School location information was not collected in France; Hong Kong SAR, China; and Liechtenstein.

A final data issue worth highlighting is that of sample coverage and representativeness. PISA samples were designed to be representative of the population of 15 year-olds who are enrolled in grade 7 or higher in any educational institution. The samples are _not,_ therefore, representative of the total population of 15 year-olds in each country: children who dropped out of school before they turned fifteen, as well as those who are so delayed that they are in grade 6 or lower at age fifteen, are purposively excluded. In addition, sampling flaws induce an additional under-coverage of enrolled 15 year olds. PISA documentation suggests that this arises from the fact that their sampling frame (a listing of schools and sampling weights) is established in the year preceding the surveys, on the basis of current school enrollment on that year. But some schools close down between the two years, and new ones are not included in the sample. Changes in the enrollment of 15 year-olds arising from this process are not taken into account.

The PISA sample coverage rate, defined as the ratio of the covered student population (using PISA expansion factors) to the total population of 15 year-olds, varies considerably across countries, and is reported in column 2 of Table 1. Although coverage is typically high in OECD countries, it is low in many developing ones: coverage rates are as low as 47% for Turkey, 53% for Indonesia, 54% for Mexico, and 55% for Brazil. Overall, coverage is less than 80% of the total population of 15 years-olds in fifteen countries. Table 2 provides a sense of the sources of exclusion for the four countries in our dataset with the lowest coverage rates, by decomposing those selected out of the sample into children no longer in school, children with excessive delays, and those missed due to PISA sampling issues. It should be obvious from these magnitudes that any international comparison of countries with vastly different coverage rates must seek to address the problem in some way, and we suggest two alternatives in Section 3.

# _Ancillary household survey data sets_

Our proposed procedure to examine the sensitivity of inequality measures to sample selection, which is described below, relies on using information on fifteen year-olds from general-purpose household surveys. While these surveys may have their own sampling issues, these are not dictated by school enrollment or delay status, or by school closures, openings and reforms. We obtained such household surveys for the four countries with the lowest coverage rates in the 2006 PISA sample: those reported in Table 2. For Brazil, we used the Pesquisa Nacional por Amostra de Domicílios (PNAD) 2006. For Indonesia, we used the SUSENAS 2005. For Mexico, the Encuesta Nacional de Ingresos y Gastos de los Hogares (ENIGH) for 2006 was used. For Turkey, the Household Budget Survey (HBS) 2006 was used.

All four are large-sample household surveys with national coverage and representative down to the regional level, which are fielded on an annual basis by each country’s national statistical authority. The PNAD 2006 collected information from a sample of about 119,000 households and 410,000 individuals; SUSENAS 2005 from 257,900 households and 1,052,100 individuals; the 7