A possible alternative would be to use an absolute measure of inequality – such as the variance, or the absolute Gini coefficient<sup>9</sup> - which are ordinally invariant in the standardization. The variance of a post-standardized distribution ( ), for example, is a monotonic (linear) function of the pre-standardization variance ( ), and does not depend on any other moment of the prestandardization distribution: The variance is seldom used as an inequality measure because it is scale-dependent: it increases with the mean. It also fails the transfer sensitivity axiom, by placing greater weight on transfers higher up the distribution than to those lower down. While these are not trivial concerns, it appears to us that in the context of distributions of educational achievement, they are less severe than violating either the transfer principle itself (like the percentile based measures) or ordinal invariance in the standardization, which allows an apparently innocuous operation to fundamentally alter distributional rankings. The variance (and the standard deviation, of course) is a meaningful measure of inequality in the precise sense that it satisfies axioms (i)-(iii) above. The variance is also additively decomposable, and shares of the variance obtained from some such decompositions can be shown to be cardinally invariant to standardization, as discussed in the next section. These properties will prove instrumental in adapting an intuitive measure of inequality of opportunity to the context of education.

For these reasons, we adopt the variance and the standard deviation as our basic measures of inequality of educational achievement. Because users of this kind of data are generally more comfortable with the standard deviation than its square, this is the variable we report. Columns 3-11 in Table 1 present the mean and standard deviation (S.D.) of the standardized test scores in reading, math and science, in that order, for all 57 countries in the 2006 PISA surveys. The column immediately to the right of each S.D. column reports its bootstrapped standard error. Among the countries with higher inequality in math scores are Western European countries such as Austria, Belgium, France, Germany, and Italy; East European ones such as Czech Republic and Bulgaria, Latin American countries such as Argentina and Uruguay, but also Israel and Taiwan, China. Among the ones with lower inequality in achievements are other European countries such as Croatia, Denmark, Estonia, Finland, Ireland, and Latvia, but also Asian countries such as Indonesia, Thailand and Jordan. Countries such as the UK, Japan, and the United States take intermediate rankings.<sup>10</sup> Figure 1 portrays the S.D. (and its confidence interval) in the mathematics test scores for all countries in the sample.

> 9 The absolute Gini coefficient, of course, is the standard (relative) Gini index scaled up by the mean.

> 10 The inequality measures obtained for Azerbaijan seem particularly small and place the country as an outlier in all the analyses. It is unclear how much of this is due to the data collection procedures in this country, but such a different pattern is not likely due to real differences only.

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