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The Gini Index and Measures of Inequality

Frank A. Farris

Abstract. The Gini index is a summary statistic that measures how equitably a resource is

distributed in a population; income is a primary example. In addition to a self-contained pre-

sentation of the Gini index, we give two equivalent ways to interpret this summary statistic:

ﬁrst in terms of the percentile level of the person who earns the average dollar, and second

in terms of how the lower of two randomly chosen incomes compares, on average, to mean

income.

1. INTRODUCTION. You hear anecdotes all the time: The poorest 20% of the peo-

ple on Earth earn only 1% of the income. A mere 20% of the people on Earth consume

86% of the consumer goods. Only 3% of the U.S. population owns 95% of the privately

held land.

The Gini index offers a consistent way to talk about statistics like these. A single

number that measures how equitably a resource is distributed in a population, the Gini

index gives a simple, if blunt, tool for summarizing economic data. It allows us to

illustrate how equity has changed in a given situation over time, such as how U.S.

family income changed over the 20th century. (The poor got poorer over the second

half.) We can also compare income or wealth across societies, and even analyze salary

structures of organizations.

Being only a single summary statistic, the Gini index has been critiqued by social

scientists [2]. It is true that no summary statistic can reveal all we need to know about

the distribution of income, wealth, or land. Even so, the Gini index deserves to be

better known in the mathematical community, as it continues to ﬁnd application in

new situations, from genetics [7] to astronomy [1].

In addition to a self-contained presentation of the Gini index, we give two equivalent

ways to interpret this summary statistic: ﬁrst in terms of the percentile level of the

person who earns the average dollar, and second in terms of how the lower of two

randomly chosen incomes compares, on average, to mean income. The ﬁrst of these

appears to be new; the second has appeared in the literature [11], but does not seem to

be well known. Beyond the inherent mathematical interest, our story draws attention

to the concept of inequity and offers readers tools to help them go beyond the factoids

of the ﬁrst paragraph.

2. DEFINING THE GINI INDEX. Though it is named for Italian statistician Cor-

rado Gini (1884–1965), the Gini index can almost be glimpsed in the diagrams from

a 1905 paper by M. O. Lorenz [12]. Gini’s original work on the subject appeared in

1912 in Italian [8]; it is not easy to access. Fortunately, the paper by Lorenz is quite

charming to read and gives an excellent historical snapshot of the seeds of this train of

thought. The ﬁrst sentence is memorable:

There may be wide difference of opinion as to the signiﬁcance of a very unequal

distribution of wealth, but there can be no doubt as to the importance of knowing

whether the present distribution is becoming more or less unequal.

doi:10.4169/000298910X523344

December 2010] THE GINI INDEX AND MEASURES OF INEQUALITY 851

Let us deﬁne a Lorenz curve, the instrument Lorenz proposed for visualizing the

distribution of a quantity in a population. Suppose that some quantity Q, which could

stand for wealth, income, family income, land, food, and so on, is distributed in a pop-

ulation. If we imagine the population to be lined up by increasing order of their shares

of Q(with ties being broken arbitrarily), then for any pbetween 0 and 1 the people

in the ﬁrst fraction pof the line represent the Q-poorest 100 p% of the population. We

then call L(p)the fraction of the totality of Qowned (or earned or controlled or eaten)

by that fraction of the population. In summary:

The Lorenz curve for a resource Qis the curve y=L(p),wheretheQ-poorest

fraction pof the population has a fraction L(p)of the whole.

Using this vocabulary, the ﬁrst sentence of the paper would be expressed as

L(.20)=.01, where Lis the Lorenz curve for world income. The variable pis

called the percentile variable.

If everyone had exactly the same amount of Q, the order of our imaginary line-up

would be completely arbitrary and we would say that L(p)=p,thecurveofperfect

equitability. In other situations where some fraction of the population all share equally

in an amount of Q, our rule for an arbitrary order of that portion of the line results in a

linear segment of the Lorenz curve. For instance, if everyone in the bottom half of the

population owned an equal share of 1/4 of the wealth, we would say that L(p)=p/2

for 0 ≤p≤1/2, so that L(1/2)=1/4.A purist might say that, in a population of N

individuals, it only makes sense for pto take on values of the form k/N. In practice,

we model Lorenz curves as being deﬁned for all p, using linear interpolation whenever

necessary. This requires us to say, for instance, that the poorest 10% of an individual

earns 10% of that person’s income, which is not too much of a stretch.

