Inequity in tertiary education systems - World...
42
Inequality in tertiary education systems: which metric should we use for measuring and benchmarking? by Béatrice d'Hombres Produced as background for World Bank “Equity of access and success in tertiary education” study with funding from the Bank Netherlands Partnership Program (BNPP).
Transcript of Inequity in tertiary education systems - World...
Inequality in tertiary education systems: which metric should we use for measuring
and benchmarking?
by Béatrice d'Hombres
Produced as background for World Bank
“Equity of access and success in tertiary education” study with funding from the Bank Netherlands Partnership Program (BNPP).
1 Introduction
There is a very large literature on income inequality as well as on some
non-income dimensions of inequality such as health inequality. A variety of
metrics have been proposed to operationalize inequality and to allow cross-
country comparisons, with each of them having its own advantages and dis-
advantages.
Studies on education inequality and inquitable attainment of education across
countries are much more limited, with the notable exceptions of Thomas et
al (2001), Zand and Li (2002), and Barros et al (2009). Drawing upon the
existing literature on income inequality, this note intends to discuss the quan-
ti�cation of disparities in tertiary education and, more precisely, to examine
which of the common metrics could be used for benchmarking the inequity
dimension of tertiary education systems. We distinguish the summary indices
that are appropriate for cross-country comparisons and those that should be
more adapted for country-speci�c studies.
Our review suggests that cross-country comparisons should focus on dispar-
ities across social groups that exist and can be compared across countries.
This restriction limits the benchmarking exercise to measuring and compar-
ing disparities in tertiary education by sex and/or income quintile. In addi-
tion, given that such a benchmarking requires comparable indicators across
countries and over time, the e¤orts should �rst be put on compiling educa-
tion indicators at the aggregated level (country level) but broken down by
equity groups. The simple measures of dispersion such as those presented in
section 3.1, therefore, may be the more appropriate indicators of disparities
in tertiary education. When social groups are de�ned on the basis of income
groups, the analysis could be complemented by employing the concentration
index or regression-based indices that are respectively presented in sections
3.2.1 and 3.2.7.
Country-speci�c studies would allow a more in-depth investigation of dis-
parities in tertiary education. First, various equity groups could be covered
by the analysis, in addition to the two groups mentioned before. It might
1
be, for instance, relevant in some countries to scrutinize disparities in ter-
tiary education across ethnic or religious groups. Second, the data at hand
will likely come from household surveys or survey on tertiary students, so
the analysis will be carried out at the individual level. As a corollary, the
datasets employed will most probably contain, in addition to the educational
status of the respondents, several variables on the personal characteristics of
the persons interviewed (age, family structure, province of residence, etc.).
As a result, it should be possible to compute scalar summary measures of
the dispersion of education across social groups, while controlling for other
individual characteristics that are simultaneously correlated with education
and the social groups to which the individuals belong to.
Then the appropriate inequality indicator for such country-speci�c studies de-
pends on the characteristics of the variable used to de�ne the equity groups.
For measuring the inequality of opportunity across economic groups, it seems
logical to rely on the concentration index or on regression-based indices. Sim-
ilarly, sex disparities might be examined in the context of regression-based
indices. Finally, when there is not an inherent ordering among social groups
(religious or ethnic groups) and the groups are numerous (more than 2 eth-
nic or religious groups), it should be appropriate to rely on entropy indices
(section 3.2.4) and/or the dissimilarity index (section 3.2.6).
The structure of the paper is a follows. Section 2 includes preliminary con-
siderations on the concept of inequity and a discussion on the characteristics
of the dataset and education indicators that should be available for estimat-
ing empirically the disparities across social groups in tertiary education. It
also presents the di¤erent equity target groups and their relevance for cross-
country comparisons and country-speci�c studies. Section 3 describes several
inequality indicators. This section seeks to (i) highlight the main advantages
and shortcomings of each of those indicators and (ii) stress in which con-
text (cross-country benchmarking or within country analysis) they could be
employed. Section 4 concludes and makes some recommendations.
2
2 Measuring inequity in tertiary education
systems: some preliminary considerations
2.1 Inequality versus Inequity
There is often confusion surrounding the concepts of inequality and inequity.
Inequality refers to di¤erences between groups without any considerations on
the fairness of these di¤erences while, in contrast, inequity presupposes an
ethical judgment.
In a recent report, the OECD (2008) de�nes equitable tertiary systems as
"those that ensure that access to, participation in and outcomes of tertiary
education are based only on individual�s innate ability and study e¤ort. They
ensure that the achievement of educational potential at tertiary level is not
the result of personal and social circumstances, including of factors such as
socio-economic status, gender, ethnic origin, immigrant status, pace of resi-
dence age or disability."
In other words, although di¤erences between individuals are interesting in
themselves, if we want to develop a policy message, it is necessary to dis-
tinguish variations in educational outcomes that are driven by di¤erences in
individuals�e¤orts from those linked to factors that are beyond the individ-
uals�control.
2.2 Data
2.2.1 Vertical versus horizontal dimensions of equity
Assessing the vertical dimension of equity requires information on the:
1. Admission and enrollment in tertiary education of students having com-
pleted a secondary education
2. Progression of students enrolled in tertiary education
3. Completion of tertiary studies of students enrolled in tertiary education
3
4. Tertiary education learning outcomes
A comprehensive understanding of the issue under study would also re-
quire information on the organization of tertiary education systems. In par-
ticular, the horizontal dimension of equity depends on the level of diversi�-
cation of tertiary education systems and refers to the type of institution or
programmes attended by the di¤erent equity groups as well as the associated
outcomes on the labour market. In the context of building an index of equity
for benchmarking, the focus should probably be �rst limited to the vertical
dimension of equity. Country-speci�c studies, however, could also consider
the horizontal dimension of equity conditional data availability.
2.2.2 Individual and aggregated data
The data used to examine the educational disparities in tertiary education
will be drawn either from surveys collecting information at the individual
level or from various aggregated indicators collected by institutions involved
in the production of statistics.
At the individual level, the information should mainly come from household
surveys or surveys on tertiary education students.
� Household surveys are available in a large number of countries. Theo-retically, such data could be used both for country-speci�c studies and
cross-country comparisons.
The population under consideration included adult individuals having
achieved their education at the time of the interview. Such data will,
unfortunately, give a picture not of the current level of inequity in ter-
tiary education systems but of the past. In addition, the sample will
include individuals having undertaken tertiary studies at di¤erent pe-
riods of time.
In terms of education-related information, we will utilize at most of
variables on the educational attainment of the respondents (number
of years of completed education, or the graduation level - primary,
secondary or tertiary grade - of respondents). In other words, while
4
household surveys might allow us to canvass the equity dimension of
tertiary education systems in terms of enrollment and completion rates,
the information on the progression of students during tertiary studies
will be more di¢ cult to obtain.
Household surveys will also contain information on the characteristics
of individuals (sex, age, region of residence, etc) and this could be taken
into account to re�ne the analysis of inequity in tertiary education. In
most of cases, however, family background-related information (socioe-
conomic status of the parents of the respondents) will not be available.
It implies that the use of such data might be problematic for cross-
country comparisons. Instead, household surveys should be of wider
interest for country-speci�c studies.
