P-Value Calculator
Turn a z-score into a left-tailed, right-tailed or two-tailed p-value, check it against the usual significance levels, or work backwards from a significance level to the critical z. Everything runs on the standard normal distribution.
A p-value is the probability of data this extreme if the null hypothesis were true — it is not the probability that the null hypothesis is true.
Formula and calculation
Worked example
A z-score of 1.96 in a two-tailed test gives a p-value of 0.05: that is 2.5% of the curve above the value and 2.5% below it, so 0.025 sits in the upper tail alone. The calculator reads this as significant at the 5% level, which is the whole reason 1.96 is the number printed on the inside cover of every statistics textbook.
What a p-value is not
A p-value is the probability of data this extreme if the null hypothesis were true — it is not the probability that the null hypothesis is true, and it says nothing about how large the effect is. A tiny p-value from a huge sample can come from a difference far too small to matter, and a large p-value does not prove the null: it may just mean the sample was too small to see the effect. Report the effect size and a confidence interval next to the p-value; the confidence interval calculator builds one from a mean, a standard deviation and a sample size. If you need the z-score first, the z-score calculator standardizes a raw value.
What the results mean
- P-value
- The p-value for the test type you picked: the probability of seeing a result at least this extreme if the null hypothesis were true. Values below 0.0001 are shown as < 0.0001 rather than rounded to zero.
- Z-score
- The test statistic in standard deviations — the z-score you entered, or the critical z that matches your significance level in the reverse mode.
- Left-tailed p
- Left-tailed p-value, the area under the curve to the left of z. Use it when the alternative hypothesis says “less than”. A negative z lands here.
- Right-tailed p
- Right-tailed p-value, the area to the right of z. Use it when the alternative says “greater than”. This is the same area the standard normal table prints in its last column.
- Two-tailed p
- Two-tailed p-value: both tails added together, which is twice the smaller of the one-tailed values. This is the number most journals expect, and the one the 1.96 cutoff belongs to.
- Verdict
- One line reading the result against the conventional 0.05 cutoff — or, in the reverse mode, the rule for rejecting the null hypothesis at your chosen level.
Frequently asked questions
How do I calculate a p-value from a z-score?
Take the area beyond z on the standard normal curve: 1 − Φ(z) for a right-tailed test, Φ(z) for a left-tailed test, and 2 × (1 − Φ(|z|)) for a two-tailed test. A z of 1.96 gives p = 0.05 two-tailed, which is why that value appears on every critical-value table.
What does p < 0.05 actually mean?
It means data this extreme would turn up less than 5% of the time if the null hypothesis were true, so the result is called statistically significant. The 0.05 line is a convention from the Fisher and Neyman–Pearson traditions rather than a law of nature — physics and genetics demand far smaller p-values.
What is the difference between a one-tailed and a two-tailed test?
A two-tailed test asks whether there is an effect in either direction and adds both tails; a one-tailed test commits in advance to one direction and uses that tail alone. The same z of 1.96 is p = 0.05 two-tailed but p = 0.025 right-tailed, so the choice has to be made before you look at the data.
What z-score gives a 95% confidence level?
z = 1.96 leaves 95% in the middle of the curve and 2.5% in each tail, which is the value behind the usual 95% confidence interval. Enter 5 in the reverse mode above to get it, or 1 for the 99% level (z = 2.576).
Can I get the critical z for a one-tailed test?
Yes. Switch to the reverse mode, enter the significance level and pick right-tailed or left-tailed. A one-tailed 5% test has a critical z of 1.645, not 1.96 — the smaller threshold is exactly why one-tailed tests reach significance more easily, and why reviewers ask which one you used.