The Gini index is a quantity calculated from a particular Lorenz curve. It is deﬁned

as an integral that summarizes how much the Lorenz curve in question deviates from

perfect equitability:

G:= 21

0

[p−L(p)]dp.(1)

The formula reveals why the Gini index sometimes appears in calculus books in the

section on the area between two curves. The reason for the factor 2 is to scale the area

in such a way that the Gini index varies between 0, perfect equitability where everyone

has the same share of the good, and 1, where one person has everything.

I downloaded data for 2006 family income from the U.S. Census Bureau [14]. Mod-

ulo some details, described in another section, it is easy to use a spreadsheet program

to create a Lorenz curve from the data and estimate the Gini index. This is shown in

Figure 1.

The Gini index for this situation works out to be about .47, which agrees with the

ﬁgure reported by the U.S. Census Bureau [13]. For perspective, the similar indices

for Brazil and Denmark are about .58 and .24 respectively [4]. Does this ﬁt with what

you know about these societies? It is also instructive to compare Gini indices over

time: The index for U.S. family income hit its 20th century low of around .36 in 1968.

Does this match your understanding of social change in America over the last several

decades?

The CIA World Factbook [4] reports that the Gini index of U.S. family income (for

2007) is .45, and other sources claim ﬁgures even below .40. These lower ﬁgures tend

to be adjusted indices, taking income adjustments into account. For instance, in the

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THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117

US Family Income 2006

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Figure 1. The Gini index is twice the area between the Lorenz curve and the curve of perfect equitability. For

U.S. family income in 2006, the data leads to an estimate of G≈.47.

U.S., the tax structure, which asks higher income citizens to pay a higher percentage

of income so that various beneﬁts (food stamps, Medicare, etc.) may be given to lower

income residents, does in fact reduce inequity, but this feature of U.S. society would

be missed if only raw income ﬁgures were used to compute a Gini index. When com-

paring Gini indices, we should take care to specify exactly what is being measured. We

will side-step any deep analysis of this point in favor of mathematics, simply warning

readers to be careful to compare Gini indices only when they are computed in the same

way.

The Gini index need not be a grim topic. When I was learning how to calculate

indices, Alex Rodriguez had just been hired by the N.Y. Yankees for an annual salary

of $22 million. I downloaded the salary schedules for the Yankees and the Red Sox

from the ESPN website and determined that the salary Gini for the Yankees during

that season was .57, noticeably higher than for the more equitable Red Sox, at .52.

I found a contrasting example in the exam scores of my calculus students. Even

with more than half the students below a mean of 84%, the Gini index for distribution

of exam points was only .06. Does this mean that my grading is especially fair?

3. CONNECTING TO PROBABILITY. As we move toward calculating Gini in-

dices, we must accept that economic data are almost always reported in aggregated

form. Except in fanciful applications like salaries of baseball players where we have

an income ﬁgure for each individual, we get tables of data where one column lists the

(very large) number of people in a given range of incomes and another gives the mean

income for this group. Income ranges are listed in convenient order from lowest to

highest. A truncated version of the data appears in Table 1.

At the ends of the spectrum of 2006 U.S. family income, there are 2,533 thousand

households with mean income $295 and 2,240 thousand households with mean income

$448,687.

In general, let us name the entries in any such table as hjunits (households) that,

on the average, have an amount xjof our good Q(income). If our table has nrows,

then jranges from 1 to nand the order of the table means that xj<xkwhen j<k.

December 2010] THE GINI INDEX AND MEASURES OF INEQUALITY 853

Tab le 1. U.S. family income, 2006, aggregated.

Number of households Average income

(in thousands) hjxj

2,533 295

1,030 3,737

2,124 6,431

3,002 8,713

3,677 11,206

3,203 13,668

3,677 16,088

3,169 18,646

3,886 21,056

3,005 23,690

.

.

.

.

.

.

13,385 119,461

4,751 169,454

1,776 219,377

2,240 448,687

As a ﬁrst step in calculating a Gini index from such a table, let us consider how to

express the Lorenz curve, L(p). First, we deﬁne numbers

N=

n

i=1

hiand T=

n

i=1

xihi.

