The standard error of a weighted mean concentration—I. Bootstrapping vs other methods

Concentrations of chemical constituents of precipitation are frequently expressed in terms of the precipitation-weighted mean, which has several desirable properties. Unfortunately, the weighted mean has no analytical analog of the standard error of the arithmetic mean for use in characterizing its statistical uncertainty. Several approximate expressions have been used previously in the literature, but there is no consensus as to which is best. This paper compares three methods from the literature with a standard based on bootstrapping. Comparative calculations were carried out for nine major ions measured at 222 sampling sites in the National Atmospheric Deposition/National Trends Network (NADP/NTN). The ratio variance approximation of Cochran (1977) gave results that were not statistically different from those of bootstrapping, and is suggested as the method of choice for routine computing of the standard error of the weighted mean. The bootstrap method has advantages of its own, including the fact that it is nonparametric, but requires additional effort and computation time.