P-Value Calculator
Turn a z test statistic into a p-value for a two-tailed, left-tailed or right-tailed test, and see the tail area shaded on the normal curve.
Last updated: October 2026
p-value
0.035729
Formula below · 3 sources (itl.nist.gov, amstat.org, Wikipedia) · Updated Oct 2026
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About this calculator
A p-value is the probability of getting a test statistic at least as extreme as the one you observed, assuming the null hypothesis is true. Small p-values mean the data would be surprising if the null hypothesis held.
For a z statistic, with Φ the standard normal cumulative distribution:
Two-tailed: p = 2 × (1 − Φ(|z|)) Right-tailed: p = 1 − Φ(z) Left-tailed: p = Φ(z)
The calculator evaluates the normal tail with a rational approximation that is accurate to about seven significant digits, so very small p-values (z beyond 5) are reported correctly rather than as 0. Values below 0.0001 are shown in scientific notation.
Reading the result
Compare p with the significance level α you chose before looking at the data, most often 0.05. If p < α, reject the null hypothesis; otherwise you do not have enough evidence to reject it. A p-value is not the probability that the null hypothesis is true, and it says nothing about how large or important an effect is, so report it alongside the effect size or a confidence interval.
This calculator is for z statistics: tests where the population standard deviation is known, large-sample tests and tests of proportions. A t statistic from a small sample needs the t distribution with its degrees of freedom, which gives a larger p-value for the same number.
How to use
Enter your z statistic and choose the test type. The default, z = 2.1 with a two-tailed test, gives p = 2 × (1 − Φ(2.1)) = 2 × 0.017864 = 0.035729. That is below 0.05, so the result is significant at the 5% level. For a left-tailed test with z = −1.5, p = Φ(−1.5) = 0.066807, which is not significant at 5%. The same z in a two-tailed test gives 0.13361. A z of 1.96 gives a two-tailed p of 0.049996, the source of the familiar 1.96 cutoff.
Frequently asked questions
What does p = 0.05 mean?
If the null hypothesis were true, a result at least this extreme would turn up about 5% of the time by chance. It does not mean there is a 5% chance the null hypothesis is true, and p = 0.049 and p = 0.051 are practically the same evidence.
Should I use a one-tailed or a two-tailed test?
Use two-tailed unless you decided before seeing the data that only one direction matters. A one-tailed p-value is half the two-tailed one for the same z, so choosing the direction after looking at the data overstates the evidence.
Can I use this for a t-test or chi-square test?
Not directly. Those statistics follow the t and chi-square distributions, not the normal. For large samples a t statistic is close to z, but for small samples using z understates the p-value; at 5 degrees of freedom, t = 2.0 has a two-tailed p of about 0.10, while z = 2.0 gives 0.046.
What significance level should I use?
0.05 is the most common convention, 0.01 is used where false positives are costly, and fields such as particle physics use much stricter thresholds. Pick α before analyzing the data and, if you run many tests, adjust for multiple comparisons.
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Sources & references
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