In words, Nis the size of the population and Tis the total amount of the good Q. With

this notation, the mean amount owned is T/N, which we call X.

In our example of 2006 U.S. family income, Nis about 116 million households, T

is almost 8 trillion dollars, and the mean income is X≈$66,570. The ﬁrst entry in

the table corresponds to a percentile value of h1/N≈.02183, the poorest 2% of the

population. In general, the numbers

pj=1

N

j

i=1

hi

give us n(not necessarily equally spaced) points along the p-axis between 0 and 1.

For convenience, we deﬁne p0=0.

The Lorenz curve is easily calculated for these particular values of p:L(0)=0and

L(pj)=1

T

j

i=1

xihi,1≤j≤n,(2)

because this is the fraction of the total earned by the poorest fraction pj.

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THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117

The simplest way to ﬁll in the rest of the Lorenz curve is by linear interpolation.

When we do this, we are assuming that every one of the hjunits with mean amount

xjhas an equal share in that amount. As we examine in more detail later, this always

results in an underestimate of the Gini index, though a small one if our aggregation

uses a ﬁne partition.

It is a little cumbersome to calculate the Gini index (1) by direct integration from

(2). Instead, we will reinterpret that equation, uncovering a surprising connection to

probability density functions.

With a little algebra, we rewrite (2) in terms of the percentile variable and recognize

the result as a Riemann sum:

L(pj)=

j

i=1

xi

T/N(pi−pi−1)

=pj

0

s(p)dp,(3)

where

s(p)=xj

X,for pj−1<p≤pj.(4)

We call the function s(p)the share density, because it tells us what share of the

whole is owned by the portion of the population that falls in a given percentile range.

In our example, the income of the poorest 2% is about 0.0044 of the mean income,

while the group at the top, above the 99th percentile, have a share that is about 6.74

times the mean.

It seem reasonable to use (3) to deﬁne the Lorenz curve for every value of p, not

just the numbers pj, which accomplishes the linear interpolation we spoke of before.

In fact, we prefer to think of the share density as the primary object here, from which

details of the Lorenz curve can be derived. One result of this approach is that the

nondecreasing nature of s(p)establishes L(p)as a convex function. For instance, the

following inequalities demonstrate midpoint convexity for any p1<p2:

p1+p2

2

p1

s(˜p)d˜p≤p2

p1+p2

2

s(˜p)d˜p,so Lp1+p2

2≤L(p1)+L(p2)

2.

We acknowledge that aggregation of data always leads to some error in using (3) to

deﬁne the Lorenz curve, and hence in the Gini index from which it is computed. We

address this brieﬂy in a later section. It has also been treated widely in the economics

literature. Indeed, the concept of share density is implicit in the work of Gastwirth

from 1971 [5].

It may happen that a Lorenz curve for a given situation is known, or proposed

theoretically as a function of some particular type [6]. In such a case, we could deﬁne

the share density, perhaps only almost everywhere, as

s(p)=d

dp L(p). (5)

Since s(p)≥0and1

0s(p)dp =L(1)=1, the function s(p)ﬁts the requirements

of a probability density function (pdf). What experiment would lead to a random vari-

able that has this pdf? We propose the following:

December 2010] THE GINI INDEX AND MEASURES OF INEQUALITY 855

Pick a dollar earned by a U.S. household at random, assuming that every dollar

is equally likely to be chosen. Record the value of the percentile variable, p,of

the person who earned that dollar.

For this experiment, pis a random variable with density s(p). To see this, look

at (4). For instance, the probability that a dollar chosen at random was earned in the

percentile range from pj−1to pjis exactly the fraction of those dollars in proportion

to the whole, which is

xjhj

T=xj

T/N(pj−pj−1)=pj

pj−1

s(p)dp.

Also, we note that a share density of s(p)≡1 would indicate a perfectly equitable

distribution, in which case each dollar has an equal chance of being earned in all per-

centiles.

