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schools. These coefficient estimates are divided by the funding level at each school to create the efficiency scores that are then compared between school types. Mathematically, consider the comparison of the efficiency of independent charter schools and regular public schools: (2) where β 0 and β 1 represent coefficients derived from equation 1, κ1 and κ0 represent the constant per-pupil expenditure in each school type, and σ2 represents the combined variance derived through the equation: (3) Where Ϫ represents the covariance of β 0 and β 1 , the estimated parameter for the indicator of school type. If there is no difference in the efficiency of independent charters and regular public schools, the expected value of Z is 0, and the variance of Z will equal σ 2 . If any of the three types of charter schools are more efficient than traditional public schools, we expect to observe a positive, significant difference in the t-test. A negative difference would indicate that the school type is less efficient, while an insignificant difference would indicate that performance in that school type is approximately equally efficient as traditional public schools. b. Summary Statistics Table 1 below presents the summary statistics for the data in our analysis. 17 In both science and math, one can observe a high degree of variability in the school-grade test average. In science, the average ranges from 1.33 to 3.57 while in math, the range is from 1.0 to 2.92. It is notable that the average scores on the Badger Math exam tend to be lower than those for the WKCE science, though we cannot determine here whether this is due to <strong>differences</strong> in the difficulty of the exam or the difficulty of the subject matter for students. One can also note the high levels of non-white students on average (more than 80 percent) in the schools under study, as well as the high level of students (also more than 80 percent) receiving free and reduced lunch. Bang for the Buck 9
Table I. Summary Statistics of the Key Variables Variable Mean (SD) Minimum Maximum Test Average-Science 2.3556 (0.4451) 1.3333 3.5714 Test Average-Math 1.6767 (0.4471) 1.0000 2.9286 Percent white 0.1642 (0.2106) 0.0000 0.9500 Free/Reduced Lunch 0.8139 (0.1939) 0.0000 1.0000 English as a Second Language 0.0707 (0.1314) 0.0000 0.6800 Traditional Public School 0.7638 (0.4256) 0.0000 1.0000 Instrumentality Charter 0.0276 (0.1640) 0.0000 1.0000 Non-instrumentality Charter 0.0787 (0.2699) 0.0000 1.0000 Independent Charter 0.1300 (0.3369) 0.0000 1.0000 Note: Table 1 depicts the mean, standard deviation and range of each key variable in our analysis. c. Analysis #1: School-Level Table 2 presents the top fifteen schools in efficiency in Milwaukee on both the WKCE and Badger Exam. The efficiency measure is averaged across grades 18 at the school level, while the efficiency measure in Analysis #2 is aggregated over all schools of each type. Note also that 10th graders did not take the Badger exam, and thus high schools do not appear in the Badger top 10 list. As a robustness check, we can see a great deal of similarity between these lists. Downtown Montessori, for example, ranks first on both lists. Table 2. Most Efficient Schools 1. DOWNTOWN MONTESSORI (Indp) 1. DOWNTOWN MONTESSORI (Indp) 2. WOODLANDS (Indp) 2. WOODLANDS (Indp) 3. MIL COLLEGE PREP-38TH ST. (Indp) 3. ROCKETSHIP S COMM PREP (Indp) 4. HIGHLAND COMMUNITY (Non-instr) 4. MIL GERMAN IMMERSION (Trad) 5. CARMEN HS (Non-Instr) 5. SEEDS OF HEALTH (Indp) 6. MARYLAND MONTESSORRI (Trad) 6. BRUCE GUADALUPE (Indp) 7. BRUCE GUADALUPE (Indp) (Indp) 7. MARYLAND MONTESSORRI (Trad) 8. VERITAS HS (Indp) 8. GOLDA MEIR (Trad) 9. MIL GERMAN IMMERSION (Trad) 9. WHITMAN ELEMENTARY (Trad) 10. 21ST CENTURY PREP (Indp) 10. MIL SPANISH IMMERSION (Trad) 11. GOLDA MEIR (Trad) 11. CENTRAL CITY CYBERSCHOOL (Indp) 12. SEEDS OF HEALTH (Indp) 12. HAPA-AMERICAN PEACE (Non-instr) 13. HAPA-AMERICAN PEACE (Non-Instr) 13. WHITTIER ELEMENTARY (Instr) 14. HONEY CREEK ELEMENTARY (Instr) 14. MIL COLLEGE PREP-38TH ST. (Indp) 15. WHITTIER ELEMENTARY (Instr) 15. HONEY CREEK ELEMENTARY (Instr) Bang for the Buck 10
Page 2 and 3: EXECUTIVE SUMMARY Background: Not a
Page 4 and 5: 2. The most inefficient schools wit
Page 6 and 7: I. Introduction Although the public
Page 8 and 9: All charter schools in Milwaukee ar
Page 12 and 13: These results are supportive of our
Page 14 and 15: Table 4. Effect of School Type and
Page 16 and 17: IV. Limitations A few limitations o
Page 18 and 19: Endnotes 1. New legislation has exp
Page 20 and 21: TENOR HS (Indp) 0.287191 IDEAL (Ins
Page 22 and 23: BUSINESS AND ECON ACAD OF MIL. (Non
Page 24 and 25: WHS INFORMATION TECHNOLOGY 0.167557
Page 26 and 27: VIEAU ELEMENTARY 0.24841 MIL. ACADE
Page 28: METCALF ELEMENTARY 0.192206 TOWSEND

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