This paper discusses distribution theory and power computations for four common "tests of combined significance." These tests are calculated using one-sided sample probabilities or p values from independent studies (or hypothesis tests), and provide an overall significance level for the series of results. Noncentral asymptotic sampling distributions are derived for four combined significance tests (the method of "adding z's," Fisher's method of "adding logs," George's logit summary, and Edgington's mean-p method), and a simulation study is used to examine the accuracy of the distributions in finite samples. The paper also presents a noncentral asymptotic sampling distribution for z(p), the standard normal deviate associated with the one-sided p value, which is used to derive the distributions of statistics used in tests of combined significance. A data set from research on teaching is used to illustrate estimation of power of the combined significance tests to detect several plausible patterns of study outcomes. Recommendations are made for the use of the tests of combined significance. Appendix A presents notation for the results of studies using two-sample comparisons, and outlines how the distributions would be applied to such results. Appendix B provides a proof of the convergence theorem. (Author/JAZ)