The share density earns its keep from the following computation, in which we sub-

stitute the integral form of the Lorenz curve into the deﬁnition of the Gini index (1)

and then switch the order of integration:

G=21

0

[p−L(p)]dp

=1−21

0p

0

s(˜p)d˜pdp

=1−21

0

(1−˜p)s(˜p)d˜p

=21

0

ps(p)dp −1.(6)

This last integral is simply the expected value of our random variable with density

s(p).Weuse pfor this expected value and call it the percentile of the average dollar

earned. This proves a theorem that gives our ﬁrst interpretation of the Gini index:

Theorem 1. Suppose G is the Gini index associated with the Lorenz curve L (p)and

the share density is deﬁned by s(p)=L(p)almost everywhere. Let p be the expected

value of the random variable on [0,1]whose density function is s(p). Then G and p

are related by

G=2p−1and p=G+1

2.(7)

Let us apply this to our examples: The average dollar earned in the U.S. in 2006

was earned at a percentile level of (.47 +1)/2, or above the 73rd percentile. For the

Yankees, with salary Gini .57, the average dollar comes in above the 78th percentile.

In my opinion, this gives a more visceral fact to share with the general public than just

the value of an index that ranges between 0 and 1.

Calculating Ginis. Interpreting the Gini index in terms of the average dollar earned

is also key to calculations. To compute p, we break up the integral into pieces where

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THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117

s(p)is constant:

p=1

0

ps(p)dp

=

n

j=1

xj

Xpj

pj−1

pdp

=

n

j=1

xj

X

pj+pj−1

2pj,(8)

where, as usual, pj=pj−pj−1. This form is ideal for entering into a spreadsheet

and it was this version that produced the values reported for the examples.

We can rearrange (8) as

G=1

T

n

j=1

xj(pj+pj−1)hj−1,(9)

for the Gini index of the Lorenz curve obtained by linear interpolation of the data. With

some work, this last expression for Gcan also be derived directly from the deﬁnition

of the Gini index by applying the trapezoid rule to ﬁnd the area under the piecewise

linear Lorenz curve.

4. THE LOWER OF TWO INCOMES. Consider this experiment: Pick two house-

holds in the U.S. and record the lower of their two incomes; call the result Y, a random

variable that takes values in [0,∞). An amusing computation shows that the expected

value of Y, in ratio to the mean income, is the complement of the Gini index relative

to 1. In symbols,

Y/X=1−G.(10)

To prove this, we need to know the pdf for Y. This in turn requires the pdf for X,

the random variable recording the income of a single household. This pdf is not so

directly available from the data, as presented in government statistics. That data could

be interpreted as requiring point probability masses placed at each aggregated mean

income. In other words, we could place a point mass of 2,533/Nat income X=$295

and a point mass of 2,240/Nat income X=$448,687. This would be both inaccurate

and mathematically cumbersome.

Instead, let us work theoretically, assuming a known piecewise smooth density func-

tion f(x)(0 ≤x<∞) for the random variable X. Knowing fallows us to compute

the cumulative density function (cdf) for X:

F(x)=P(X<x)=x

0

f(˜x)d˜x.

A standard computation in order statistics gives a cdf for Y,H(x), as follows:

H(x)=P(Y<x)=1−P(Y>x)

=1−P(ﬁrst income >x)·P(second income >x)

=1−(1−F(x)) ·(1−F(x)).

December 2010] THE GINI INDEX AND MEASURES OF INEQUALITY 857

This gives the pdf for Yas H(x)=2f(x)(1−F(x)),since F(x)=f(x)(almost

everywhere). Therefore, the expected value of Yis

Y=∞

0

2xf(x)(1−F(x)) dx.(11)

Let us connect this to the Gini index. The percentile variable pis easily related

to the cdf for X. For a speciﬁc value of x, the probability that an income chosen at

random is less than xis exactly the size of the fraction of the population earning less

than x. In symbols, this means that

F(x)=x

0

f(˜x)d˜x=p,

which means that we can express xin terms of pand pin terms of x. The share density

s(p)is simply x/X, recording the proportion of the mean income at percentile level p.

We interpret (11) in terms of the variable p, using the substitution p=F(x),dp =

f(x)dx.

Y/X=∞

0

2(x/X)f(x)(1−F(x)) dx

=1

0

2s(p)(1−p)dp

=2−2p=2−2G+1

2=1−G.(12)

This manipulation allows us to interpret any known Gini index in an approachable,

conversational way: Assuming that the Gini index for U.S. family income is .47, we

conclude that the lower of two U.S. family incomes, chosen at random, is about 53% of

the mean; on the average, the poorer of two families earns only about half the national

mean.