� Surveys on tertiary students: such data should o¤er much more infor-mation on the educational performance of the respondents, with, in
particular, assessments of higher education learning outcomes through
scores obtained in di¤erent kinds of tests on key competencies. Surveys
on tertiary students usually contain detailed information on the per-
sonal characteristics and the family environment of the respondents,
however, such surveys are implemented in a limited number of coun-
tries. The OECD initiative (AHELO) which will eventually allow for
future cross-country comparisons of higher education learning outcomes
is still at an early stage. For the time being, when available, this source
of information can thus only be employed for country-speci�c studies.
At the macro level, aggregated indicators provided by national statistical
o¢ ces and/or tertiary institutions might be the most useful. The indicators
on education can be aggregated at various unit levels. Such indicators are
appropriate for cross-country and time comparisons if they are broken down
by equity groups (e.g. country mean level of educational attainment by equity
group).
5
2.2.3 Education variables
At the individual level, the education variables will generally take the form
of discrete variables, i.e. variables which can be broken down into separate
categories but where no fractions are possible. For instance, participation
in tertiary education is a discrete variable since there are only two possible
educational status for a given individual: the individual is participating or
not participating in tertiary education. Similarly the completion of tertiary
studies is a discrete variable. When the number of categories is equal to 2,
the discrete variable is called a binary discrete variable. When the discrete
variable can take more than 2 di¤erent values, then the variable is more gen-
erally considered to be a categorical variable.
The use of surveys on tertiary students could also give access to continuous
educational variables, i.e. variables that can take on any value from an inter-
val of real numbers. An education variable de�ned by the score obtained by
tertiary students in a writing skill test is a continuous variable. However, as
we mentioned earlier, survey data on tertiary students are only available for a
very limited number of countries, and so can only be employed in the context
of country-speci�c studies. If we rely on individual datasets for carrying out
cross-country comparisons of educational disparities, then the educational
variables at hand will be discrete variables.1
At the macro level, the education variables will be continuous variables. In
other words, if we use indicators aggregated at the country level, the educa-
tional variables employed for the empirical analysis could be, for instance, the
enrollment rate in tertiary education, the completion rate of tertiary studies,
or the survival rate in tertiary education, with the education variable taking
on the country mean value by equity group.
1This is problematic for several of the inequality indicators with in particular the fol-lowing two consequences. First, as we will see later, we will not be able to measure pureinequality among individuals and second, we will often have to work with grouped data(i.e aggregated indicators) in order to express the education variables on a ratio scale.
6
Table 1: Dataset, education indicators and level of analysisDataset Education variable Type of analysis
Individual data1 - Household surveys - Discrete variable - Country-speci�c studies
- Cross-country comparisons
2 - Surveys on tertiary - Discrete variable - Country-speci�c studiesstudents - Continuous variable
Aggregated indicators - Continuous variable - Cross-country comparisons
2.2.4 Equity groups
There are several personal and social circumstances that can lead to in-
equitable access to tertiary education, including economic status, gender,
ethnic, origin, or immigrant status. For constructing an index on inequality
opportunity in tertiary education, the equity group should be (1) compa-
rable across countries and (2) measured with an indicator available on an
international scale.
The concept note underpinning this entire project, Equity and Access to
Tertiary Education, considers the following 4 equity target groups: (1) in-
dividuals from the lower income groups, (2) individuals from groups with
a minority status de�ned on the basis of their ethnic, linguistic, religious,
cultural or age characteristics, (3) females, and (4) people with disabilities.
The conditions above limit cross-country analysis to the �rst and third equity-
target groups.
The four equity groups can be examined in the context of country-speci�c
studies on inequity. Indeed, while it might be relevant to study ethnic, reli-
gious, or linguistic disparities in the access to tertiary education in a given
country, this is more problematic for international comparisons, simply be-
cause the ethnic or linguistic heterogeneity is country-speci�c.
7
Table 2: Equity groups and level of analysisEquity groups Type of analysis
1 - Individuals from the lower - Country-speci�c studiesincome groups - Cross country-comparisons
2 - Individuals from groups with - Country-speci�c studiesa minority status
3 - Females- Country-speci�c studies- Cross country-comparisons
4 - People with disabilities - Country-speci�c studies
Note that the �rst equity group considers the educational disparities among
individuals belonging to di¤erent categories of income. The variable used to
de�ne the equity groups is thus an ordinal variable measured on an interval
scale. It means that there is an inherent rankings between the income groups
and that the groups can be ranged from the poorest to the richest. This is a
fundamental di¤erence with the variables employed to de�ne the other equity
groups. Indeed, there is not an inherent ordering among racial or religious
groups and thus the variables used for de�ning those equity groups are non
ordinal categorical variables.
2.3 Quality criteria
The legitimacy and applicability of each disparity indicator should be evalu-
ated along the following dimensions:
� Is the indicator easy to compute and understand for non statisticians?
� Is the indicator adapted to the variables used for monitoring the tertiaryeducation systems?
�Variable used for measuring the educational performance.
�Variable used for de�ning the equity groups.
8
� Is the indicator sensitive to the social gradient in education? Meaning,does the indicator provide a summary measure of inequality or does
the indicator also tell us which are the most disadvantaged groups?
� Does the indicator provide us with information about the whole distri-bution of the education variable?
� Does the indicator meet the "desirable "statistical properties under-lined in the literature on income inequality? There are three properties
that are of particular interest:
�Principle of transfers (Pigou-Dalton condition), which says in theincome literature that a transfer of income from a richer to a
poorer individual will result in a reduction in the indicator of
disparity, assuming that the income of other individuals remains
unchanged and the transfer is not large enough to reverse anyone�s
relative position.
� Scale independence which says that if the value of the educationindicator doubles for each of the equity groups, the value associ-
ated with the inequality indicator does not change.
�Boundness of the indicator: the interpretation of the value as-signed to the inequality indicator will be easier if this indicator
has a lower bound and an upper bound.
3 Measures of disparity
3.1 Simple mesures of dispersion
Most of the studies that discuss education disparity rely on simple compar-
isons between speci�c groups as those that are presented below.
The easiest way to measure education disparities is to use range and ra-
tio measures of dispersion. Consider J the number of equity groups, with
j = 1; :::; J and Edj the education variable for the equity group j. Range
9
measures (RM1) are based on the distance in education between pairs of
equity groups as follows:
RM1 = Edj � Edk; with j 6= k: (1)
while ratio measures (RM2) are equal to:
RM2 =EdjEdk
; with j 6= k: (2)
Those two measures di¤er in the sense that RM1 is expressed in absolute
terms and RM2 is a relative measure of dispersion. Very often, Edj and
Edk are the two extreme groups. It is also possible to produce j pairwise
comparisons as follows:
RM1j = Edj � Ed and RM2j =Edj
Ed; j = 1; :::; J: (3)
with Ed = 1J
PJj=1Edj the mean educational performance of the total popu-
lation. We can also replace Edk by Ed� the "ideal" educational performance:
RM1j = Edj � Ed� and RM2j =EdjEd�
; j = 1; :::; J: (4)
Example: Consider that the education variable is the enrollment rate in
tertiary education and the population of interest is divided by income quin-
tile. In such a situation, RM1 would be the di¤erence in the enrollment rate
between the highest and lowest socio-economic groups and RM2 the percent-
age of the enrollment rate of the lowest socio-economic group with respect
to the highest socio-economic group.
# Pros :
� Such indicators are very easy to compute and have a straightforwardinterpretation.
