Evaluating Quantitative Indices of Ethnic Composition in School Districts

• In Event: Excellence in Education Research: Early Career Scholars and Their Work

Abstract

The ethnic composition of school districts in states like California is becoming increasingly diverse. For example, the ethnic composition of the Los Angeles Unified School District is 73.4% Hispanic, 10% African American, 8.8% White, and 3.9% Asian (Ed-Data, 2011). In many school districts like this one, White Non-Hispanic students are in fact the minority. As such, systematic measures of ethnic composition are necessary to compare the increasingly diverse profiles of students across districts. This research introduces an existing measure of ethnic diversity known as Simpson’s index of diversity, and suggests two new applications of multidimensional scaling to this field of research.
Comparing ethnic composition profiles across districts is difficult when there are multiple ethnic groups. To summarize the ethnic diversity of school districts as a single number, the Simpson’s index of diversity D_i can be calculated (Simpson, 1949) for every district. The values of this index can range from 0 to 1, with values closer to 1 indicating a greater spread of proportions across ethnic groups. This is known as the Simpson’s diversity index, and although it has traditionally found application in ecological studies, researchers in education are beginning to study the extent of ethnic diversity across schools (Lee, 2007; Graham, Bellmore, Nishina, & Juvonen, 2009). Multidimensional scaling (MDS) is contrasted with one-dimensional methods such as Simpson’s index of diversity in that functions of ethnic composition can be projected visually onto a two dimensional space by first generating what is known as a dissimilarity metric (not unlike Simpson’s index), whose dimensions can be further reduced using principal components (Gower, 1966).
Two methods of MDS, using student counts and student proportions as data points are explored. MDS with proportions as data points combines the advantages of proportional weighting used in Simpson’s index with the benefits of visualizing a two-dimensional plot of relative similarities. Essentially each element in this new dissimilarity matrix is the squared difference in proportions summed across m ethnic groups for a corresponding pair of school districts. For example, to create the (2,1) element in the S matrix of the pair of districts Los Angeles Unified and San Gabriel, subtract the difference in proportion between the two districts for Hispanics, Asians and African Americans, and then sum all the squared distances together. Much like the procedure for Euclidean distances described above, repeat the process for all n^2 pairs, noting that diagonals should be 0 and the upper triangle should be symmetric with the lower one.

Author

• Johnny Lin, Educational Testing Service