P-Value from Z Calculator

Converting a z-score to a p-value is an evaluation of the standard normal cumulative distribution function, Phi(z), which gives the area under the bell curve to the left of z. The right-tail probability is 1 minus Phi(z), the left-tail probability is Phi(z) itself, and the two-tailed value is twice the smaller of the two. Because the standard normal has mean 0 and variance 1, the same table serves any test statistic that has been standardised by subtracting its expected value and dividing by its standard error. Landmark pairs worth memorising are z = 1.645 for a one-tailed p of 0.05, z = 1.960 for a two-tailed 0.05, z = 2.576 for 0.01 and z = 3.291 for 0.001. No elementary closed form exists for Phi, so software uses series or rational approximations; the five-term polynomial used here follows the Zelen and Severo formulation and is accurate to better than 1e-7 in absolute probability, which is ample for reported p-values but degrades in the far tail beyond about z = 6. The z-test itself is only appropriate when the variance is known or the sample is large enough for the central limit theorem to apply; with small samples and an estimated standard deviation the t distribution gives noticeably larger p-values. Thresholds vary by field: genome-wide association studies use 5e-8 to absorb roughly a million independent tests, while a single confirmatory clinical endpoint may use 0.01. A p-value is the probability of the data given the null, never the probability that the null is true.

The normal curve itself predates significance testing by two centuries. Abraham de Moivre derived it in 1733 as the limiting form of the binomial distribution, publishing the result in later editions of The Doctrine of Chances; Pierre-Simon Laplace generalised it in 1810 into what is now the central limit theorem, and Carl Friedrich Gauss tied it to the theory of measurement error in 1809. Isolated tail-probability arguments appeared even earlier, as when John Arbuthnot argued in 1710 from London christening records that the excess of male births was too consistent to be chance. The modern p-value emerged from Karl Pearson's chi-squared paper of 1900 and from Ronald Fisher's Statistical Methods for Research Workers in 1925, which popularised the 0.05 convention. Fisher published tables of critical values rather than exact probabilities, partly a practical necessity and partly because Pearson controlled the more detailed tables already in print, and that format entrenched the habit of reporting a result as above or below a cutoff. Neyman and Pearson formalised alpha as a fixed error rate in 1933. Computing removed the need for tables once polynomial approximations such as those published by Zelen and Severo in the 1964 Handbook of Mathematical Functions became standard. In 2016 the American Statistical Association issued a formal statement, authored by Wasserstein and Lazar, warning against mechanical use of the 0.05 threshold.