Graphic 3 summarizes the regression results for all four models (see Appendix A for the robust standard error values, confidence intervals, and other details of the regression results).

While the coefficients on the size terms are statistically insignificant in the linear and quadratic models, they are significant at the 1 percent level in the logarithmic and checkmark models. The checkmark model is superior overall, however, explaining an additional two percentage points of the variance in district size (its R-squared value is roughly 79 compared to 77 for the logarithmic model).

Detailed regression diagnostics are presented in Appendix B, but it is sufficient here to observe that the reason why both the logarithmic and the checkmark models are statistically significant is that both accurately fit the decline in per pupil spending that occurs as enrollment rises from a handful of students to a few thousand students. The reason why the logarithmic model is somewhat inferior is that it incorrectly predicts that spending will continue to fall indefinitely as district size increases. The checkmark model, by contrast, correctly predicts that, after a certain size, further increases in district size are associated with higher spending per pupil.