1 n ( n + 1 − i ) y i ∑ i = 1 n y i ) ) . displaystyle G= frac 1 n left(n+1-2left( frac sum _ i=1 ^ n (n+1-i)y_ i sum _ i=1 ^ n y_ i right)right). This may be simplified to: G = 2 ∑ i = 1 n i y i n ∑ i = 1 n y i − n + 1 n . displaystyle G= frac 2sum _ i=1 ^ n iy_ i nsum _ i=1 ^ n y_ i - frac n+1 n . This formula actually applies to any real population, since each
person can be assigned his or her own yi.[17]
Since the
G ( S ) = 1 n − 1 ( n + 1 − 2 ( ∑ i = 1 n ( n + 1 − i ) y i ∑ i = 1 n y i ) ) displaystyle G(S)= frac 1 n-1 left(n+1-2left( frac sum _ i=1 ^ n (n+1-i)y_ i sum _ i=1 ^ n y_ i right)right) is a consistent estimator of the population Gini coefficient, but is not, in general, unbiased. Like G, G(S) has a simpler form: G ( S ) = 1 − 2 n − 1 ( n − ∑ i = 1 n i y i ∑ i = 1 n y i ) . displaystyle G(S)=1- frac 2 n-1 left(n- frac sum _ i=1 ^ n iy_ i sum _ i=1 ^ n y_ i right). There does not exist a sample statistic that is in general an unbiased estimator of the population Gini coefficient, like the relative mean absolute difference. Discrete probability distribution[edit]
For a discrete probability distribution with probability mass function
f ( yi ), i = 1 to n, where f ( yi ) is the fraction of the
population with income or wealth yi >0, the
G = 1 2 μ ∑ i = 1 n ∑ j = 1 n f ( y i ) f ( y j )
y i − y j
displaystyle G= frac 1 2mu sum limits _ i=1 ^ n sum limits _ j=1 ^ n ,f(y_ i )f(y_ j )y_ i -y_ j where μ = ∑ i = 1 n y i f ( y i ) . displaystyle mu =sum limits _ i=1 ^ n y_ i f(y_ i ). If the points with nonzero probabilities are indexed in increasing order (yi < yi+1) then: G = 1 − ∑ i = 1 n f ( y i ) ( S i − 1 + S i ) S n displaystyle G=1- frac sum _ i=1 ^ n f(y_ i )(S_ i-1 +S_ i ) S_ n where S i = ∑ j = 1 i f ( y j ) y j displaystyle S_ i =sum _ j=1 ^ i f(y_ j ),y_ j , and S 0 = 0. displaystyle S_ 0 =0. These formulae are also applicable in the limit as n → ∞ . displaystyle nrightarrow infty . Continuous probability distribution[edit]
When the population is large, the income distribution may be
represented by a continuous probability density function f(x) where
f(x) dx is the fraction of the population with wealth or income in the
interval dx about x. If F(x) is the cumulative distribution function
for f(x), then the
B = ∫ 0 1 L ( F ) d F . displaystyle B=int _ 0 ^ 1 L(F),dF. The
G = 1 − 1 μ ∫ 0 ∞ ( 1 − F ( y ) ) 2 d y = 1 μ ∫ 0 ∞ F ( y ) ( 1 − F ( y ) ) d y displaystyle G=1- frac 1 mu int _ 0 ^ infty (1-F(y))^ 2 ,dy= frac 1 mu int _ 0 ^ infty F(y)(1-F(y)),dy The latter result comes from integration by parts. (Note that this
formula can be applied when there are negative values if the
integration is taken from minus infinity to plus infinity.)