10
� In addition RM1 and RM2 do not pose a lot of restrictions on the data.
The variable used to de�ne the J groups does not need to be an ordinal
variable. The population could, for instance, be divided according to
the ethnic origin, the place of residence, or the sex, etc. It is a clear
advantage with respect to some of the indicators that will be presented
later on.
� Statistical properties: the scale invariance property is satis�ed by RM2
(but not by RM1).
# Cons :
� If the two groups Edj and Edk are small in size (in the case they arenot de�ned in terms of the highest and lowest quintiles), then RM1
and RM2 would only re�ect the disparity between two small groups
and the results will be instable. In general, RM1 and RM2 do not take
into account the size of the groups being compared.
� Such pairwise comparisons as in (1) and (2) ignore a large part of theinformation (intermediary groups) and may conceal important hetero-
geneity. In order to take into account the educational performance of
the other groups, we have already noticed that it is possible to pro-
duce j pairwise comparisons as expressed in (3) and (4). If the number
of groups is important, however, it becomes tricky to summarize the
information.
� Not taking the population share in each equity group into account maymake time and cross-country comparisons problematic.
� These 2 measures provide di¤erent types of information and, if they areused for comparisons over time, they can lead to opposite conclusions.
It is because RM2 does not inform about changes in absolute rates. We
might observe an increase in RM1 if the enrollment rate (for instance)
increases at the same speed in the two groups but RM2 will remain
identical.
11
� Statistical properties: the principle of transfers is not satis�ed and theindicators are not bounded between 2 values.
In Practical terms:
Cross-country comparisons: Despite the fact that these measures of disper-
sion have several shortcomings they could be appropriate for cross-country
comparisons because they are easy to compute, not data demanding and they
can be used with equity groups de�ned both by ordinal (income) and non
ordinal (sex) variables. RM2 is preferred to RM1:
Country-speci�c analysis of inequity: These summary measures of dispersion
taken alone do not permit an in-depth analysis of disparities across social
groups.
3.1.1 Regression based analysis
A convenient way of taking into account educational di¤erences between the
intermediary groups and to produce a summary index of dispersion is to
apply a regression analysis in which the educational performance is related
to the equity group (SEG) of the individual i; i = 1; :::; N . The equation to
be estimated will be the following:
Edi = �o + �1SEGi + �i (5)
with �i the disturbance term of equation (5), �o and �1 are the parameters to
be estimated. The variable SEGi must be measured with an ordinal variable
on an interval scale. The estimated parameter �1 provides us with a measure
of disparity in educational performances between SEGs.
Example: Depending on the characteristic of the right-hand side variable of
equation (5), this approach produces a relative e¤ect index or a regression-
based absolute e¤ect. If, Edi is a binary discrete variable measured at the
individual level, for instance a variable taking on the value one if the indi-
vidual has completed a tertiary degree and zero otherwise, while SEGi tells
us in which income quintile the family of the individual i belongs to, then
12
equation (5) should be estimated with a probit (or logit) model and �1 will
be (an odd-ratio) the e¤ect of moving up to the next quintile on the proba-
bility to complete a tertiary degree. On contrary, if we work on grouped data
and Edi is a continuous variable measuring the completion rate of a tertiary
degree of group i, then �1 is the e¤ect of one unit change in the absolute
increase in Edi:
# Pros:
� The indicator is sensitive to the social gradient in education. Stayingwith our previous example, the indicator tells us what is the direction
of the correlation (positive or negative) between the educational status
of the individual and the income group to which the family of the
respondent belongs to.
� This approach allows to take into account the whole distribution of theeducation variable and to produce a summary measure of educational
disparities.
� Equation (5) can be estimated with both individual and grouped data.On grouped data, eq(5) just becomes
Edj = �o + �1SEGj + �j; for the equity group j; j = 1; :::J: (6)
� The regression analysis will also produce con�dence intervals for eachof the estimated parameters, so it will be possible to know whether
the disparities between SEGs are signi�cantly di¤erent from zero. The
precision of the estimate will depend on the number of observations.
� It is possible to include in the equation (5) additional covariates in orderto control for other individual or group characteristics (age, location of
residence, family structure, etc) that are simultaneously correlated with
Edi and SEGi. It constitutes a great advantage over other measures of
dispersion, and it is particularly interesting if we work with individual
data.
13
� On grouped data: �1 is a valid measure of educational disparitiesbetween social groups at one point in time and for a given country.
However, it is also possible take into account changes or di¤erences in
the distribution of equity groups if equation (6) is estimated through
weighted least squares and SEGj measures the average relative ranking
of the equity group j. In such a situation c�1 would tell us the e¤ecton education of moving from the bottom to the top of the social group
distribution.2
# Cons:
� We need to assume that the relationship between Edi and SEGi islinear.
� The groups are ranked by their SEG which means that SEGi must
be an ordinal variable measured on an interval scale ( such as the socio
economic status). If the equity groups are de�ned by a non ordinal
binary variable (gender status), then we can also compute a regression-
based summary index (of gender disparities). If we are interested, for
instance, in ethnic disparities, then the equity group will be de�ned
by a non ordinal categorical variable. A regression based analysis can
be carried out by replacing SEGi with (J � 1) ethnic dummies withJ being the number of ethnic groups. It would not, however, estimate
one summary index of ethnic disparities with such a speci�cation.
# Review of the literature:
� Such a speci�cation is adopted in most of the country-speci�c studies,based on individual data, which examine the determinants of educa-
tional performances in primary and secondary schools. See, for in-
stance, Alderman et al (1997), Filmer and Pritchett (2001), Fuchs and
Woosman (2007), Pontili and Kassouf (2008).
If we work with comparable household surveys and adopt a econometric
2Note that from this estimate, it is possible to compute a relative disparity metricwhich is easy to interpret and comparable over time and across countries.
14
speci�cation (equation (5)) similar for each country, it is then possible
to compute an index of equality of opportunity to be used for bench-
marking.
In Practical terms:
Cross-country comparisons: If we use comparable household surveys and
adopt an econometric speci�cation (equation (5)) similar for each country, it
is then possible to measure disparities across economic groups or by gender.
As already said, it is also possible to use such indices with aggregated data.
Country-speci�c analysis of inequity: The regressions-based approach is par-
ticularly useful when we carry out an in depth country speci�c study of
educational disparities while using individual data. It is, therefore, possi-
ble to take advantage the main pros of regression-based indices listed above
(estimation in a multivariate context, con�dence bounds associated with the
summary index). Such indices are not appropriate when the social group is
de�ned by a non ordinal categorical variable (ethnic or linguistic groups for
instance).
3.1.2 Population attributable risk
The population attributable risk (PAR) is a summary index of the di¤erence
between the educational performance of each group and the one of the best
group. Mathematically, it can be expressed as follows:
PAR% =
PJj=1 pj(Edj � Edref )PJ
j=1 pj(Edj � Edref ) + Edref(7)
with pj being the share of group j in the population and Edref the value of
the education variable for the best performing group.
Example: If the education variable is the enrollment rate in tertiary educa-
tion and the population is divided in three ethnic groups, then (7) would be
the percent improvement in the enrollment rate for the total population that
15
would be necessary to assure that the three ethnic groups have an enrollment
rate corresponding to the one of the best performing ethnic group.
