Moving beyond the mean: Identification, aggregation, properties, and axioms Choices of measurement can significantly shape understanding the size of a problem and its variation over time and space. For example, _GDP per capita_ can be used to assess a population’s prosperity, how it compares with other populations, and how it progresses through time. The _Growth Report_ used _mean income_ to identify countries with high and sustained growth.<sup>3</sup> The Sarkozy Commission, which had Joseph Stiglitz as the president, Amartya Sen as the Advisor, and J. P. Fitoussi as the coordinator, c <mark>h ose the</mark> _median income_ since it corresponds more naturally to the middle of the income distribution.<sup>4</sup> The World Bank, in the context of its twin goals to end extreme poverty and foster income growth in _the bottom 40 percent_ of the population by 2030, focused on that group.<sup>5</sup> So, the arithmetic mean is not the only possible standard for measuring a distribution’s size, and examining several standards at once can clarify the quality of growth—including whether it is shared or not.

Similarly, policy makers and researchers commonly compare educational systems over time or across countries in terms of their _average scores_ on large-scale learning assessments. An average performance can be viewed as how much learning each student could expect to achieve if all students learned the same amount. The limitation of such an approach is evident: it is hard to imagine a situation in which every child in a population achieves the same performance, given the heterogeneity of schools, teachers, peers in a classroom, family backgrounds, and intrinsic motivations. To have a richer diagnostic and a better understanding of differences between systems or of changes in performance, it is critical to move beyond the mean.

This discussion is not new. Much of the literature on measuring poverty and inequality can be seen as a systematic effort to move beyond measures of the distribution’s size (that is, the mean) and look at its spread (inequality) and base (poverty), understanding that size, spread, and base are fully interrelated, jointly determined, and complementary.<sup>6</sup> In October 2019, the World Bank and UNESCO Institute for Statistics (UIS) launched a new synthetic indicator, _learning poverty,_ based on the concept that every child should be in school and be able to read an age-appropriate text by age 10.<sup>7</sup> This formulation reflects the aspiration of Sustainable Development Goal (SDG) 4 that all children must be in school and learning.<sup>8</sup> This paper will introduce and justify other aggregation approaches complementary to the existing learning poverty indicator. This new class of measures is shown to be relevant to understand the simulation results. The new aggregations can also help describe a country’s performance over time and improve comparisons between countries.<sup>9</sup> Measuring poverty has been conceptualized as following two main steps since Sen (1976): the identification of who the poor are and the aggregation of information about poverty across a society. The first problem requires defining a standard, or poverty line, and identifying those who fall below that line and those who do not. The second problem requires a method of combining different individuals’

> 3 Commission on Growth and Development 2008.

> 4 Commission on the Measurement of Economic and Social Progress 2009.

> 5 World Bank 2015.

> 6 For a more detailed discussion, see Foster et al. 2013.

> 7 World Bank 2019.

> 8 SDG 4 makes this commitment: by 2030, signatories will “ensure inclusive and equitable quality education and promote lifelong learning opportunities for all.” The goal’s target 4.1 is to “ensure that all girls and boys complete free, equitable, and quality primary and secondary education leading to relevant and effective learning outcomes.”

> 9 Other authors have also applied the welfare measurement concept to education. Denny (2002) used the traditional Foster, Greer, and Thorbecke (FGT) class of poverty measures to compare adult literacy across 12 countries using the International Adult Literacy Survey, and Thomas, Wang, and Fan (2000) estimated education inequality using the Gini index.

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