The Meta-Distribution of Standard P-Values

We present an exact probability distribution (meta-distribution) for p-values across ensembles of statistically identical phenomena, as well as the distribution of the minimum p-value among m independents tests. We derive the distribution for small samples 2 P-values are shown to be extremely skewed and volatile, regardless of the sample size $n$, and vary greatly across repetitions of exactly same protocols under identical stochastic copies of the phenomenon; such volatility makes the minimum p value diverge significantly from the "true" one. Setting the power is shown to offer little remedy unless sample size is increased markedly or the p-value is lowered by at least one order of magnitude.The formulas allow the investigation of the stability of the reproduction of results and "p-hacking" and other aspects of meta-analysis. From a probabilistic standpoint, neither a p-value of .05 nor a "power" at .9 appear to make the slightest sense.