There is reason to suspect that with historical Census data, such as those employed in this study, error would not be a constant element of each observation, but rather would increase in proportion to the size of the total count. For example, counts of church membership in larger cities probably contain more spurious entries than statistics of religious affiliation for smaller places. Any ordinary least squares (OLS) model, then, will lose its predictive power as observations on the dependent variable increase in size.

As a consequence, variance around the fit to these data of an OLS model will be correlated with predictors in the model that are also functions of scale. This problem is especially acute in equations where, to avoid resort to ratio measures, population is entered as an explicit control for the effects of scale on other variables (Bollen and Ward, 1979: 437, 441-443; Kasarda and Nolan, 1979: 222–223, 225n; Pendleton et al., 1979: 470). Residual variance around predictions of the dependent variable will increase as the size of observations increases. The presence of this relation, however, violates one of the fundamental assumptions under which any OLS regression analysis is conducted, namely that individual disturbance terms are, as a group, homoscedastic (i.e., they exhibit a constant variance across the range of values of the dependent variable).

In short, larger values are troublesome under such conditions in that they yield greater residuals from regression; they are, by implication, empirically less trustworthy and practically less useful in estimating true regression slopes than are the smaller observations.