The
G = 1 2 μ ∫ 0 1 ∫ 0 1
Q ( F 1 ) − Q ( F 2 )
d F 1 d F 2 . displaystyle G= frac 1 2mu int _ 0 ^ 1 int _ 0 ^ 1 Q(F_ 1 )-Q(F_ 2 ),dF_ 1 ,dF_ 2 . For some functional forms, the Gini index can be calculated explicitly. For example, if y follows a lognormal distribution with the standard deviation of logs equal to σ displaystyle sigma , then G = erf ( σ 2 ) displaystyle G=operatorname erf left( frac sigma 2 right) where erf displaystyle operatorname erf is the error function ( since G = 2 ϕ ( σ 2 ) − 1 displaystyle G=2phi left( frac sigma sqrt 2 right)-1 , where ϕ displaystyle phi is the cumulative standard normal distribution).[18] In the table below, some examples are shown.[citation needed] The Dirac delta distribution represents the case where everyone has the same wealth (or income); it implies that there are no variations at all between incomes.
Dirac delta function δ ( x − x 0 ) , x 0 > 0 displaystyle delta (x-x_ 0 ),,x_ 0 >0 0 Uniform distribution 1 b − a a ≤ x ≤ b 0 o t h e r w i s e displaystyle begin cases frac 1 b-a &aleq xleq b\0&mathrm otherwise end cases ( b − a ) 3 ( b + a ) displaystyle frac (b-a) 3(b+a) Exponential distribution λ e − x λ , x > 0 displaystyle lambda e^ -xlambda ,,,x>0 1 / 2 displaystyle 1/2 Log-normal distribution 1 x σ 2 π e − 1 2 ( ln ( x ) − μ σ ) 2 displaystyle frac 1 xsigma sqrt 2pi e^ - frac 1 2 left( frac ln ,(x)-mu sigma right)^ 2 erf ( σ / 2 ) displaystyle textrm erf (sigma /2) Pareto distribution α k α x α + 1 x ≥ k 0 x < k displaystyle begin cases frac alpha k^ alpha x^ alpha +1 &xgeq k\0&x<kend cases 1 0 < α < 1 1 2 α − 1 α ≥ 1 displaystyle begin cases 1&0<alpha <1\ frac 1 2alpha -1 &alpha geq 1end cases Chi-squared distribution 2 − k / 2 e − x / 2 x k / 2 − 1 Γ ( k / 2 ) displaystyle frac 2^ -k/2 e^ -x/2 x^ k/2-1 Gamma (k/2) 2 Γ ( 1 + k 2 ) k Γ ( k / 2 ) π displaystyle frac 2,Gamma left( frac 1+k 2 right) k,Gamma (k/2) sqrt pi Gamma distribution e − x / θ x k − 1 θ − k Γ ( k ) displaystyle frac e^ -x/theta x^ k-1 theta ^ -k Gamma (k) Γ ( 2 k + 1 2 ) k Γ ( k ) π displaystyle frac Gamma left( frac 2k+1 2 right) k,Gamma (k) sqrt pi Weibull distribution k λ ( x λ ) k − 1 e − ( x / λ ) k displaystyle frac k lambda ,left( frac x lambda right)^ k-1 e^ -(x/lambda )^ k 1 − 2 − 1 / k displaystyle 1-2^ -1/k Beta distribution x α − 1 ( 1 − x ) β − 1 B ( α , β ) displaystyle frac x^ alpha -1 (1-x)^ beta -1 B(alpha ,beta ) ( 2 α ) B ( α + β , α + β ) B ( α , α ) B ( β , β ) displaystyle left( frac 2 alpha right) frac B(alpha +beta ,alpha +beta ) B(alpha ,alpha )B(beta ,beta ) Other approaches[edit]
Sometimes the entire
Xk is the cumulated proportion of the population variable, for k =
0,...,n, with X0 = 0, Xn = 1.
Yk is the cumulated proportion of the income variable, for k =
0,...,n, with Y0 = 0, Yn = 1.
