P-value calculator
Statistical software gives you a test statistic; a printed table only gives you a range. This converts t, z, F, or chi-square into the exact probability.
A p-value is the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true; the smaller it is, the stronger the evidence against that hypothesis. This calculator converts a t, z, F, or chi-square statistic into its exact one- and two-tailed p-value by evaluating the correct sampling distribution, so you are not stuck reading a critical value between two rows of a textbook table.
Two-tailed p value
.0208
t(28) = 2.45 is significant at the .05 level.
How the p-value is computed
A p-value is a tail area under the test statistic's distribution. For a two-tailed t or z test it is twice the area beyond the absolute value of the statistic:
p = 2 × P(T > |t|)
The F and chi-square statistics are one-sided by construction, so their p-value is the single upper-tail area, P(F > f) or P(χ² > x). The calculator obtains each area from the cumulative distribution function of the relevant family, using the regularised incomplete beta function for t and F and the regularised incomplete gamma function for chi-square. These are the same routines statistical packages call, so the values match the p-values in SPSS, R, and Stata output to the displayed precision.
The guides behind this tool
Frequently asked questions
- What does a p-value of 0.05 mean?
A p-value of 0.05 means there is a 5% probability of obtaining a result at least as extreme as the one observed if the null hypothesis were true. It is the conventional threshold for calling a result statistically significant, but the cut-off is a convention, not a law of nature. A p-value just below or just above 0.05 should be read in the context of the effect size and the study design, not treated as a hard pass or fail.
- What is the difference between a one-tailed and a two-tailed p-value?
A two-tailed p-value tests for a difference in either direction and counts extreme results in both tails of the distribution, while a one-tailed p-value tests for a difference in one specified direction only. The one-tailed value is half the two-tailed value for a symmetric distribution. Use a two-tailed test unless you have a strong directional hypothesis stated before seeing the data, because two-tailed is the more conservative and more widely expected default.
- How do you find a p-value from a test statistic?
You locate the test statistic on its sampling distribution and read off the tail area beyond it, which is the p-value. The distribution depends on the test: a t statistic uses the t distribution with its degrees of freedom, a z uses the standard normal, an F uses the F distribution with two degrees of freedom, and a chi-square uses the chi-square distribution. This calculator evaluates the exact distribution rather than interpolating from a printed table.
- Can a p-value be exactly zero?
No, a p-value can be extremely small but never exactly zero, because there is always some non-zero probability of an extreme result under the null hypothesis. Software often displays very small values as p less than .001 rather than printing many leading zeros, and reporting guidelines such as APA recommend writing p < .001 in that case. A reported p of 0 is a rounding artefact, not a true probability.