# Pros:
� Summary index of education inequality easy to compute and not datademanding
� There are no restrictions on the characteristics of the grouping variablewhich can be an ordinal variable (income groups) or a non ordinal
variable (gender, ethnic or religious groups).
� PAR is sensitive to the proportion of individuals in each group: con-venient for time comparisons.
� Statistical property: indicator bounded between 0 and 100.
# Cons:
� The indicator fails to re�ect the socioeconomic dimension of inequal-ities in education: the indicator does not tell us which are the most
disadvantaged and the most advantaged groups.
# Review of the literature:
� To the best of our knowledge, there are not studies in education usingthe PAR as indicator of inequality. For a discussion about the use
of this indicator in public health, see, for instance, Mackenback and
Kunst (1997), Krokstad et al (2002) or Regidor (2004).
� Statistical property: PAR does not satisfy the principle of transfers
and is not scale invariant.
In Practical terms:
Cross-country comparisons: This use of this indicator for cross-country com-
parisons is a possibility but might be problematic if the best performing
16
group only represents a very low proportion of the population. In that case,
comparing the education performance of each equity group with respect to
this group does not really make sense.
Country-speci�c analysis of inequity: The population attributable risk indi-
cator taken alone does not allow for an in-depth analysis of disparities across
social groups.
3.1.3 Education Gini coe¢ cient
The Gini coe¢ cient as well as the Theil and Atkinson measures are standard
metrics for measuring pure income inequality among individuals. Such indi-
cators fundamentally require continuous variable collected at the individual
level. As we have discussed before, the indicators that we might have at our
disposal are most probably binary variables at the individual level that can
be converted into continuous variables when one works with grouped data.
In this context, the information provided by the Gini coe¢ cient or entropy
indices is about disparities across groups of individuals.
Education Lorenz Curve: The education Lorenz Curve maps the cu-
mulative educational share on the y-axis against the cumulative population
share ordered from the least educated to the most educated on the x -axis.
When the education variable is a "positive" variable such as the the number
of years of education (higher is the level of education, better it is), the Lorenz
curve will lie below the diagonal, with the diagonal representing a uniform
distribution of education as shown in �gure (1). In contrast, the Lorenz
curve will be above the diagonal if the education variable is a "negative"
variable such as the retention ratio (lower is the retention ratio, better it
is). In case of perfect equality in the distribution of education, the Lorenz
Curve and the diagonal coincide. The larger the distance of the curve from
the diagonal line, the larger the inequality. When for two countries, A and
B, the Lorenz curve of country A lies in any point below the Lorenz curve
of country B, we can conclude that education disparities in country A are
higher than in country B. If the two Lorenz curves cross each other, then we
cannot conclude which distribution is more equitable and we need to rely on
17
Figure 1: Education Lorenz Curve
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1
Cum
ulat
ive
enro
llmen
t sha
re
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1Cumulative percentage of the tertiary age population
Lorenz curveLine of perfect equality
the gini coe¢ cient. Furthermore, while the Lorenz curve gives a graphical
representation of disparity over the whole distribution of education, its use
for comparisons across many countries is not su¢ cient. Instead, the educa-
tion gini coe¢ cient provides a summary index of education disparities that
can be easily used for international comparisons.
Education Gini coefficient: Adapted from Thomas et al (2002), the
Education gini coe¢ cient can be written as follows:
EG =1
Ed
JXj=1
j�1Xk=1
pjpk jEdj � Edkj (8)
where pj and pk are the proportions of the population that respectively
belong to the equity groups j and k: Edj and Edk are the values taken
by the educational variable for the two corresponding equity groups. The
coe¢ cient varies between 0, which re�ects complete equality and 1, which
indicates complete inequality.
18
Example. Suppose the enrollment rate in tertiary education at the region
level for a given country known. The lorenz curve for that country will map
the cumulative enrollment share on the y-axis against on the cumulative
percentage of the tertiary age population, region by region, with the regions
being sorted by enrollment rate from the region with the lowest enrollment
rate to the one with the highest enrollment rate. The Gini coe¢ cient corre-
sponds to twice the area between the Lorenz curve and the diagonal.
# Pros:
� It is the most well-known inequality metric.
� Among its advantages with respect to entropy measures, the outcomevariable can include negative and null values.
� When the gini coe¢ cient is used with grouped data, the grouping vari-able can be an ordinal or a non ordinal variable.
� Statistical properties: Pigou-Dalton Transfer sensitivity, scale invari-ant, bounded between 0 and 1.
# Cons:
� The educational variable must be a continuous variable.
� Using individual data, supposing the education outcome is continu-ous, the Gini coe¢ cient measures pure inequality between individual
(overall level of inequality between individuals) but fails to provide in-
formation on the "unfair" component of inequality (part of the overall
inequality due to the social circumstances).
� When the Gini coe¢ cient is computed with data aggregated by socialgroup, what matters is how the share of each social group in the pop-
ulation with a given education condition compares with its share in
the total population. In other words, and staying with the previous
example, the result is a summary index of disparities across regions
but it is not possible to identify the regions in the most (less) favorable
situation in terms of enrollment rate.
19
� Although the level of inequality is given by the value of the educationgini coe¢ cient, the interpretation of the coe¢ cient can only be done in
comparative terms.
# Review of the literature:
� The education Gini coe¢ cient has only been used in few occasions toquantify and explore cross-country variations in education inequality
(Maas and Criel 1982, Thomas et al., 2002, Zhang and Li, 2002, Sahn
and Younger, 2007). Among the most cited studies, there are Maas
and Criel (1982) who use enrollment data by province to estimate the
education Gini coe¢ cients of 16 East African countries and Thomas et
al. (2001) who rely on the schooling distribution data of Barro and Lee
(1993 and 1997) and the schooling cycle data of Psacharopoulos and
Arriagada (1986) to measure the education Gini coe¢ cient based on
educational attainment of 140 countries from 1960 to 1990. Similarly,
Zang and Li (2002) explore the international educational inequality and
convergence in educational attainment over the period 1960-2000. See
also SITEAL (2005) for a very comprehensive overview of the di¤erent
inequality indices with empirical examples.
In Practical terms:
Cross-country comparisons: We believe that the Gini coe¢ cient is of limited
interest for the reasons outlined before, but it could be useful if we are inter-
ested in cross-regional comparisons within a given country.
Country-speci�c analysis of inequity: The Gini coe¢ cient utilizing individual
data can be a good starting point to measure overall inequality among in-
dividuals. But as noted earlier, its use for measuring educational disparities
between equity groups is limited.
3.1.4 Generalized Entropy indices and Atkinson index
Theil and mean logarithm deviation indices
The property of additive decomposition between and within exclusive groups
20
valid for entropy indexes, but not for Gini index, greatly contributed to in-
creasing the use of entropy indexes in income inequality studies. Additive
decomposition means that it is possible to decompose the overall inequality
among individuals into two components: the �rst component (between social
groups index) is the unfair part of inequality driven by factors beyond the
control of the individual (gender, race, income groups, etc) while the second
one results form the individual�s e¤ort (within social groups entropy indices).
On individual data, the generalised entropy (GE) class of metrics can be ex-
pressed as follows:
GE(�) =1
�2 � �
"1
N
NXi=1
�Edi
Ed
��� 1#
(9)
where N is the number of individuals, Edi is the value of the education
variable for individual i, Ed is the mean value of the education variable in the
total population and � represents the weight given to the distance between
Edi and Ed at di¤erent part of the education distribution. The most common
entropy indices, the Theil index and the Mean Logarithm Deviation index
(MLD) correspond to (9) when respectively � = 1 and � = 0.3 The more
positive is the sensitivity parameter �; the more sensitive is the entropy index
to inequalities at the top of the education distribution.