Yk should be indexed in non-decreasing order (Yk > Yk – 1)
If the
G 1 = 1 − ∑ k = 1 n ( X k − X k − 1 ) ( Y k + Y k − 1 ) displaystyle G_ 1 =1-sum _ k=1 ^ n (X_ k -X_ k-1 )(Y_ k +Y_ k-1 )
is the resulting approximation for G. More accurate results can be
obtained using other methods to approximate the area B, such as
approximating the
k = A + N ( 0 , s 2 / y k ) displaystyle k=A+ N(0,s^ 2 /y_ k ) Ogwang showed that G can be expressed as a function of the weighted least squares estimate of the constant A and that this can be used to speed up the calculation of the jackknife estimate for the standard error. Giles (2004) argued that the standard error of the estimate of A can be used to derive that of the estimate of G directly without using a jackknife at all. This method only requires the use of ordinary least squares regression after ordering the sample data. The results compare favorably with the estimates from the jackknife with agreement improving with increasing sample size.[19] However it has since been argued that this is dependent on the model's assumptions about the error distributions (Ogwang 2004) and the independence of error terms (Reza & Gastwirth 2006) and that these assumptions are often not valid for real data sets. It may therefore be better to stick with jackknife methods such as those proposed by Yitzhaki (1991) and Karagiannis and Kovacevic (2000). The debate continues.[citation needed] Guillermina Jasso[20] and Angus Deaton[21] independently proposed the following formula for the Gini coefficient: G = N + 1 N − 1 − 2 N ( N − 1 ) μ ( ∑ i = 1 n P i X i ) displaystyle G= frac N+1 N-1 - frac 2 N(N-1)mu (sum _ i=1 ^ n P_ i X_ i ) where μ displaystyle mu is mean income of the population, Pi is the income rank P of person i, with income X, such that the richest person receives a rank of 1 and the poorest a rank of N. This effectively gives higher weight to poorer people in the income distribution, which allows the Gini to meet the Transfer Principle. Note that the Jasso-Deaton formula rescales the coefficient so that its value is 1 if all the X i displaystyle X_ i are zero except one. Note however Allison's reply on the need to divide by N² instead.[22] FAO explains another version of the formula.[23] Generalized inequality indices[edit]
See also: Generalized entropy index
The
r j = x j / x ¯ displaystyle r_ j =x_ j / overline x , equals 1 for all j units in some population (for example, there is perfect income equality when everyone's income x j displaystyle x_ j equals the mean income x ¯ displaystyle overline x , so that r j = 1 displaystyle r_ j =1 for everyone). Measures of inequality, then, are measures of the average deviations of the r j = 1 displaystyle r_ j =1 from 1; the greater the average deviation, the greater the inequality. Based on these observations the inequality indices have this common form:[24] Inequality = ∑ j p j f ( r j ) , displaystyle text Inequality =sum _ j p_ j ,f(r_ j ), where pj weights the units by their population share, and f(rj) is a function of the deviation of each unit's rj from 1, the point of equality. The insight of this generalised inequality index is that inequality indices differ because they employ different functions of the distance of the inequality ratios (the rj) from 1. Gini coefficients of income distributions[edit]
See also: List of countries by income equality
Derivation of the
Regional income Gini indices[edit] According to UNICEF, Latin America and the Caribbean region had the highest net income Gini index in the world at 48.3, on unweighted average basis in 2008. The remaining regional averages were: sub-Saharan Africa (44.2), Asia (40.4), Middle East and North Africa (39.2), Eastern Europe and Central Asia (35.4), and High-income Countries (30.9). Using the same method, the United States is claimed to have a Gini index of 36, while South Africa had the highest income Gini index score of 67.8.[30] World income Gini index since 1800s[edit]
Taking income distribution of all human beings, worldwide income
inequality has been constantly increasing since the early 19th
century. There was a steady increase in the global income inequality
Gini score from 1820 to 2002, with a significant increase between 1980
and 2002. This trend appears to have peaked and begun a reversal with
rapid economic growth in emerging economies, particularly in the large
populations of
Year World Gini coefficients[10][30][33] 1820 0.43 1850 0.53 1870 0.56 1913 0.61 1929 0.62 1950 0.64 1960 0.64 1980 0.66 2002 0.71 2005 0.68 More detailed data from similar sources plots a continuous decline since 1988. This is attributed to globalization increasing incomes for billions of poor people, mostly in India and China. Developing countries like Brazil have also improved basic services like health care, education, and sanitation; others like Chile and Mexico have enacted more progressive tax policies.[34] Year World Gini coefficient[35] 1988 .80 1993 .76 1998 .74 2003 .72 2008 .70 2013 .65 Gini coefficients of social development[edit]
Gini coefficients and income mobility[edit]
In 1978,
Features of Gini coefficient[edit]
The
Anonymity: it does not matter who the high and low earners are.