5. GINI ESTIMATES. Gini indices, in practice, must be computed from incomplete

data. It helps to have some idea of the errors introduced when we infer a Gini index

from partial data. This topic has received extensive treatment in the economics liter-

ature [6]; our self-contained treatment is meant to be accessible to mathematicians.

One important way in which we depart from practical concerns is that we consider

aggregate data, as for example in Table 1, as representing the exact averages for each

group reported. In other words, we do not consider the reality that this data contains

reporting errors. In this section, we make statements about the conclusions that can be

drawn from the given data, assuming that it is accurate.

We begin with the case where our knowledge is limited to a single nontrivial point

on a Lorenz curve. This is the situation of the ﬁrst sentences of the paper. For instance,

knowing that 20% of the people on Earth consume 86% of the consumer goods, what

can we say about the Gini index?

Proposition 1. If G is the Gini index associated with a Lorenz curve L (p)and we

know that L (a)=b, where 0<b<a<1,then

a−b≤G<1−2b(1−a). (13)

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THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117

Smaller upper bounds are possible, but less easy to state. For instance,

if b <1

2<a,then G ≤1−4b(1−a).

Proof. The least possible area between y=pand y=L(p)occurs when L(p)is the

piecewise linear function shown in Figure 2, which simply connects (0,0)to (a,b)

andthento(1,1). In that case, the Gini index is easily calculated to be 1 −(ab +

(1−a)(1+b)) =a−b.

The rather blunt upper bound in (13) arises from geometric considerations. In Figure

2,theareaofthebby 1 −arectangle in the lower right corner is clearly off limits.

The estimate follows by removing twice this area from the highest possible Gini of 1.

(1,1)

(a,b)

(0,0)

L

(p)

p

Figure 2. Estimates for the Gini index obtained from a single data point are possible, but not especially

accurate.

The better estimate comes from recognizing that Lorenz curves must be convex.

Extreme behavior arises from linear functions, at the edge of convexity; some readers

may recognize that we are talking about support lines for the Lorenz curve.

To get the largest possible Gini index, we should cut the smallest possible area from

below y=x. It turns out that among all lines through (a,b), the one that forms the

smallest triangle (together with the x-axis and the line x=1) is parallel to the diago-

nal of the b-by-(1−a)rectangle mentioned earlier. The only issue is whether this line

crosses y=x. As long as b<1/2<a, this line cuts a triangle of area 2b(1−a)from

the large triangle, resulting in a Gini index of 1 −4b(1−a). (Alert readers may recog-

nize that we are talking about a discontinuous Lorenz curve with limp→1−L(p)<1,

which requires the share density to have a point mass at p=1. This models a situation

where a very small portion of the population has a very large share of Q.)

An economic interpretation of this proposition holds some interest. The lowest pos-

sible Gini estimate comes from assuming that a fraction bof the good Qis distributed

absolutely equally through the poorest fraction aof the population, with the remaining

portion shared equally among the remainder.

For instance, when we conclude from the estimate about consumer goods (where

L(.80)=.14) that .66 ≤G,thevalue.66 would arise from the poorest 80% sharing

equally in 14% of the goods—an unlikely situation.

December 2010] THE GINI INDEX AND MEASURES OF INEQUALITY 859

The highest possible Gini index consistent with our single data point is 1 −

4(.14)(.2)=.888. This corresponds to a distribution under which 60% have no

goods at all, 40% have an equal share in 28%, and one person has all the remaining

72%—again, unlikely. (As mentioned earlier, this requires the share density to have

a point mass at p=1.) The range of the estimate is wide, but we can still say that

consumer goods are less equitably distributed than U.S. family income.

Golden [9] offers a related estimate, based on knowledge of the Lorenz curve at a

point that is known to be farthest from the line of perfect equity, in which case the

relevant support line has slope 1. This is done in the special case of quintile data.

Readers may wish to derive this estimate as an exercise.

