P-Value from Z Calculator
Free Pvalue zcalculator Calculator for biostatistics. Enter variables to compute results with formulas and detailed steps.
Reviewed by Daniel Agrici, Founder & Lead Developer
Formula
P(Z > z) = 1 - Phi(z), where Phi is the standard normal CDF
For a right-tailed test, p = 1 - Phi(z). For a left-tailed test, p = Phi(z). For a two-tailed test, p = 2 * min(Phi(z), 1-Phi(z)). Phi(z) represents the cumulative distribution function of the standard normal distribution, giving the probability that a standard normal random variable takes a value less than or equal to z.
Worked Examples
Example 1: Clinical Trial Outcome
Problem:A clinical trial comparing drug vs placebo yields a test statistic z = 2.45. What is the two-tailed p-value and is it significant at alpha = 0.05?
Solution:Using the standard normal CDF:\nPhi(2.45) = 0.99286\nRight-tail p = 1 - 0.99286 = 0.00714\nTwo-tailed p = 2 * 0.00714 = 0.01428\nSince 0.01428 < 0.05, we reject the null hypothesis.
Result:Two-tailed p-value = 0.0143, which is significant at alpha = 0.05 (reject H0)
Example 2: Gene Expression Z-Score
Problem:A gene shows a z-score of -1.5 in a differential expression analysis. Calculate the left-tailed p-value at alpha = 0.05.
Solution:Using the standard normal CDF:\nPhi(-1.5) = 0.06681\nLeft-tail p = 0.06681\nSince 0.06681 > 0.05, we fail to reject the null hypothesis.\nThe gene is not significantly downregulated at the 5% level.
Result:Left-tailed p-value = 0.0668, not significant at alpha = 0.05 (fail to reject H0)
Frequently Asked Questions
What is a p-value and what does it tell us?
A p-value is the probability of observing a test statistic as extreme as (or more extreme than) the one calculated from your data, assuming the null hypothesis is true. It does NOT tell you the probability that the null hypothesis is true or false. A small p-value (typically < 0.05) suggests the observed data would be unlikely under the null hypothesis, providing evidence against it. In biostatistics, p-values help researchers determine whether observed differences between groups (e.g., treatment vs control) are likely due to chance or reflect real biological effects.
Why is the p-value often misinterpreted?
The most common misinterpretation is thinking that p = 0.03 means there is a 3% probability the null hypothesis is true. In reality, it means that if the null hypothesis were true, there would be a 3% chance of seeing data this extreme. Other misconceptions include: (1) A non-significant p-value does not prove the null hypothesis. (2) A significant p-value does not prove the alternative hypothesis. (3) P-values do not measure effect size; a tiny meaningless difference can be highly significant with large samples. (4) P = 0.049 and p = 0.051 are practically identical, despite falling on different sides of the 0.05 cutoff.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy