_Progress in International Reading Literacy Study_ (PIRLS), which is applied to younger students, shares a number of common features.<sup>2</sup> As anyone who has been to school may recall, performance in a test, while probably preferable to a simple indicator of enrollment or attendance, is not a perfect measure of learning either. For one thing, tests and test items (i.e. questions) vary in difficulty. The final result is known to measure scholastic ability or learning achievement only imperfectly. For this reason, all of the aforementioned surveys present scores constructed from the raw results by means of Item Response Theory (IRT) models, which attempt to account for “test parameters”, so as to better infer true learning. This process generates an arbitrary metric for test scores, which are then typically standardized to some arbitrary mean and standard deviation.

Using these standardized test scores, a number of studies have attempted to provide international comparisons of educational inequality on the basis of _achievement_ , rather than _attainment_ . Micklewright and Schnepf (2007) and Brown et al. (2007) examine the robustness of measures of central tendency and dispersion in the distribution of student achievement obtained using different surveys, by comparing the measures and country rankings across them. They find broad agreement across surveys, but also some evidence that the specific statistical models used to estimate IRT adjustments do affect results, in particular for less developed countries. Marks (2005), Schultz, Ursprung and Wossmann (2008), and Macdonald et al. (2010) examine the question of intergenerational persistence in educational achievement, which is closely related to that of inequality of opportunity, and present cross-country comparisons of measures of the association between student achievement and certain family characteristics.

This paper seeks to contribute to that literature by proposing two simple and closely-related measures of inequality - one for educational achievement and another for opportunity to education – and reporting them for all countries that participated in the 2006 wave of PISA surveys. To measure inequality in achievement, we propose simply using the variance or the standard deviation of test scores. But we arrive at this simple proposal by considering the implications of two issues specific to the distribution of test scores for the measurement of inequality. These two issues are: (i) the fact that many common inequality indices are not ordinally invariant in the standardization to which IRT-adjusted test scores are generally subjected; and (ii) the fact that PISA student samples are likely to suffer from non-trivial selection biases in a number of countries. The choice of the variance (or the standard deviation) addresses the first issue. We also propose two alternative two-sample non-parametric procedures to assess the robustness of the inequality measure to the sample selection biases, and implement them in the four countries for which PISA sample coverage (as a share of the total population of 15 year-olds) is smallest.

The proposed measure of inequality of educational opportunity draws on the recent literature on inequality of opportunity in the income space, but is also adapted to the specificities of

> 2 There is also an International Adult Literacy Survey (IALS), which is applied to adults long after they have left school.

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