Our simple estimate becomes more useful when we apply the same geometric ideas

to individual summands in our trapezoid rule approximation for the Gini index, (9). Let

us use GTto denote the Gini index for the Lorenz curve obtained by linear interpola-

tion of aggregated data and ﬁnd estimates for the Gini index for the situation where we

have one data point for every individual in the population. In each term of the sum, the

simplest estimate we can make is that the Gini index could increase by twice the area

of a triangle with base pj−pj−1and height L(pj)−L(pj−1). Working out formulas

for these gives a potential positive contribution to the error of xjh2

j/(TN).Wehave

proved a proposition:

Proposition 2. If GTis the approximation for a Gini index from (9), the actual index,

G, satisﬁes

GT≤G<GT+1

TN

n

j=0

xjh2

j.(14)

In our computations for U.S. family income, the error is bounded by 0.042 and we

conclude that .467 ≤G<.509.

For a more precise upper bound on the Gini index, we now focus on one particular

interval in which mean data has been given. In terms of the notation established earlier,

we are talking about the population between pj−1and pj, where at ﬁrst we assume that

1<j<n, leaving discussion of the ﬁrst and last intervals for later.

Over this interval, the Lorenz curve has slope sj, which is intermediate between

the slopes on the left, sj−1, and right, sj+1. This is shown in Figure 3 where segments

are labeled with their slopes. The most extreme Lorenz curve that still connects points

(pj−1,L(pj−1)) and (pj,L(pj)) consists of the two line segments shown, continuing

the line of slope sj−1as far as the point where it meets the line of slope sj+1.Callthe

p-coordinate of this point of intersection p∗.

To understand this in economic terms, recall that in this portion of the population,

hjhouseholds have an average income of xj, which gives them a share density of

sj=xj/¯

X=xj/(T/N). (Although our notation favors the interpretation of family

income, the discussion applies to any situation.) To achieve the extreme Lorentz curve

mentioned, we could redistribute the hjxjdollars earned, taking away income from

one group to push a fraction p∗−pj−1down one bracket to have share density sj−1,

pushing the remainder up to a share density of sj+1. Note that this results in one fewer

income bracket than the data original suggested.

The effect on the Gini index would be to increase it by twice the area of the triangle

in Figure 3.

860 c

THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117

sj–1

sj+1

sj

pj–1 p* pj

b

Figure 3. Better estimates for the Gini index involve an interval-by-interval analysis.

A simple computation, writing the area as the difference of two triangles whose

base is labeled bin the diagram, shows the maximal increase in the Gini index to be

(sj−sj−1)(sj+1−sj)

(sj+1−sj−1)(pj−pj−1)2=1

NT

(xj−xj−1)(xj+1−xj)

(xj+1−xj−1)h2

j.(15)

At the lowest endpoint, there is no lower income bracket to push population into;

in the highest bin, there is no possibility to redistribute upward, so the estimates work

out a bit differently. Still, if we set x0=0andxn+1=xn, the formulas are correct.

Of course, pushing population from one bracket into neighboring brackets affects

the analysis of those regions; we cannot capture this extra area in every interval while

still maintaining a convex curve. Even so, we know that the Gini index cannot be

increased by more than twice the sum of areas of all these triangles. We have sketched

the proof of a theorem:

Theorem 2. If a Lorenz curve is generated from aggregate data, where the jth bin

consists of h jindividual units in possession of x junits of the resource Q, then the ac-

tual Gini index G, the one that would result from analyzing every individual’s portion,

must satisfy

GT≤G<GT+1

NT

n

j=1

(xj−xj−1)(xj+1−xj)

(xj+1−xj−1)h2

j,(16)

wherewesetx

0=0and xn+1=xn.

Applying this analysis to U.S. family income from 2006 gives .467 ≤G<.472,

showing that .47 is the correct Gini index to two-digit accuracy. (Remember that this

is a raw Gini index, ignoring the effect of taxes, Medicare, and Social Security.)

Efforts to ﬁnd the best upper bound have shown this to be a complicated question.

Experiments attempting to minimize total area using a variable support line at each

data point suggest that the minimum is realized only at endpoints of the intervals of

possible slopes, except in simple cases like the example with a single point. I con-

jecture that the largest possible Gini index consistent with data from 2nbins arises by

December 2010] THE GINI INDEX AND MEASURES OF INEQUALITY 861

applying the redistribution method outlined above in alternate bins, pushing population

into neighboring bins to create nnew bins and a maximally less equitable distribution.