Neither indices can be used with binary education indicators, so unless there
are continuous education outcome data (through, for instance, the OECD
initiative on assessing higher education learning), cross-country comparisons
will utilize grouped data.
The Theil index can mathematically be expressed, with grouped data, as
follows:
Theil =JXj=1
pjEdj
Edln
�Edj
Ed
�(10)
3Note that when � = 2; the GE index corresponds to half the squared coe¢ cient ofvariation.
21
while the Mean Logarithm Deviation index is equal to:
MLD =JXj=1
pj ln
�Ed
Edj
�(11)
where pj is the proportion of the population that belongs to the equity
group j; Edj is the value of education for group j and Ed is the mean value
of the education variable in the total population.
Atkinson Index
The Atkinson�s measure is another well-known inequality measure which can
be de�ned as follows:
A� = 1�"
JXj=1
pjEdj
Ed
1��# 1(1��)
, � > 0 (12)
The extent of disparity depends on the value of �, which indicates the degree
of aversion to disparity. When � > 0 , there is a preference for equality (i.e.
an aversion to inequality). As � rises, more weight is attached to education
transfers at the lower end of the education distribution and less weight to
transfers at the top of the education distribution. The Atkinson index ranges
between 0 and 1 with 0 indicating perfect equality and 1maximum inequality.
Example. Supposing that the educational variable of interest is the enroll-
ment rate in tertiary education and that the population is divided by income
groups, the Atkinson index and the two entropy indices are all function of
each group proportion and the ratio (i.e. distance between) of the enrollment
rate of each social group to the enrollment rate in the total population.
# Pros:
� Main advantage: the generalized entropy index for the entire popula-tion can be decomposed into a weighted average of each social group�s
generalized entropy index (within social group entropy index) and a
between social group index ("unfair" component of inequality) as pre-
viously mentioned. In other words, if the education variable is the score
22
obtained in a reading test by tertiary students and that population is
divided into J ethnic groups, the property of additive decomposition
should permit to measure the level of inequality between individuals
belonging to the same ethnic group (i.e. within social groups) and
the level of inequality between the ethnic groups (i.e. between social
groups). The property of additive decomposition requires continuous
education outcomes de�ned at the individual level however.
� Convenient for cross-country and time comparisons.
� Statistical properties: Pigou-Dalton transfer sensitivity, scale invariant.
# Cons:
� See comments on the Gini coe¢ cient.
� Statistical properties: no upper bound for the two entropy indices.
# Review of the literature:
� Only few studies have used entropy indices to compare across countrieseducation inequalities:
Sahn and Younger (2007) compare world education inequality in math
and science knowledge, using scores on math and science achievement
tests collected by the 1999 round of Trends in International Mathemat-
ics and Science Study (TIMSS) to entropy indices.
Thomas et al (2001) investigate cross-country inequalities in educa-
tional attainment using the education Theil index.
SITEAL (2005) computes the Gini, Theil and Atkinson (for di¤erent
values of �) indices for Argentina and Mexico.
Barros et al (2009) have recently examined inequality of opportunity
in educational achievement for 5 Latin American countries using 2000
PISA surveys. The education variables are reading and mathematics
test scores, and the inequality measure is the mean log deviation in-
dex. They �nd that inequality of opportunity accounts for a substantial
23
amount of observed education inequality in Latin America ranging be-
tween 14% and 28% of total inequality in reading scores and between
15% and 29% of total inequality in mathematics achievement.
In Practical terms:
Cross-country comparisons: As noted above, the main advantage of those
indicators is their decomposability property. It would be di¢ cult to take
advantage of this property for cross-country comparisons given that individ-
ual data with an education indicator measured by a continuous variable is
required for such comparisons.
Country-speci�c analysis of inequity: The three indicators presented above
might be particularly appealing for country-speci�c studies when relying on
surveys of tertiary students.
3.1.5 Education Standard Deviation and Coe¢ cient of Variationof Schooling between groups.
When social groups are unordered groups, i.e. groups without an inherent
ordering (such as ethnic or religious groups), the education standard devia-
tion (ESD) and the coe¢ cient of variation of schooling (CV S) might be two
useful indices of inequality.
The education standard deviation between J equity groups (j = 1; :::; J) is
given by
ESD =
vuut JXj=1
pj(Edj � Ed)2 (13)
and the coe¢ cient of variation of schooling corresponds to the ESD divided
by the Ed :
CV S =
qPJj=1 pj(Edj � Ed)2
Ed(14)
24
with pj the share of group j in the population and Ed the mean level of the
education variable in the population.4
Example. If the education variable is the completion rate of tertiary studies
and the population is divided by income groups, then ESD is the standard
deviation in completion rate, between income groups, while the coe¢ cient of
variation expresses the standard deviation as a percentage of the population
mean.
# Pros:
� Both indicators are commonly used and very easy to compute.
� There are no restrictions on the characteristics of the variable used toregroup the individuals.: the grouping variable does not need to be
ordinal.
# Cons:
� Neither indicator is sensitive to the direction of the social gradient ineducation: for example, it is not known whether the education status
either increases or decreases with increasing socioeconomic position. A
given value of the CVS could simultaneously correspond to a positive
or a negative association between the socioeconomic position and the
educational performance.
� Statistical properties: The CV S is preferred to the ESD because the
second one is not scale-invariant. Both indicators do not satisfy the
principle of transfers. While both indicators have a lower limit equal
to 0 and corresponding to 0 dispersion, they do not have an upper
limiting value equal to 1.
# Review of the literature:
4Similarly, we could compute the variance or the absolute mean deviation index betweengroups.
25
� Zang and Li (2002) and SITEAL (2005) compute and then compare var-ious indicators of education inequality, the ESD and CV S are among
them. Note also that Ram (1990) examines the relationship between
educational expansion and schooling inequality for about 100 countries.
Schooling inequality is measured by the standard deviation of the edu-
cational attainment. In the robustness section, the author tests whether
the results change when the schooling coe¢ cient of variation instead of
the standard deviation is used for measuring education inequality.
In Practical terms: The CV S is preferred to the ESD.
Cross-country comparisons: The coe¢ cient of variation is useful for cross-
country comparisons or cross-regional comparisons within a given country,
but this indicator does not convey information on the sense of the correlation
between education and the grouping variable and does not have as appealing
an interpretation capacity as other indicators presented in this note.
Country-speci�c analysis of inequity: The coe¢ cient of variation taken alone
does not allow for an in-depth analysis of disparities across social groups.
3.1.6 Dissimilarity index
The index of dissimilarity has been widely used in the literature on segrega-
tion. Supposing that the population is divided into J groups, the index of
dissimilarity can be expressed as follows:
ID =1
2
JXj=1
jSj � pjj (15)
where Sj is the proportion of group j in the population with a level of
education equal to Ed and pj is the share of group j in the total population.
Exemple. Supposing that the education variable is the completion rate
of tertiary studies and that the grouping variable refers to the sex of the
individual, the index of dissimilarity tells us about the proportion of all
cases that needs to be redistributed across the population to insure that, for
26
each sex, the male (female) group�s share of the population having completed
tertiary studies is equal to the male (female) group�s population share.