Scale independence: the
Limitations of Gini coefficient[edit]
The
Table A. Different income distributionswith the same Gini Index[23] HouseholdGroup
Country AAnnual
1 20,000 9,000 2 30,000 40,000 3 40,000 48,000 4 50,000 48,000 5 60,000 55,000 Total Income $200,000 $200,000 Country's Gini 0.2 0.2 Different income distributions with the same Gini coefficient
Even when the total income of a population is the same, in certain
situations two countries with different income distributions can have
the same Gini index (e.g. cases when income Lorenz Curves
cross).[23] Table A illustrates one such situation. Both
countries have a
Extreme wealth inequality, yet low income Gini coefficient
A Gini index does not contain information about absolute national or
personal incomes. Populations can have very low income Gini indices,
yet simultaneously very high wealth Gini index. By measuring
inequality in income, the Gini ignores the differential efficiency of
use of household income. By ignoring wealth (except as it contributes
to income) the Gini can create the appearance of inequality when the
people compared are at different stages in their life. Wealthy
countries such as Sweden can show a low
Table B. Same income distributionsbut different Gini Index Householdnumber
Country AAnnual
1 20,000 1 & 2 50,000 2 30,000 3 40,000 3 & 4 90,000 4 50,000 5 60,000 5 & 6 130,000 6 70,000 7 80,000 7 & 8 170,000 8 90,000 9 120,000 9 & 10 270,000 10 150,000 Total Income $710,000 $710,000 Country's Gini 0.303 0.293 Small sample bias – sparsely populated regions more likely to have
low Gini coefficient
Gini index has a downward-bias for small populations.[58]
Counties or states or countries with small populations and less
diverse economies will tend to report small Gini coefficients. For
economically diverse large population groups, a much higher
coefficient is expected than for each of its regions. Taking world
economy as one, and income distribution for all human beings, for
example, different scholars estimate global Gini index to range
between 0.61 and 0.68.[10][11]
As with other inequality coefficients, the
Table C. Household money incomedistributions and Gini Index, US[61]
Under $15,000 14.6% 13.7% $15,000 – $24,999 11.9% 12.0% $25,000 – $34,999 12.1% 10.9% $35,000 – $49,999 15.4% 13.9% $50,000 – $74,999 22.1% 17.7% $75,000 – $99,999 12.4% 11.4% $100,000 – $149,999 8.3% 12.1% $150,000 – $199,999 2.0% 4.5% $200,000 and over 1.2% 3.9% Total Households 80,776,000 118,682,000 United States' Ginion pre-tax basis 0.404 0.469
Inability to value benefits and income from informal economy affects
Alternatives to Gini coefficient[edit]
Given the limitations of Gini coefficient, other statistical methods
are used in combination or as an alternative measure of population
dispersity. For example, entropy measures are frequently used (e.g.
