There are a number of federal laws and regulations, such as the Individuals with Disabilities Education Act and the Title I funding program for low-income children, that require or encourage districts to provide additional services to certain students. Since this federal funding is attached to categories of children and is, at least in principle, entirely independent of school district size, it is necessary to control for the impact of federal categorical spending on district expenditures. Specifically, this study controlled for the total federal revenues per pupil received by each district. This control also takes into account school spending on free and reduced-price lunches for low-income students, because these lunch programs also receive federal funding.

It might be argued, however, that since some federal funding programs are elective rather than mandated (for example, gifted and talented programs), and since larger districts have more administrative personnel available to apply for federal grants, larger districts may bring in more federal money at least partially due to their size. If that were the case, then including total federal revenues as a control variable in the regression would understate the true significance and magnitude of the district-size term, since some of the size term’s effect would be subsumed by the federal-revenue term.

While this hypothesis is certainly plausible, it does not appear to be a cause for concern in practice. The standard correlation coefficient (“Pearson’s r”) between district size and total federal revenues per pupil is quite low (0.075),[*] as is the Spearman’s Rank correlation coefficient (-0.175), which suggests that a district’s size and a district’s ability to raise per-pupil federal grant funding are not strongly correlated.

[*] A Pearson’s correlation coefficient of 0 indicates the absence of a linear relationship between the two variables and a correlation coefficient of 1 (or -1) indicates perfect positive (or negative) linear correlation (i.e., one variable can be expressed as a linear function of the other).