If inequality rises among each sub-group of the population, overall inequality should also rise (and vice versa). This means that if the educational attainment gap between groups remains stable, but inequality rises within the groups, overall inequality should also rise.

The most commonly used measures of inequality are: variance, decile ratios, relative mean deviation, coefficient of variation, the Gini coefficient, and the Theil index. Other possibilities for consideration are the standard deviation of logarithms and the mean log deviation; however, I have not included them as these measures are only defined for strictly positive values, and cannot be calculated from my data due to cases in which some proportion of the population has 0 education. First, I will give the explanations and calculations for these indices. Next, I will examine how well they conform to the axioms laid out above.

#### Relative Mean Deviation

$$ RMD=\frac{\frac{1}{N}\sum \limits_{i=1}^N\left|{x}_1-\overline{x}\right|}{\left|\overline{x}\right|} $$

This measure takes the average absolute deviation and divides it by the mean. Dividing by the mean allows for comparison across distributions with different means or different ranges. For example, primary and secondary school last only 10 years in Nepal, but 13 years in Sri Lanka. In order to compare inequality across the two systems, it is necessary to standardize the measure to account for the bigger range of educational attainment in Sri Lanka.

In evaluating these measures, a few immediately fail to achieve the desired qualities. Variance is dependent on scale (fails the test of scale independence), meaning that if educational attainment were to increase by 50 percent for everyone in society, the variance would automatically increase as well. This makes variance an inappropriate choice for comparing inequality among distributions with different ranges.

Decile ratios only capture what is going on at the specific cut-off levels chosen, but fail to take into account transfers in the rest of the distribution. For example, in a 90/10 decile ratio, even if the 20th percentile had to give some of their assets to those in the 80th percentile, the decile ratio could remain unchanged as long as the 90th and 10th percentile groups were not changed. This measure therefore fails the Pigou-Dalton Transfer Principle.

Similarly, relative mean deviation is not sensitive to transfers that are made on the same side of the mean (Atkinson 1970). Transfers could be made from the very worst off to those in the 55th percentile with no recorded change in inequality, thus again failing the Pigou-Dalton Transfer Principle.

The Gini coefficient satisfies Litchfield’s axioms 1–4 and axiom 5 as long as the population sub-groups do not overlap (Litchfield 1999). This means that as long as the worst-off people in one group are all still better than the best-off people in another group, the Gini coefficient will still pass the test of decomposability. Where this might be a problem would be if data were grouped according to criteria other than the distribution of the good itself, such as ethnicity or gender. The way my dataset is built, based on the attainment of different levels of education, this is not a problem.

The coefficient of variation and the Theil index also satisfy all of the principles named above. These two measures are both special cases in the family of Generalized Entropy (GE) index measures, based on information theory. The general formula for GE measures is:

$$ GE\left(\alpha \right)=\frac{1}{\alpha^2-\alpha}\left[\frac{1}{N}\sum \limits_{t=1}^N{\left(\frac{y_i}{\overline{y}}\right)}^{\alpha }-1\right] $$

GE(0) is the mean log deviation, GE(1) is the Theil index and GE(2) is 1/2 the squared coefficient of variation (Litchfield 1999). This class of indicators, unlike the Gini coefficient, has the advantage of being decomposable/additively separable—that is, total inequality can be broken down into the sum of the inequality of different sub-groups (De Maio 2007). Ways of decomposing the Gini coefficient have been found (Araar 2006), but it is a considerably more complex process.

Some critics have argued that the Gini coefficient is not an appropriate measure of inequality for a few reasons. Atkinson argues that all inequality measures include inherent value judgments, and that the best way to approach this is to explicitly specify a social welfare function that includes an inequality aversion parameter, ε. The higher the value of ε, the more sensitive the index becomes to inequalities at the bottom of the distribution. He argues that this approach is more transparent than using an index that purports to be value-free, but is in fact based on particular assumptions about what constitutes more or less inequality (1970). The GE measures, like Atkinson index measures, include a parameter to weight inequalities in different parts of the distribution. For the Gini coefficient, however, not only is such a parameter not part of the formula, but Newberry has shown that the Gini coefficient does not rank inequality in a manner consistent with any additive utility function (Newbery 1970, p. 264). In other words, Atkinson and Newberry argue that if we assume that equality is valued and contributes to social welfare, then we should rank distributions according to a utility function whereby greater equality gives greater social welfare. Additivity simply means that the whole is equal to the sum of the parts.

Nevertheless, Sen (1976) and others defend the use of the Gini coefficient as a good measure of inequality . Furthermore, although the Gini coefficient on its own cannot be written as a utility function as described by Rothschild and Stiglitz (1973) show that the Gini coefficient can still be “part of an equality preferring social welfare function” (p. 199). For this reason and due to its place as the most widely used and widely recognized measure of inequality, I will retain the use of the Gini coefficient, while supplementing it with other measures, such as the coefficient of variation and the Theil index. By employing all three, I can test the robustness of my findings and get a more nuanced picture than could be obtained by sticking with a single measure of inequality (De Maio 2007).

Table

4.2 illustrates the correlation between the Gini index, the Theil index, the coefficient of variation, and the educational attainment of various deciles. The Gini index and the Theil index are highly correlated, at 0.92, but the Gini and the coefficient of variation are only correlated at 0.56. The Theil index is very closely correlated with the coefficient of variation: 0.98.

Table 4.2 Correlation of measures of education inequality

As mentioned before, these measures have different sensitivities to inequality in different parts of the distribution. The Gini coefficient is more sensitive to transfers in the middle of the distribution, the Theil index is sensitive to transfers at either the lower end or the upper end, and the coefficient of variation is equally sensitive to transfers anywhere in the distribution (Atkinson

1970). When the Lorenz curves are far apart, this does not matter much, and all three measures will agree on inequality rankings. When the Lorenz curves intersect, however, different inequality measures may give different rankings. In practice, the three indicators agree on rankings in the vast majority of cases. Nevertheless, as a robustness check, I will run different models to include each one of these measures as the dependent variable. See Table

4.3 for Education Gini coefficients by country and year.

Table 4.3 Education Gini coefficient by country and year