the
Relation to other statistical measures[edit]
The
A U C = ( G + 1 ) / 2 displaystyle AUC=(G+1)/2 .[70] The
1 / λ displaystyle 1/lambda is used to quantify diversity, and this should not be confused with the Simpson index λ displaystyle lambda . These indicators are related to Gini. The inverse Simpson index increases with diversity, unlike Simpson index and Gini coefficient which decrease with diversity. The Simpson index is in the range [0, 1], where 0 means maximum and 1 means minimum diversity (or heterogeneity). Since diversity indices typically increase with increasing heterogeneity, Simpson index is often transformed into inverse Simpson, or using the complement 1 − λ displaystyle 1-lambda , known as Gini-Simpson Index.[73] Other uses[edit]
Although the
G 1 displaystyle G_ 1 , in calculation section above, may be used for the final model and also at individual model factor level, to quantify the discriminatory power of individual factors. It is related to accuracy ratio in population assessment models. See also[edit] Diversity index
Economic inequality
Great Gatsby curve
Human
References[edit] ^ Gini (1912). ^ Gini, C. (1909). "Concentration and dependency ratios" (in Italian). English translation in Rivista di Politica Economica, 87 (1997), 769–789. ^ "Current Population Survey (CPS) – Definitions and Explanations". US Census Bureau..mw-parser-output cite.citation font-style:inherit .mw-parser-output .citation q quotes:"""""""'""'" .mw-parser-output .citation .cs1-lock-free a background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center .mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center .mw-parser-output .citation .cs1-lock-subscription a background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center .mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration color:#555 .mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span border-bottom:1px dotted;cursor:help .mw-parser-output .cs1-ws-icon a background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center .mw-parser-output code.cs1-code color:inherit;background:inherit;border:inherit;padding:inherit .mw-parser-output .cs1-hidden-error display:none;font-size:100% .mw-parser-output .cs1-visible-error font-size:100% .mw-parser-output .cs1-maint display:none;color:#33aa33;margin-left:0.3em .mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format font-size:95% .mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left padding-left:0.2em .mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right padding-right:0.2em ^ Note:
^ Gini, C. (1936). "On the Measure of Concentration with Special
Reference to
^ a b c "
^ "South Africa Snapshot, Q4 2013" (PDF). KPMG. 2013. Archived from the original (PDF) on 24 May 2014. ^ "Gini Coefficient". United Nations Development Program. 2012. Archived from the original on 12 July 2014. ^ Schüssler, Mike (16 July 2014). "The Gini is still in the bottle". Money Web. Retrieved 24 November 2014. ^ a b c d Hillebrand, Evan (June 2009). "Poverty, Growth, and Inequality over the Next 50 Years" (PDF). FAO, United Nations – Economic and Social Development Department. Archived from the original (PDF) on 20 October 2017. ^ a b c The Real Wealth of Nations: Pathways to Human Development, 2010 (PDF). United Nations Development Program. 2011. pp. 72–74. ISBN 978-0-230-28445-6. ^ Yitzhaki, Shlomo (1998). "More than a Dozen Alternative Ways of Spelling Gini" (PDF). Economic Inequality. 8: 13–30. ^ Sung, Myung Jae (August 2010). "Population Aging, Mobility of
Quarterly Incomes, and Annual
^ a b Blomquist, N. (1981). "A comparison of distributions of annual
and lifetime income: Sweden around 1970". Review of
^ Sen, Amartya (1977), On Economic Inequality (2nd ed.), Oxford: Oxford University Press ^ Treanor, Jill (13 October 2015). "Half of world's wealth now in hands of 1% of population". The Guardian. ^ "Gini Coefficient". Wolfram Mathworld. ^ Crow, E. L., & Shimizu, K. (Eds.). (1988). Lognormal distributions: Theory and applications (Vol. 88). New York: M. Dekker, page 11. ^ Giles (2004). ^ Jasso, Guillermina (1979). "On Gini's Mean Difference and Gini's Index of Concentration". American Sociological Review. 44 (5): 867–870. doi:10.2307/2094535. JSTOR 2094535. ^ Deaton (1997), p. 139. ^ Allison, Paul D. (1979). "Reply to Jasso". American Sociological Review. 44 (5): 870–872. doi:10.2307/2094536. JSTOR 2094536. ^ a b c d Bellù, Lorenzo Giovanni; Liberati, Paolo (2006). "Inequality Analysis – The Gini Index" (PDF). Food and Agriculture Organization, United Nations. ^ Firebaugh, Glenn (1999). "Empirics of World
^ Kakwani, N. C. (April 1977). "Applications of Lorenz Curves in Economic Analysis". Econometrica. 45 (3): 719–728. doi:10.2307/1911684. JSTOR 1911684. ^ a b Chu, Ke-young; Davoodi, Hamid; Gupta, Sanjeev (March 2000).