# Pros:
� The indicator is easy to compute and interpret.
� The social groups can be ordered or unordered groups, i.e they can bede�ned by ordinal or non ordinal variables.
� Statistical properties: bounded between 0 and 1.
# Cons:
� The index of dissimilarity is not sensitive to the socioeconomic dimen-sion of inequalities in education. What matters is how each socioeco-
nomic group�s share of the population�s education compares with its
population share, not how this disparity compares with the group�s
socioeconomic status.
� Statistical properties: not scale invariant, does not satisfy the principleof transfers.
# Review of the literature:
� Recently, Barros et al (2009) have used a version of the dissimilarityindex for analyzing children�s inequality of opportunity in education,
electricity, and improved water and sanitation in 19 Latin America
and Carribean countries. For education, they use the probability of
having completed the sixth grade on time for children age 12 to 16
and school attendance for children ages 10 to 14. Social groups are
de�ned by parents�education, family per capita income, gender, age,
family structure and area of residence. Their results suggest that, on
average, 11% of education, as measured with the �rst indicator, needs
to be reallocated in order to remove di¤erences between the di¤erent
social groups. The estimated value of the dissimilarity index for the
second education indicator is less than 5%.
27
In Practical terms: .
Cross-country comparisons: The index of dissimilarity is an attractive metric
for cross-country comparisons, but this indicator is not sensitive to the social
gradient in education. Staying with the example given above, it implies that
a given value of the indicator could correspond to a situation where females
are disadvantaged in terms of the completion rate of tertiary studies or to a
situation where males are those that experience inquitable opportunity..
Country-speci�c analysis of inequity: This index would not be enough for an
in-depth country speci�c studies of educational disparities.
3.1.7 Concentration index
The concentration curve and concentration index are derived from the bivari-
ate distribution of education and the social group ranking. The concentration
curve plots the cumulative percentage of the education variable on the y-axis
against the cumulative percentage on the x -axis of the individuals ordered
according to their socioeconomic status, beginning with the poorest and end-
ing with the richest. The concentration curve is di¤erent from the Lorenz
curve in that the x -axis for the Lorenz curve represents the cumulative per-
centage of individuals ordered according to their educational level.
If the socioeconomic status has no e¤ect on the probability of enrolling at
university, the concentration curve will correspond to the diagonal line as
shown in �gure 2.
When the education variable corresponds to grade attainment, test scores, or
enrollment ratios, the concentration curve will lie below the diagonal; while
when the education variable is the retention ratio, then the concentration
curve will be above the diagonal line. The area between the diagonal and
the concentration curve represents the extent of disparities across socioeco-
nomic groups.
Like the Lorenz curve, the Concentration curve (CC) is not a summary mea-
sure of the magnitude of inequality, so it is not useful for comparisons of socio
28
Figure 2: Education Concentration Curve
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1
Cum
ulat
ive
enro
llmen
t sha
re
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1Cumulative population ranked by SES
Concentration curvePerfect equality
economic-related education inequalities among many countries.5 Instead, the
concentration index (CI) is preferred.
The concentration index, which is based on the CC in the same way as the
Gini coe¢ cient is related to the Lorenz curve. The index is negative if the
CC lies below the diagonal and positive when the curve is situated below the
diagonal. On grouped data, with j = 1; :::J equity groups, the concentration
index is de�ned as follows:
CI =2
Ed
JXj=1
pjEdjrj � 1 with rj =j�1Xk=1
pk + pj=2 (16)
where , pj is the proportion of the jth group in the total population; Edj is
the average of the Education variable in the jth group and rj is the fractional
5In addition if the CCs of two countries A and B cross each other, it it not possibleanymore to compare the degree of inequality in those 2 countries, given that neitherdistribution dominate the other.
29
rank of the jth group.6 The index is ranged between �1 and 1. The concen-tration index can also be de�ned in terms of covariance between the education
variable and the rank in the living standards distribution as follows:
CI =2
Edcov(Ed; r) (17)
Example. In plotting the cumulative percentage of enrollment at university
accruing to the poorest quintile of the population, the concentration curve
would tell us that the 25% poorest students represent x% of the students
enrolled in tertiary education. The concentration index is equal to twice the
area between the concentration curve and the diagonal.
# Pros
� The CI is a good measure of socio-economic inequalities in education.
� Given equation (17), it can be showed that the CI could be obtained,on individual data (i = 1; :::N), from the estimate of the following
equation:
2�2r
�Edi
Ed
�= c+ �ri + �Xj + �i (18)
with b� an estimate of the CI, �2r the variance of the fractional rankand Xi a set of covariates to control for potential confounding e¤ects
(i.e. if Xi is simultaneously correlated with Edi and ri and not in-
cluded in equation (18), the estimated coe¢ cient b� will capture thee¤ect of socioeconomic status on education but also the impact of the
other covariates Xi). In addition, a con�dence interval of the estimated
concentration index can be easily computed.
� The value of the index does not change if the living standard variablechanges but does not a¤ect the rank.
� Convenient for time and cross-country comparisons.6Wagsta¤ (2002) has proposed an extended CI capturing di¤erent levels of aversion to
inequality.
30
� Statistical properties: scale invariance, bounded between -1 and 1.
# Cons:
� The equity group must be measured with an ordinal variable.
� The CI has the disadvantage of lacking a straightforward interpretation
# Review of the literature:
� The concentration curve and its associated concentration index havebeen widely used to measure socio economic inequalities in health
(Wagsta¤, 2002, 2003). However, to the best of our knowledge, only
SITEAL (2005) estimates education concentration indices in Chili and
Costa Rica for the year 2000.
In Practical terms:
Cross-country comparisons: The concentration index is particularly interest-
ing for cross-country comparisons when the population is divided by income
groups. If the equity group is not de�ned by an ordinal variable, however, it
is not possible to rely on this indicator.
Country-speci�c analysis of inequity: This index can also be integrated into
a country-speci�c study on disparities in tertiary education when examining
educational disparities across economic groups.
3.1.8 An intuitive interpretation of inequality indices
Most of the indicators presented above are indices of disproportionality and,
as such, have an interesting underlying interpretation which is often omitted.
The disproportionality between social groups means the over-represention/under-
representation of speci�c groups with a given education condition in relation
to their representation in the overall population.
Consider J the number of equity groups, with j = 1; :::; J and Sjand pj
31
being respectively the share of group j with a given level of education and
the share of population of group j
Sj=
NEdjJXj=1
NEdj
and pj =NjJXj=1
Nj
with NEdj being, for instance, the number of students of group j enrolled in
tertiary education, Nj the number of individuals of group j in age of being
enrolled in tertiary education. A perfectly fair representation of each equity
group in the students�population is such that for each group j:
Sj= pj (19)
which also corresponds to
NEdjNj
=
JXj=1
NEdj
JXj=1
Nj
=) Edj = Ed
Measures of inequality that are based on the notion of average departure of
sj from pj or equivalently of Edj from Ed (or EdjEd
from 1) are measures of
disproportionality.7 If education is evenly distributed across the J groups,
then the ratio of students enrolled in tertiary education to the number of
students in age of being enrolled for each group is equal to the ratio obtained
for the whole population, and the disproportionality indices are equal to 0.
