About the P-value Calculator
A p-value is the probability of observing data at least as extreme as yours if the null hypothesis were true. It is the number that decides statistical significance across most of empirical science — and one of the most consistently misinterpreted quantities in it.
The formula
Two-tailed: p = 2 · (1 − Φ(|z|)) One-tailed: p = 1 − Φ(|z|)Φ is the cumulative standard normal distribution and z your test statistic. A two-tailed test looks for a difference in either direction; a one-tailed test only in a specified direction.
How to use this calculator
- 1Enter your Z-Score (test statistic). The field starts at
1.96, which you can overwrite. - 2Enter your Tails (1 = one-tailed, 2 = two-tailed). The field starts at
2, which you can overwrite. - 3Read the result straight away — it recalculates as you type, so there is no button to press. Use Share to copy a link that reopens the page with your exact numbers filled in.
Worked example
| Input | Value |
|---|---|
| Z-Score (test statistic) | 1.96 |
| Tails (1 = one-tailed, 2 = two-tailed) | 2 |
Result
P-value: 0.0500
One-tailed p: 0.0250
Two-tailed p: 0.0500
Significance: *
Result at α=0.05: Reject H₀ (Significant)
Those are the values the page loads with, so you can reproduce this result yourself and then change one field at a time to see what drives the outcome.
Understanding your result
A p-value of 0.03 means that if there were genuinely no effect, data this extreme would arise 3 percent of the time by chance. It does not mean there is a 97 percent chance the effect is real, and it says nothing about the size or importance of the effect. A trivial difference in a very large sample can produce a tiny p-value.
The conventional 0.05 threshold is a convention, not a law of nature — Fisher proposed it as a rough guide. Treating it as a bright line has caused real damage through p-hacking: testing many hypotheses and reporting only those that cross it. With twenty independent tests, one significant result at p < 0.05 is exactly what chance predicts.
Things worth knowing
- Always report effect size and confidence intervals alongside the p-value. Significance without magnitude is not a finding.
- Choose one-tailed versus two-tailed before seeing the data. Switching afterwards doubles your effective false positive rate.
- A non-significant result is not evidence of no effect. It may only mean the sample was too small to detect one.
- Correct for multiple comparisons — Bonferroni or false discovery rate — when running many tests.
- Very large samples make trivial differences significant. Ask whether the effect matters, not just whether it is detectable.
Frequently asked questions
What does a p-value actually mean?+
The probability of seeing results at least as extreme as yours if the null hypothesis were true. A p of 0.03 means such data would arise 3 percent of the time purely by chance under the null.
Does p < 0.05 prove my hypothesis?+
No. It indicates the data would be unlikely under the null hypothesis, which is weaker than proof. It also says nothing about whether the effect is large enough to matter.
Should I use a one-tailed or two-tailed test?+
Two-tailed unless you have a strong prior reason to test only one direction, and you must decide before looking at the data. One-tailed tests halve the p-value, which is why choosing after the fact is a form of cheating.
What does a non-significant p-value tell me?+
That your data does not provide strong evidence against the null hypothesis. That is not the same as showing no effect exists — an underpowered study routinely misses real effects.