The random variable p and its functions figure in several "tests of combined significance," meta-analysis summaries based on sample significance values, and ps have been used singly, as well as in other tests for evaluating the outcomes of individual research studies. In this work, asymptotic distributions of the sample one-sided significance value, p, and several functions of p were examined. Asymptotic normal distributions of p, of the standard normal deviate associated with the one-sided p value, z(p), and of the natural logarithm of the significance value, log(p), were studied. Cumulative normal distribution functions of the transformed variables were calculated and compared to the computed exact distributions of ps associated with one-sample t tests. Distributions based on the z(p) transformation appeared quite accurate even for very small sample sizes, while others were quite inaccurate. One of the distributions based on z(p) was particularly accurate and can be applied to ps not based on t tests. However, the standard normal approximation which is much simpler, was accurate for small effect sizes. (Author/JGL)