In other words, such indices give information about the distance, for a given
education condition, between the representation of each social group and the
average representation or equivalently between the share of group j in the
total student and the share of group j in a given education condition with
respect to the total number of students in that situation.
7See Firebaugh (1999) for additional information.
32
While disparities across social groups can be assimilated with the concept
of disproportionality, most of the indicators used for measuring inequality -
gini coe¢ cient, entropy indices, standard deviation, coe¢ cient of variation
of schooling, concentration index dissimilarity index - can be expressed as
metrics of average disproportionality.
4 Recommendation and conclusion
This note discusses the indices of inequity that could be used for monitoring
and benchmarking the equity dimension of tertiary education systems, pre-
senting an overview of the di¤erent summary indices of inequality that would
allow for (i) in-depth country speci�c studies and (ii) large cross-country com-
parisons.
Constructing an index to measure and compare equality of opportunity in
tertiary education across countries implies studying disparities in tertiary
education among equity groups that are comparable across countries. This
will limit the analysis of disparities to income and gender groups. Country-
speci�c studies could be used to cover the two remaining groups (minority
groups and people with disabilities, see section 2.2.4).
The education indicators (see section 2.2.3) that will be used will most likely
be discrete variables available at the individual level (e.g. educational status
of the respondents) or continuous variables available at the aggregated level
(e.g. enrollment rate in tertiary education by country but broken down by
equity group).
When the object of the study is to make country-speci�c studies and when
the social groups are ordered groups with an inherent ordering (e.g. income
groups), the concentration index (section 3.2.7) or regression-based indices
(section 3.2.1) are preferred to alternative indices for three reasons. First, the
concentration index and regression-based indices are sensitive to the direc-
tion of the social gradient in education and, as such, measure how educational
status varies with socio-economic position. Second, both set of indicators can
33
be derived from an estimate in a multivariate context, which means that it is
possible to control for factors that are simultaneously correlated with educa-
tional performances in tertiary education and the equity group to which the
individual belongs. In addition, con�dence intervals associated with the esti-
mated disparity indices can be easily computed. Third, they are appropriate
for time comparisons given that in both cases, they depend on disparities
between groups and on the proportion of the population in each of these
groups.
When social groups are unordered groups without an inherent ordering, de-
�ned by a binary discrete variable (e.g. gender), for country-speci�c studies,
regression-based indices are most useful because of the second and third rea-
sons mentioned above. When social groups are unordered groups and de�ned
by a categorical variable (e.g. minority groups such as ethnic or religious
groups) or by several circumstances (sex, gender, family background), en-
tropy indices and/or the dissimilarity index should be preferred. The dis-
similarity index (section 3.2.6) is particularly appealing because it is a well-
known metric and can be easily understood. Entropy indices (section 3.2.4)
are less intuitive, yet, the generalized entropy index for the entire population
can be decomposed into a weighted average of each social group�s generalized
entropy index (within social group entropy index) and a between social group
index ("unfair" component of inequality) in order to assess the contribution
to overall inequality of inequality within and between social groups of the
population.
When the purpose of the analysis is to make cross-country comparisons and
the social groups are ordered groups (e.g. income groups), the concentration
index (section 3.2.7) and regression-based indices (section 3.2.1) are the pref-
ered indicators. Given that these indices are only suitable for social groups
with an inherent ordering, however, simple measures of dispersion (section
3.1) could also be convenient because they are easy to compute and have
a straightforward interpretation. Moreover, since simple measures of dis-
persion can be used with both ordered (e.g. income groups) and unordered
groups (gender), a similar framework could be applied to examine dispari-
34
ties in tertiary education across various types of social groups. In addition,
these indicators are not data demanding (aggregated education indicators
broken down by equity groups) and could, in consequences, be used for large
cross-country and time comparisons. Note, however, that these scalar sum-
mary measures of the dispersion of the distribution of education across social
groups disregard a lot of important information about the distribution.
As done by Barros et al (2009) for analyzing the inequality of opportunity
for children in education, electricity, and improved water and sanitation, it
is possible to combine into a single composite indicator a metric of inequal-
ity with an indicator of coverage of tertiary education, should the desired
result of this work be building an index capturing both the supply of tertiary
education and the distribution of opportunities in tertiary education. To
move ahead to the empirical analysis and decide precisely which inequality
indicator to use, it is vital to know better which data are available and the
characteristics of the education outcomes.
References
[1] Alderman, H. and Berhman, J.R, Kahn, S., Ros, D.R. and R.Sabot
(1997): The Income Gap in Cognitive Skills in Rural Pakistan, Economic
Development and Cultural Change 46(1): 97-122.
[2] Barro, R.J.and J.L. Lee (1993): International Comparisons of Educa-
tional Attainment, Journal of Monetary Economics, 32(3), 363-394.
[3] Barro, R.J.and J.L. Lee (1997): International Measures of Schooling
Years and Schooling Quality, American Economic Review, Papers and
Proceedings, 86(2), 218-223.
[4] Barros, R. P. de, Chanduvi, J. S., Ferreira, F. H. G., and J. R. Molinas
Vega (2009). Measuring Inequality of Opportunities in Latin America
and the Caribbean. Washington DC: the World Bank.
35
[5] Barros, R.P., Molinas Vega, J.R. and J. Saavedra (2008). Measuring In-
equality of Opportunities for Children. Background Paper. Washington
DC: World Bank
[6] Birdsall, N. and J.L. Londono (1997): Asset Inequality Matterss: An
Assessment of the World Bank�s Approach to Poverty Reduction, Amer-
ican Economic Review 87(2), 32-37.
[7] Bishop, J. A., K. V. Chow, and J. P. Formby. (1994):.Testing for Mar-
ginal Changes in Income Distributions with Lorenz and Concentration
Curves, International Economic Review, 35(2): 479�88.
[8] Ferreira, F.H.G. and J. Gignoux (2008): Inequality of Economic Op-
portunity in Latin America. Background Paper. Washington DC: World
Bank.
[9] Ferreira, F.H.G. and J. Gignoux (2008): Towards an Understanding of
Socially-Inherited Inequalities in Educational Achievement: Evidence
from Latin America and the OECD. Background Paper. Washington
DC: World Bank.
[10] Filmer, D. and L.H. Pritchet (2001): Estimating Wealth E¤ects With-
out Expenditure Data�Or Tears: An Application to Educational Enroll-
ments in States of India, Demography, 38(1), 115-132.
[11] Firebaugh G (1999): Empirics of world income inequality,.American
Journal of Sociology,104(6):1597�630.
[12] Gallagher, M. (1991): Proportionality, Disproportionality and Electoral
Systems, Electoral Studies, 10, 33-51.
[13] Grant, M. J. and J.R. Behrman (2008) �Gender Gaps in Educational
Attainment in Less Developed Countries,�Philadelphia, PA: University
of Pennsylvania, memo.
[14] Grilli di Cortona, P., C. Manzi, A. Pennisi, F. Ricca and B. Simeone
(1999), �Evaluation and Optimization of Electoral Systems�, SIAM,
Philadelphia.
36
[15] Krokstad, S, Kunst, A. E. and S Westin (2002). Trends in health in-
equalities by educational level in a Norwegian total population study,
Journal of Epidemiology and Community Health, 56, 375-380.
