P-Value Calculator
Turn any z-score into a p-value for a one- or two-tailed hypothesis test.
Example
A two-tailed test produces a z-score of 1.96 at α = 0.05:
Φ(1.96) = 0.97500
p (two-tailed) = 2 × (1 - 0.97500)
= 0.05000
0.05000 ≤ 0.05 → Reject H0For a right-tailed test the same z gives p = 1 - 0.97500 = 0.02500.
How it works
Enter a z-score and pick the tail type; the tool applies the standard normal CDF to return the p-value. One-tailed gives the upper-tail area in the score's direction; two-tailed doubles the area beyond |z|.
Good to know
This P-Value Calculator converts a z-score from a standard normal test into a p-value and tells you whether to reject the null hypothesis at your chosen significance level. You enter the z-statistic, pick a left-tailed, right-tailed, or two-tailed test, set your alpha (default 0.05), and it returns the p-value along with the cumulative probability Φ(z) and a plain reject / fail-to-reject decision. It is built for students, researchers, analysts, and anyone reporting results from a z-test, A/B test, or proportion comparison who already has a z-score in hand.
Reach for it once your test statistic is computed and you need the corresponding tail probability without flipping through a printed z-table. The two-tailed option doubles the area beyond the absolute value of z, so use it when you only care whether a difference exists; choose right- or left-tailed when your hypothesis predicts a specific direction. Because the same magnitude of z produces a two-tailed p that is exactly twice the one-tailed p, the tail setting can change your conclusion at the margin.
To read the output: the p-value is the probability of seeing a result at least this extreme if the null hypothesis were true, so smaller values are stronger evidence against it. If p is less than or equal to your alpha, the tool shows "Reject H0"; otherwise "Fail to reject H0," which is not the same as proving the null true. The Φ(z) figure is the cumulative area to the left of your z-score and is mostly useful as a sanity check.
A practical caveat: this calculator assumes a standard normal sampling distribution, so it fits large-sample z-tests rather than small-sample situations that call for a t-distribution. It also takes a z-score as input and does not compute one from raw data, and a "significant" p-value speaks to statistical detectability, not the practical size or importance of an effect.
Frequently asked questions
What is the difference between one-tailed and two-tailed p-values here?
A one-tailed (right or left) p-value is the area in a single tail beyond your z-score, used when your hypothesis predicts a specific direction. A two-tailed p-value doubles the area beyond |z|, used when you only care that there is a difference, not its direction. That is why a two-tailed p is exactly twice the corresponding one-tailed p for the same magnitude.
How accurate is the p-value computed by this calculator?
It uses the Abramowitz & Stegun rational approximation of the error function (erf) to evaluate the standard normal CDF, with a maximum absolute error under 1.5e-7. For example, z = 1.96 returns 0.04999 (two-tailed) and z = 2.576 returns 0.00999, matching standard z-tables to four decimal places.
Is my data uploaded anywhere?
No — this calculator runs entirely in your browser; nothing is uploaded.
Is it free?
Yes, completely free with no sign-up and no limits.
People also ask
How do I calculate a z-score to enter into this calculator?
For a single mean, subtract the hypothesized value from your observed statistic and divide by its standard error: z = (x bar minus mu) / (sigma / sqrt(n)). The resulting number is the z-score you type into the calculator.
What does a p-value of 0.05 actually mean?
It means there is a 5 percent probability of observing a result at least as extreme as yours if the null hypothesis were true. It is a measure of evidence against the null, not the probability that the null is true.
Should I use a one-tailed or two-tailed test?
Use a two-tailed test when you only want to know if there is a difference in either direction, and a one-tailed test when your hypothesis specifies a direction beforehand. Choosing one-tailed after seeing the data is generally considered improper.
Is a smaller p-value better?
A smaller p-value indicates stronger evidence against the null hypothesis, but it does not measure how large or important an effect is. A tiny p-value with a trivial effect size can still be practically meaningless.
What is the difference between a z-test and a t-test p-value?
A z-test uses the standard normal distribution and assumes a known population standard deviation or a large sample, which is what this tool computes. A t-test uses the t-distribution and is appropriate for small samples with an estimated standard deviation.
Can a p-value be greater than 1 or negative?
No. A valid p-value is always between 0 and 1 because it represents a probability. This calculator clamps its output to that range to avoid rounding artifacts.
What does 'fail to reject the null hypothesis' mean?
It means your data did not provide enough evidence to conclude the null is false at your chosen significance level. It does not prove the null hypothesis is true; the effect may simply be too small or the sample too limited to detect.
What z-score corresponds to a 0.05 significance level?
For a two-tailed test at alpha 0.05 the critical z is about 1.96, and for a one-tailed test it is about 1.645. A z-score beyond those thresholds yields a p-value at or below 0.05.
Related calculators