P Value Calculator
Enter the test statistic, pick its distribution and give the degrees of freedom. The p value calculator returns the two-tailed, right-tailed and left-tailed p-values and the critical value at your alpha.
p-value (two-tailed)
- p-value, two-tailed P(|z| ≥ 1.9600): 5% of the standard normal lies beyond ±1.9600. Twice the smaller tail, for an alternative of "not equal".
- 0.0500
- p-value, right-tailed P(z ≥ 1.9600): 2.5% of the standard normal lies above it.
- 0.0250
- p-value, left-tailed P(z ≤ 1.9600): 97.5% of the standard normal lies below it.
- 0.9750
- Critical value at alpha = 0.05 Reject when |z| is larger.
- ±1.9600
- Decision p = 0.0500 is below alpha = 0.05.
- Significant: reject the null
| Alpha | One-tailed critical value | Two-tailed critical value |
|---|---|---|
| 0.1 | 1.2816 | ±1.6449 |
| 0.05 | 1.6449 | ±1.9600 |
| 0.025 | 1.9600 | ±2.2414 |
| 0.01 | 2.3263 | ±2.5758 |
| 0.005 | 2.5758 | ±2.8070 |
| 0.001 | 3.0902 | ±3.2905 |
- A p-value is the probability of a statistic at least this extreme if the null hypothesis were true. It is not the probability that the null hypothesis is true.
- Choose the tail and alpha before looking at the data. Switching to one tail after seeing the direction halves the p-value without any new evidence.
How the P Value Calculator works
A p-value is the probability, if the null hypothesis were true, of getting a test statistic at least as extreme as the one you got. Every test ends the same way: it turns the data into one number, a z, t, chi-square or F statistic, and the p-value is the area of that statistic's distribution beyond it. So there is no single p-value formula: the formula is the tail area of whichever distribution the test uses, and the p value calculator finds that area.
Which distribution. A z statistic, from a z test or a test of proportions with a large sample, is read against the standard normal curve. A t statistic, from a t test or a regression coefficient, is read against Student's t with its degrees of freedom, so to use it as a p value calculator from t, enter the t and its df. A chi-square statistic, from a goodness-of-fit or independence test, uses the chi-square distribution with its degrees of freedom. An F statistic, from ANOVA or a comparison of two variances, uses the F distribution with two degrees of freedom: the numerator's and the denominator's.
Which tail. A two-tailed p-value counts extreme values in both directions and fits an alternative of "not equal". For the default, z = 1.96, the two-tailed p is 0.0500: 2.5% of the normal curve lies above 1.96 and 2.5% below -1.96. A right-tailed p counts only the upper side, 0.0250 here, and fits an alternative of "greater than". Chi-square and F tests of fit, independence and ANOVA reject only for large values, so their p-value is the right tail, and the calculator leads with that tail when two-tailed is picked for them.
Worked examples. A t statistic of 2.228 on 10 degrees of freedom has a two-tailed p of 0.0500, which is why 2.228 is the 5% entry in the t table for 10 df. A chi-square of 3.841 on 1 degree of freedom has a right-tailed p of 0.0500. An F of 3.326 on 5 and 10 degrees of freedom has a right-tailed p of 0.0500, matching the NIST/SEMATECH e-Handbook's F table. The table under each result lists the critical values for the distribution and degrees of freedom you entered.
Significance. Compare p with alpha, the cutoff chosen before the test, usually 0.05. If p is below alpha, the result is statistically significant and the null hypothesis is rejected. That is the same decision as checking whether the statistic is beyond the critical value. A small p-value says the data would be surprising if the null were true; it does not measure how large or important the effect is, and p = 0.04 and p = 0.06 are much closer in evidence than "significant" and "not significant" suggest.
The areas are computed from the exact distributions: the normal by Hart's rational approximation, and t, chi-square and F through the regularized incomplete beta and gamma functions. Each tail is computed directly, so a p-value of 1e-12 keeps its digits.
P Value Calculator FAQ
How do I find a p-value from a z score?
For a two-tailed test, double the area beyond |z| under the standard normal curve. z = 1.96 gives 2 × 0.025 = 0.0500. For a one-tailed test use the area on one side only: 0.0250 for z = 1.96 on the right.
How do I find a p-value from a t statistic?
Use Student's t distribution with the test's degrees of freedom. t = 2.228 on 10 df has a two-tailed p of 0.0500. The same t on more degrees of freedom gives a smaller p, because the t distribution's tails thin out as df grows.
Is a chi-square p-value one-tailed or two-tailed?
Right-tailed for goodness-of-fit and independence tests, because only a large chi-square counts against the null. χ² = 3.841 on 1 df gives p = 0.0500. A two-sided test of a single variance uses both tails.
What p-value is significant?
One below the alpha you chose before the test, most often 0.05. Some fields use 0.01 or 0.001. The cutoff is a convention, and the p-value itself is worth reporting rather than only significant or not.
What is the p-value for an F statistic?
The right-tail area of the F distribution with the numerator and denominator degrees of freedom. F = 3.326 on (5, 10) df gives p = 0.0500. In ANOVA the numerator df is the number of groups minus 1 and the denominator df is the total sample size minus the number of groups.