[16] Loosemore, J. and V.J. Hanby (1971): The Theoretical Limits of Max-
imum Distortion: Some Analytic Expressions for Electoral Systems
British Journal of Political Science, 1, 467-77.
[17] Lorenz (1905): «Methods of Measuring the Concentration of Wealth»
Publications of the American Statistical Association
[18] Maas, J. and G. Criel (1982): Distribution of primary school enrollments
in eastern africa, Working Bank Sta¤ Working Paper 511.
[19] Mackenback, J.P. and A.E. Kunst (1997): Measuring the magnitude
of socio-economic inequalities in health: an overview of available mea-
sures illustrated with two examples from Europe, Social Science and.
Medicine, 44(6), 757-771.
[20] OECD (2008): Tertiary Education for the Knowledge Society, Volume
2 V, by Paulo Santiago, Karine Tremblay, Ester Basri,and Elena Arnal
[21] Pennisi, A. (1998): Disproportionality Indexes and Robustness of Pro-
portional Allocation Methods, Electoral Studies, 17(1), 3-19.
[22] Pontili, R.M. and Kassouf, A.L (2008): Is age-grade distorsion in Brazil�s
primary public education system more closely associated to school in-
frastructure or to family characteristics?, Well-Being and Social Policy,
4(1), 29-54
[23] Ram, R. (1990): Educational Expansion and Schooling Inequality: in-
ternational edidence and some implications, Review of Economics and
Statistics, 72(2), 266-74.
[24] Reardon SF and G. Firebaugh (2002). Measures of multigroup segrega-
tion. Sociol Methodology, 32, 33�67.
37
[25] Regidor, E. (2004): Measures of health inequalities: part 2, Journal of
Epidemiology and Community Health,58, 900�903.
[26] Sahn, D.E. and S.D. Younger (2007): Decomposing World Education
Inequality, SAGA Working Paper.
[27] Shorrocks A. F. (1980): The class of additively decomposable inequality
measures, Econometrica, 48, 613-625.
[28] SITEAL (2005): Medidas de Desigualdad para Variables Educativas.
Boletín No. 4. Santiago, Chile: SITEAL.
[29] Thomas, V., Wang, Y. and X. Fan (2001): Measuring Education In-
equality: Gini Coe¢ cients of Education, World Bank Policy Research
Working Paper 2525.
[30] Usher, A (2004): A New Measuring Stick. Is Access to Higher Education
in Canada More Equitable? Toronto, ON: Educational Policy Institute.
[31] Wagsta¤, A. and N. Watanabe (2003): What di¤erence does the choice
of SES make in health inequality measurement?, Health Economics,
12(10), 885-890
[32] Wagsta¤, A. (2002): Inequality Aversion, Health Inequalities and Health
Achievement, Journal of Health Economics, 21(4), 627-641.
[33] Wagsta¤, Adam, P. Paci and E. van Doorslaer (1991). On the Measure-
ment of Inequalities in Health, Social Science and Medicine, 33.
[34] Zhang, J. and T. Li (2002). International Inequality and Convergence in
Educational Attainment, 1960-1990, Review of Development Economics,
6, 383-392.
5 Appendix
38
METRIC SOCIAL GROUP (SG)
EDUCATIONAL VARIABLE (EV)
DATA P CROS ONS LEVEL OF ANALYSIS FINAL RECOMMENDATIONS1 INDICATOR TO BE USED FOR:
Range measure RM1 Ratio measure RM2
Ordered SG Unordered SG
Continuous EV Discrete EV
Data aggregated by social group
- Easy to compute and interpret - No restrictions on the characteristics of the grouping variable
- If more than 2 SGs, the intermediary groups are not taken into consideration - Not sensitive to the distribution of the population between SGs
- Appropriate for cross country comparisons - Not enough for country-specific studies
Cross-country comparisons / benchmarking
Regression based index
Ordered group
Continuous EV Discrete EV
Data aggregated by social group Individual data
- Summary index of educational disparities - Reflects the socioeconomic dimension of inequalities in education - Possibility to control for other confounding factors - Sensitive to the distribution of the population between SGs
- The relationship between EV and SG must be linear - Ordered SG
- Appropriate for cross-country comparisons and country-specific studies when the equity group is defined by an ordinal variable (ordered SG such as income groups) or by a binary discrete variable (sex)
Country-specific studies when - ordered SG - unordered SG defined by a binary variable
Population attributable risk
Ordered SG Unordered SG
Continuous EV Discrete EV
Data aggregated by social group
- Summary index of educational disparities easy to compute and interpret - No restrictions on the characteristics of the grouping variable - Sensitive to the distribution of the population between SGs
- Fails to reflects the socioeconomic dimension of inequalities in education: it does not tell us which groups are disadvantaged - Problematic is the best performing group is a very small group
- Appropriate for cross-country comparisons - Not enough for country-specific studies
Gini coefficient
Ordered SG Unordered SG
Continuous EV
Data aggregated by social group Individual data
- Summary index of education disparities and well known metric - Sensitive to the distribution of the population between SGs - Good statistical properties
- The EV must be continuous - Fails to reflects the socioeconomic dimension of inequalities in education: it does not tell us which social groups are disadvantaged - Gini computed on individual data: measure of pure inequality between individuals but not of inequality between social groups
- Not appropriate for cross-country comparisons - Appropriate for country-specific studies only when the object of analysis is to measure the overall inequality among individuals
1 See the conclusion for additional information
Entrop indices Atkinson index
Ordered SG Unordered SG
Continuous EV
Data aggregated by social group Individual data
- The entropy indices are decomposable into within and between components: property very interesting if we can work with individual data - Atkinson index: possibility to take into account the level of aversion for inequality - Sensitive to the distribution of the population between SGs - Good statistical properties
- Fails to reflects the socioeconomic dimension of inequalities in education: it does not tell us which social groups are disadvantaged - The EV must be continuous
- Not appropriate for cross-country comparisons - Appropriate for country-specific studies
Country-specific studies when - unordered SG or - SG defined by several circumstances
Education Standard Deviation (ESD) between groups Coefficient of Variation of Schooling (CSV) between groups
Ordered SG Unordered SG
Continuous EV Discrete EV
Data aggregated by social group
- Both indicators are very easy to compute. - No restrictions on the characteristics of the grouping variable - Sensitive to the proportion of the population in each group
- Not sensitive to the direction of the social gradient in education - Compare each data point to the mean and not all data points with each other
- Appropriate for cross-country comparisons - Not enough for country-specific studies
Concentration index
Ordered SG
Continuous EV
Data aggregated by social group Individual data
- Summary index of education disparities - Reflects the socioeconomic dimension of inequalities in education - Possibility to adjust the CI for the effect of other confounding factors: particularly interesting if one works with individual data - Good statistical properties - Sensitive to the proportion of the population in each group
- The equity group must be defined on an interval scale.
- Appropriate for cross-country comparisons and country-specific studies when the SG is an ordered SG (income)
Country-specific studies when ordered SG
Dissimilarity index
Ordered SG
Continuous EV
Data
- Summary index of
- Fails to reflects the
- Appropriate for cross-
Country-specific studies
Unordered SG
Discrete EV aggregated by social group
educational disparities easy to compute and interpret - No restrictions on the characteristics of the grouping variable
socioeconomic dimension of inequalities in education: it does not tell us which social groups are disadvantaged
country comparisons - Not enough for country-specific studies
when - unordered SG or - SG defined by several circumstances
