The formula
How it works
Convert a z-score into a p-value — the probability of seeing a result this extreme if nothing is really going on. Choose a one- or two-tailed test to match your hypothesis.
FAQ
What counts as “significant”?
A common threshold is p < 0.05, meaning a result this extreme would happen by chance less than 5% of the time. Some fields use stricter cut-offs like 0.01. The threshold should be chosen before running the test.
One-tailed or two-tailed?
Use two-tailed when you only care whether there is a difference in either direction — the usual choice. Use one-tailed when your hypothesis specifies a direction, such as “greater than”, which puts all the probability in one tail.
Does a p-value tell me the null hypothesis is true?
No. The p-value is the probability of the observed data (or something more extreme) assuming the null hypothesis is true — it is not the probability that the null hypothesis itself is true. That distinction is one of the most common misunderstandings in statistics.
What does “fail to reject the null” mean?
A p-value above your threshold means the data does not provide strong enough evidence against the null hypothesis — it does not prove the null hypothesis is true. Absence of significance can also simply reflect a small sample size or weak effect.
Where does the z-score come from?
A z-score usually comes from a separate calculation, such as a one-sample, two-sample, or proportion test, that standardizes your observed effect relative to its expected variability. This calculator starts from that z-score rather than computing it from raw data.
How does a p-value relate to a confidence interval?
They are two views of the same test: if a 95% confidence interval for an effect excludes zero (or the null value), the corresponding two-tailed p-value will be below 0.05, and vice versa. Confidence intervals additionally show the size and direction of the effect, which a p-value alone does not.
Does a small p-value mean the effect is large or important?
Not necessarily. Statistical significance depends on sample size as well as effect size, so a tiny, practically meaningless effect can still produce a very small p-value in a large enough sample. Always look at the effect size alongside the p-value.
About the p-value calculator
This calculator turns a z-score into a p-value, the number at the heart of hypothesis testing. The p-value is the probability of getting a result at least as extreme as the one you observed, assuming the null hypothesis — the idea that there is no real effect — is true. A small p-value suggests your result would be unlikely by chance alone, which is the evidence researchers look for before claiming an effect is real.
How to use it
Enter the z-score from your test and choose whether it is a one-tailed or two-tailed test. The calculator returns the p-value. For example, a z-score of 1.96 gives a two-tailed p-value of about 0.05 — the classic borderline for significance — or about 0.025 for a one-tailed test. Compare the p-value with your chosen threshold: if it is smaller, the result is statistically significant.
The formula
The p-value comes from the standard normal distribution, . For a two-tailed test it is , the combined area in both tails beyond your z-score. For a one-tailed test it is just , the area in the single tail. This calculator computes with an accurate approximation of the error function, so no statistical tables are needed.
Where it is used
P-values are everywhere in research, from medicine and psychology to economics and A/B testing. Scientists use them to decide whether an experimental result is worth taking seriously, and journals often require them for publication. They are powerful but easy to misread — a p-value is not the chance the hypothesis is true, and a significant result is not automatically an important one, so they are best paired with effect sizes and confidence intervals.