differences autonomy included “efficiency” outcomes better generate

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 differences 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