"
^ "Monitoring quality of life in Europe – Gini index". Eurofound. 26 August 2009. Archived from the original on 1 December 2008. ^ Wang, Chen; Caminada, Koen; Goudswaard, Kees (2012). "The redistributive effect of social transfer programmes and taxes: A decomposition across countries". International Social Security Review. 65 (3): 27–48. doi:10.1111/j.1468-246X.2012.01435.x. ^ Sutcliffe, Bob (April 2007). "Postscript to the article 'World inequality and globalization' (Oxford Review of Economic Policy, Spring 2004)" (PDF). Retrieved 13 December 2007. ^ a b Ortiz, Isabel; Cummins, Matthew (April 2011). "Global Inequality: Beyond the Bottom Billion" (PDF). UNICEF. p. 26. ^ Milanovic, Branko (September 2011). "More or Less". Finance & Development. 48 (3). ^ Milanovic, Branko (2009). "Global Inequality and the Global Inequality Extraction Ratio" (PDF). World Bank. ^ Berry, Albert; Serieux, John (September 2006). "Riding the Elephants: The Evolution of World Economic Growth and Income Distribution at the End of the Twentieth Century (1980–2000)" (PDF). United Nations (DESA Working Paper No. 27). ^ "What The Stat About The 8 Richest Men Doesn't Tell Us About Inequality". ^ World Bank. "
^ Sadras, V. O.; Bongiovanni, R. (2004). "Use of Lorenz curves and Gini coefficients to assess yield inequality within paddocks". Field Crops Research. 90 (2–3): 303–310. doi:10.1016/j.fcr.2004.04.003. ^ Thomas, Vinod; Wang, Yan; Fan, Xibo (January 2001). "Measuring education inequality: Gini coefficients of education" (PDF). Policy Research Working Papers. The World Bank. CiteSeerX 10.1.1.608.6919. doi:10.1596/1813-9450-2525. Archived from the original (PDF) on 5 June 2013. Cite journal requires |journal= (help) ^ a b Roemer, John E. (September 2006). Economic development as opportunity equalization (Report). Yale University. CiteSeerX 10.1.1.403.4725. SSRN 931479. ^ John Weymark (2003). "Generalized Gini Indices of Equality of Opportunity". Journal of Economic Inequality. 1 (1): 5–24. doi:10.1023/A:1023923807503. ^ Milorad Kovacevic (November 2010). "Measurement of Inequality in Human Development – A Review" (PDF). United Nations Development Program. ^ Atkinson, Anthony B. (1999). "The contributions of
^ Roemer; et al. (March 2003). "To what extent do fiscal regimes equalize opportunities for income acquisition among citizens?". Journal of Public Economics. 87 (3–4): 539–565. CiteSeerX 10.1.1.414.6220. doi:10.1016/S0047-2727(01)00145-1. ^ Shorrocks, Anthony (December 1978). "
^ Maasoumi, Esfandiar; Zandvakili, Sourushe (1986). "A class of generalized measures of mobility with applications". Economics Letters. 22 (1): 97–102. doi:10.1016/0165-1765(86)90150-3. ^ a b Kopczuk, Wojciech; Saez, Emmanuel; Song, Jae (2010). "Earnings Inequality and Mobility in the United States: Evidence from Social Security Data Since 1937" (PDF). The Quarterly Journal of Economics. 125 (1): 91–128. doi:10.1162/qjec.2010.125.1.91. JSTOR 40506278. ^ Chen, Wen-Hao (March 2009). "Cross-national Differences in Income
Mobility: Evidence from Canada, the United States, Great Britain and
Germany". Review of
^ Sastre, Mercedes; Ayala, Luis (2002). "Europe vs. The United States: Is There a Trade-Off Between Mobility and Inequality?" (PDF). Institute for Social and Economic Research, University of Essex. ^ Litchfield, Julie A. (March 1999). "Inequality: Methods and Tools" (PDF). The World Bank. ^ Ray, Debraj (1998). Development Economics. Princeton, NJ: Princeton University Press. p. 188. ISBN 978-0-691-01706-8. ^ "Country Comparison: Distribution of family income – Gini index". The World Factbook. CIA. Retrieved 8 May 2017. ^ Garrett, Thomas (Spring 2010). "U.S.