6. HIGHER ORDER GINIS. Atkinson [2] rightly pointed out in 1970 that the Gini

index is no universal measure of society. Sometimes it helps little in judging whether

one distribution of income is preferable to another. It is easy to give a mathematical

reason for this: Many Lorenz curves give rise to the same Gini index.

Atkinson uses utility functions to weight income disparities, asking those who

would judge inequity: Is extreme poverty more socially harmful than extreme wealth?

Of course, no single utility function will serve all purposes.

My response on reading Atkinson was to place the Gini index as ﬁrst in a family of

indices, each weighting the percentile range differently. Introducing a weighting factor

of (1−p)k−1for k≥1 yields an index where extreme poverty is weighted more for

higher values of k. In fact, this idea originally appeared in a paper by Kakwani in

1982 [10], and many other economists and social scientists have taken the ball and run

with it.

We deﬁne the kth Gini index (or perhaps kth Gini poverty index) by the formula

Gk:= k(k+1)1

0

(p−L(p)) ·(1−p)k−1dp.

The factor k(k+1), analogous to the 2 in Gini’s original deﬁnition, forces each index

in the sequence to lie between 0 and 1. Note that G1is simply Gfrom (1).

It is a simple matter to mimic earlier computations to produce a spreadsheet-friendly

formula to approximate Gkfrom aggregated data. Probabilists may recognize that we

are really talking about moments of the share density. An analog of (9) is

Gk=1−(1/X)

n

j=1

xj(1−pj−1)k+1−(1−pj)k+1,(17)

though the resemblance may not be immediately evident.

Using (17) with data from U.S. family income suggests that G2is about .61. What

does this mean? To answer this, we return to order statistics.

As before, the random variable Xdenotes the income of a single household chosen

at random; the pdf for Xis f(x)and the cdf is F(x). Now we independently choose

k+1 households and record the lowest of their incomes as Ymin

k. A computation just

like the one that led to (11) shows that the expected value of Ymin

kis

Ymin

k=(k+1)∞

0

x(1−F(x))kf(x)dx.

The reasoning that led us to (12) in this case gives

Ymin

k

X=1−Gk.(18)

If the second-order Gini index of U.S. family income is G2=.61, this means that,

on average, the lowest of three incomes randomly selected is only 39% of the mean

income.

We should mention that (18), though new to the author, appears in a paper by

Kleiber and Kotz [11]. We hope that MONTHLY readers ﬁnd this presentation easier to

follow than anything in the extensive economics literature on the subject.

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THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117

7. CLOSING THOUGHTS. Higher-order Gini indices can be useful in calibrating

models. Social scientists (and authors of calculus texts) often model Lorenz curves

with a variety of Paret o functions, which are convex combinations of power functions,

such as

L(p)=ap +(1−a)pb.

Since this model has two free parameters, aand b, it is natural to calibrate it to match

values of G1and G2derived from data. The degree to which this model ﬁts the data

can be judged by the difference between the value of G3calculated from the model

and the one calculated from the data using (17).

Though it arose in the study of poverty, the Gini index is a ﬂexible idea that deserves

to be better known. As mentioned in the introduction, scientists in many ﬁelds [1, 7]

have found occasion to apply the Gini index. These are certainly not the only examples,

and the reader may enjoy ﬁnding others.

A search of the literature turns up many efforts to explain away the Gini index as

inaccurate or incomplete. It seems to me that none of these objections should pre-

vent us from using the Gini index to analyze data for ourselves and share the results

with those we know. It matters to me to know a few summary statistics, though they

be mere summary statistics, and to know how to relate them to average outcomes of

thought experiments about who earns the average dollar and about the poorer of two

households.

Our country is rich with diverse opinions about what one ought to do about eco-

nomic facts, but perhaps we are not sufﬁciently armed with facts that we have checked

for ourselves. I hope that this article will inspire you to dig into the mountains of data

that are available and reﬁne some summary statistics for yourself.

ACKNOWLEDGMENTS. The author is grateful for the support that the MAA provides to editors, allowing

them to supervise a generous and thorough review process, which greatly improved this paper.

