heterogeneity in education systems or pupil populations. They are only meant to illustrate the potential use of indicators of inequality of opportunity for future studies of the distributive impacts of education policies. Future extensions – notably involving the use of panel data - might allow for causal analysis of these relationships.

# **6. Conclusions**

Internationally comparable information on learning outcomes, such as the standardized test scores collected by PISA surveys, represents a revolution in the quality of data available for research on education. It allows for potentially much greater insight into the determinants of educational achievement, and might therefore contribute to the design of policies that raise average learning levels, or that reduce educational disparities.

The measurement of educational disparities using this kind of data is not, however, a trivial extension of inequality measurement in years of schooling, or in other variables like income. This paper has highlighted two issues that require special attention in the measurement of inequality in educational achievement, and which appear to have been overlooked so far. The first is the standardization of test scores, to which all meaningful measures of inequality are cardinally sensitive. More importantly, many common measures of inequality, including the Gini coefficient and the Theil indices, are not event ordinally invariant to standardization, invalidating country rankings that are based on them.

We show that the simple variance (or the standard deviation) of test scores is ordinally invariant to standardization, and present estimates for all 57 countries that took part in the 2006 round of PISA surveys, in all three subjects for which tests are carried out: reading, mathematics and science. There is considerable international variation in educational inequality thus measured. The standard deviation in Math scores ranges from around 80 in Indonesia, Estonia and Finland, to nearly 110 in Belgium and Israel.

The second measurement issue that may compromise international inequality comparisons based on PISA test scores is the possibility of sample selection. The surveys are designed to be representative of the population of 15 year-olds enrolled in school, and attending grades 7 or above. While this stipulation covers most of the population of that age group in OECD countries, it purposively excludes substantial numbers in poorer countries. Selection into the sample is clearly correlated with determinants of test scores, leading to a classic problem of sample selection bias. Using information on characteristics of fifteen year-olds included in other, ancillary household surveys, we use sample re-weighting methods to assess the implications of the selection bias for our measures of educational inequality in achievement and opportunity. Results for Brazil, Indonesia, Mexico and Turkey suggest that the inequality measures are relatively robust to selection on the basis of three observed variables (gender, mother’s diverting some individuals from progress to tertiary education, it seems to reduce the impact of family background on adult literacy and promote further on-the-job training by offering more effective curricula to less well performing students.

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