^ Mellor, John W. (2 June 1989). "Dramatic
^ a b c KWOK Kwok Chuen (2010). "
^ a b "The Real Wealth of Nations: Pathways to Human Development (2010 Human Development Report – see Stat Tables)". United Nations Development Program. 2011. pp. 152–156. ^ De Maio, Fernando G. (2007). "
^ Domeij, David; Flodén, Martin (2010). "Inequality Trends in Sweden 1978–2004". Review of Economic Dynamics. 13 (1): 179–208. CiteSeerX 10.1.1.629.9417. doi:10.1016/j.red.2009.10.005. ^ Domeij, David; Klein, Paul (January 2000). "Accounting for Swedish wealth inequality" (PDF). Archived from the original (PDF) on 19 May 2003. ^ Deltas, George (February 2003). "The Small-Sample Bias of the Gini
Coefficient: Results and Implications for Empirical Research". The
Review of
^ Monfort, Philippe (2008). "Convergence of EU regions: Measures and evolution" (PDF). European Union – Europa. p. 6. ^ Deininger, Klaus; Squire, Lyn (1996). "A New Data Set Measuring
^ a b "Income, Poverty, and Health Insurance Coverage in the United States: 2010 (see Table A-2)" (PDF). Census Bureau, Dept of Commerce, United States. September 2011. ^ Congressional Budget Office: Trends in the Distribution of Household
^ Schneider, Friedrich; Buehn, Andreas; Montenegro, Claudio E. (2010). "New Estimates for the Shadow Economies all over the World". International Economic Journal. 24 (4): 443–461. doi:10.1080/10168737.2010.525974. hdl:10986/4929. ^ The Informal Economy (PDF). International Institute for Environment and Development, United Kingdom. 2011. ISBN 978-1-84369-822-7. ^ Feldstein, Martin (August 1998). "Is income inequality really the problem? (Overview)" (PDF). US Federal Reserve. ^ Taylor, John; Weerapana, Akila (2009). Principles of Microeconomics: Global Financial Crisis Edition. pp. 416–418. ISBN 978-1-4390-7821-1. ^ Rosser, J. Barkley Jr.; Rosser, Marina V.; Ahmed, Ehsan (March
2000). "
^ Krstić, Gorana; Sanfey, Peter (February 2010). "Earnings inequality and the informal economy: evidence from Serbia" (PDF). European Bank for Reconstruction and Development. ^ Schneider, Friedrich (December 2004). The Size of the Shadow Economies of 145 Countries all over the World: First Results over the Period 1999 to 2003 (Report). hdl:10419/20729. SSRN 636661. ^ Hand, David J.; Till, Robert J. (2001). "A Simple Generalisation of the Area Under the ROC Curve for Multiple Class Classification Problems" (PDF). Machine Learning. 45 (2): 171–186. doi:10.1023/A:1010920819831. ^ Eliazar, Iddo I.; Sokolov, Igor M. (2010). "Measuring statistical heterogeneity: The Pietra index". Physica A: Statistical Mechanics and Its Applications. 389 (1): 117–125. Bibcode:2010PhyA..389..117E. doi:10.1016/j.physa.2009.08.006. ^ Lee, Wen-Chung (1999). "Probabilistic Analysis of Global
Performances of Diagnostic Tests: Interpreting the Lorenz Curve-Based
Summary Measures" (PDF).