REFERENCES

1. R. Abraham, S. van den Bergh, and P. Nair, A new approach to galaxy morphology, I: Analysis of the

Sloan digital sky survey early data release, Astrophysical Journal 588 (2003) 218–229. doi:10.1086/

373919

2. A. B. Atkinson, On the measurement of inequality, J. Econom. Theory 2(1970) 244–263. doi:10.1016/

0022-0531(70)90039- 6

3. , On the measurement of poverty, Econometrica 55 (1987) 749. doi:10.2307/1911028

4. Central Intelligence Agency World Factbook, available at https://www.cia.gov/library/

publications/the-world- factbook/geos/us.html, accessed April 20, 2008.

5. J. L. Gastwirth, A general deﬁnition of the Lorenz curve, Econometrica 39 (1971) 1037–1039. doi:

10.2307/1909675

6. , The estimation of the Lorenz curve and Gini index, Rev. Econom. Statist. 54 (1972) 306–316.

doi:10.2307/1937992

7. D. Gianola, M. Perez-Enciso, and M. A. Toro, On marker-assisted prediction of genetic value: Beyond

the ridge, Genetics 163 (2003) 347–365.

8. C. Gini, Variabilit`

a e mutabilit`

a; reprinted in Memorie di Metodologica Statistica, E. Pizetti and T.

Salvemini, eds., Libreria Eredi Virgilio Veschi, Rome, 1955.

9. J. Golden, A simple geometric approach to approximating the Gini coefﬁcient, Journal of Economic

Education 39 (2008) 68–77. doi:10.3200/JECE.39.1.68-77

10. N. Kakwani, On a class of poverty measures, Econometrica 48 (1980) 437–446. doi:10.2307/1911106

11. C. Leiber and S. Kotz, A characterization of income distributions in terms of generalized Gini coeeﬁ-

cients, Social Choice and Welfare 19 (2001) 789–794. doi:10.1007/s003550200154

12. M. O. Lorenz, Methods of measuring the concentration of wealth, J. Amer. Statist. Assoc. 9(1905) 209–

219.

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13. C. DeNavas-Walt, B. D. Proctor, and J. Smith, Income, Poverty, and Health Insurance Coverage in the

United States: 2006, Current Population Report, U.S. Census Bureau, August, 2007.

14. U.S. Census Bureau, Current Population Survey (CPS), available at http://pubdb3.census.gov/

macro/032007/hhinc/new06_000.htm, accessed April 20, 2008.

15. B. H. Webster, Jr. and A. Bishaw, Income, Earnings, and Poverty Data From the 2006 American Com-

munity Survey, American Community Survey Report, U. S. Census Bureau, August, 2007.

FRANK A. FARRIS received his B.A. from Pomona College in 1977 and his Ph.D. from M.I.T. in 1981. He

has taught at Santa Clara University since 1984 and recently ﬁnished a second stint as editor of Mathematics

Magazine. His article “The Edge of the Universe” in Math Horizons was honored with the Trevor Evans Award.

Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, CA 95053

ffarris@scu.edu

A Fourth Proof of an Inequality

Motivated by the paper “Three Proofs of the Inequality e<1+1

nn+0.5”[2],

we provide a fourth proof and many other reﬁnements. By truncating the well-

known continued fraction [3, p. 342]

ln(1+x)=x

1+12x

2+12x

3+22x

4+22x

5+32x

6+···

,

we see that for x>0,

x>6x+x2

6+4x>··· >ln(1+x)>··· >6x+3x2

6+6x+x2>2x

2+x.

Dividing by ln(1+x), inverting, substituting 1/nfor x, and applying the expo-

nential function, we obtain

1+1

nn

<1+1

nn+3n

6n+1

<··· <e<··· <1+1

nn+3n+1

6n+3

<1+1

nn+1

2

.

A similar argument can be found in [1].

REFERENCES

1. M. D. Hirschhorn, Response to note 88.39, Math. Gaz. 89 (2005) 304.

2. S. K. Khattri, Three proofs of the inequality e<1+1

nn+0.5,Amer. Math. Monthly 117 (2010)

273–277. doi:10.4169/000298910X480126

3. H. S. Wall, Analytic Theory of Continued Fractions, AMS Chelsea Publishing, Providence, RI,

2000.

—Submitted by Li Zhou, Polk State College, Winter Haven, FL 33881

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