with error, the variance of the achievement distribution aggregated from Maximum Likelihood estimates of individual knowledge overestimates the true variance. An alternate method, outlined by Mislevy, Beaton, Kaplan & Sheehan (1992), draws from the posterior of every student’s achievement distribution to obtain an unbiased measure of the full learning distribution. These draws—known as “plausible values”—are interpreted as individual achievement with the property that when aggregated to a population distribution they recover the correct population moments. We show that the variance of the distribution is sensitive to the estimation method used (i.e. Maximum Likelihood, Bayesian, or Plausible Values), primarily because the TIMSS test is too difficult for a large fraction of Indian children.<sup>1</sup> The method of plausible values offers an alternative for the calculation of higher moments in any setting—such as poverty mapping—where individuals attributes are estimated with a known standard error.

Linking scores to an international distribution contributes to the literature on education in low-income countries in several ways. First, linked test scores are comparable across space and time. Despite increasing worldwide testing using standardized methods—e.g. TIMSS (51 countries), PIRLS (35 countries), IALS (22 countries) and PISA (49 countries)—the Indian government, like many others, is reluctant to participate in such large-scale testing exercises. As a result, what little is known about learning achievement in India, and most low-income countries, arises from an ad-hoc collection of criterion-referenced exams.<sup>2</sup> These tests, administered by independent agencies, are typically not validated using standard testing tools, cannot be equated over time or across countries, and are not subject to a battery of robustness checks that accompany large-scale testing in the OECD countries. The methods applied here allow independent researchers to report achievement distributions for the tests they control that are directly

> 1Brown & Micklewright (2004) also highlights the importance of using a consistent methodology. They find, for instance, that rankings of countries by within-country difference in TIMSS changed substantially for some countries when the scoring model used in 1999 was retrospectively applied to 1995 data.

> 2Examples for India include a large national study by the National Center for Educational Research and Training (NCERT) in 1994, which found that children scored an average of 47 percent in language and 41 percent in mathematics (Shukla et al. 1994), and state-wide studies with smaller samples in Bihar, Tamil Nadu, Delhi, Uttar Pradesh, Madhya Pradesh and Rajasthan(Bashir 1994, Hasan 1995, Govinda & Varghese 1993, Aggarwal 2000, Goyal 2007). In a major recent effort, the NGO Pratham tested children from almost all districts and found low levels of learning: 52 percent of children between the ages of 7 and 10 could read a small paragraph with short sentences at first grade difficulty levels, 32 percent could read a story text and 54 per cent were unable to divide or subtract (Pratham 2006). Similar results have been reported for Africa. In a relatively large effort, the Monitoring Learning Achievement Project (Chinapah et al. 2000, Strauss & Burger 2000)covered 13 African countries and found literacy, numeracy, and life-skills scores for fourth graders between 30 and 70 percent.

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