^ Peet, Robert K. (1974). "The Measurement of Species Diversity". Annual Review of Ecology and Systematics. 5: 285–307. doi:10.1146/annurev.es.05.110174.001441. JSTOR 2096890. ^ Wittebolle, Lieven; Marzorati, Massimo; 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. ^ Asada, Yukiko (2005). "Assessment of the health of Americans: the average health-related quality of life and its inequality across individuals and groups". Population Health Metrics. 3: 7. doi:10.1186/1478-7954-3-7. PMC 1192818. PMID 16014174. ^ Halffman, Willem; Leydesdorff, Loet (2010). "Is Inequality Among Universities Increasing? Gini Coefficients and the Elusive Rise of Elite Universities". Minerva. 48 (1): 55–72. arXiv:1001.2921. doi:10.1007/s11024-010-9141-3. PMC 2850525. PMID 20401157. ^ Graczyk, Piotr (2007). "Gini Coefficient: A New Way To Express Selectivity of Kinase Inhibitors against a Family of Kinases". Journal of Medicinal Chemistry. 50 (23): 5773–5779. doi:10.1021/jm070562u. PMID 17948979. ^ Shi, Hongyuan; Sethu, Harish (2003). "Greedy Fair Queueing: A Goal-Oriented Strategy for Fair Real-Time Packet Scheduling". Proceedings of the 24th IEEE Real-Time Systems Symposium. IEEE Computer Society. pp. 345–356. ISBN 978-0-7695-2044-5. ^ Christodoulakis, George A.; Satchell, Stephen, eds. (November 2007). The Analytics of Risk Model Validation (Quantitative Finance). Academic Press. ISBN 978-0-7506-8158-2. ^ Chakraborty, J; Bosman, MM. "Measuring the digital divide in the United States: race, income, and personal computer ownership". Prof Geogr. 57 (3): 395–410. doi:10.1111/j.0033-0124.2005 (inactive 20 August 2019). ^ van Mierlo, T; Hyatt, D; Ching, A (2016). "Employing the Gini coefficient to measure participation inequality in treatment-focused Digital Health Social Networks". Netw Model Anal Health Inform Bioinforma. 5 (32): 32. doi:10.1007/s13721-016-0140-7. PMC 5082574. PMID 27840788. Further reading[edit]
.mw-parser-output .refbegin font-size:90%;margin-bottom:0.5em
.mw-parser-output .refbegin-hanging-indents>ul
list-style-type:none;margin-left:0 .mw-parser-output
.refbegin-hanging-indents>ul>li,.mw-parser-output
.refbegin-hanging-indents>dl>dd
margin-left:0;padding-left:3.2em;text-indent:-3.2em;list-style:none
.mw-parser-output .refbegin-100 font-size:100%
Amiel, Y.; Cowell, F. A. (1999). Thinking about Inequality.
Cambridge. ISBN 978-0-521-46696-7.
Anand, Sudhir (1983). Inequality and
External links[edit] Wikimedia Commons has media related to Gini coefficient. Deutsche Bundesbank: Do banks diversify loan portfolios?, 2005 (on
using e.g. the
Category Business portal vteDeprivation and poverty indicatorsSocialTopics:
Social exclusion
Social vulnerability
Relative deprivation
Disadvantaged
Fushūgaku
Hikikomori
Social determinants of health in poverty
Measures:
Social Progress Index
PsychologicalTopics:
psychological poverty
vteIndices of DeprivationNational (general deprivation)
Carstairs index
Environment portal Category Commons Organizations Authority contro |