P-Value Calculator
Enter a z, t, chi-square or F test statistic to get its p-value, one-tailed or two-tailed, and whether it is significant at your α level. The area that makes up the p-value is shaded on the curve of the test statistic.
Related guides: what p < 0.05 actually means, Type I vs Type II errors and p-values in Excel.
Chosen before looking at the data, usually 0.05
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P-Value Explained
What p < 0.05 actually means: the precise definition, a worked coin-flip example, the decision rule against α, and the five misreadings to avoid.
What a p-value tells you
The p-value is the probability of getting a test statistic at least as extreme as the one you observed if the null hypothesis is true. A small p-value means the data would be surprising under the null hypothesis, which is evidence against it; a large one means the data are consistent with it. If the p-value is at most your significance level α, the result is called statistically significant and you reject the null hypothesis.
Which tail counts as extreme depends on the alternative hypothesis. A two-tailed test looks for a difference in either direction, a right-tailed test for a value above the null, and a left-tailed test for a value below it. Pick the direction before seeing the data.
How each p-value is calculated
Right-tailed: p = P(X ≥ x)
Left-tailed: p = P(X ≤ x)
Two-tailed, z and t: p = 2 × P(X ≥ |x|)
Two-tailed, χ² and F: p = 2 × min( P(X ≤ x), P(X ≥ x) ), at most 1
Here X is the test statistic under the null hypothesis and x the value you observed. Each tail is computed directly from its own end of the distribution rather than as 1 minus the other tail, so a very small p-value keeps all its digits; one below 1e-300 is reported as a bound. The z and t distributions are symmetric, so the two-tailed p-value is twice the tail beyond |x|. Chi-square and F are skewed and have no mirror image, so the two-tailed p-value doubles the smaller tail, the same rule R uses in var.test. Chi-square and F tests are normally one-sided, looking at the upper tail.
Which test statistic to use
| Statistic | Typical use | Tail usually reported |
|---|---|---|
| Z | Means with a known standard deviation or a large sample, proportions (z-test) | Two-tailed, or one-tailed for a directional claim |
| t | Means with an unknown standard deviation, paired data, regression coefficients | Two-tailed, or one-tailed for a directional claim |
| Chi-square | Goodness of fit, tests of independence, a variance against a target | Right-tailed |
| F | ANOVA, comparing two variances, the overall regression test | Right-tailed |
Need the test statistic first? See the z-test, t-test, chi-square and ANOVA calculators, which return the p-value together with the statistic. To work backwards from a significance level to the cutoff statistic, use the critical value calculator.
Worked example
A one-sample t-test on 11 measurements gives t = 2.5 with 10 degrees of freedom. The right tail beyond 2.5 is 0.015723, so the two-tailed p-value is 2 × 0.015723 = 0.031447. Because 0.031447 is below α = 0.05 the result is significant at the 5% level, but it is not below 0.01 so it is not significant at the 1% level. Load example fills in these numbers. A right-tailed test on the same statistic would report 0.015723 and a left-tailed test 0.984277.
| Statistic | Direction | P-value |
|---|---|---|
| z = 2 | Two-tailed | 0.0455 |
| z = 1.645 | Right-tailed | 0.049985 |
| t = 2.5, df = 10 | Two-tailed | 0.031447 |
| χ² = 11.07, df = 5 | Right-tailed | 0.05001 |
| F = 3.5, df₁ = 3, df₂ = 20 | Right-tailed | 0.034493 |
How to read a p-value
| P-value | Reading |
|---|---|
| Below 0.001 | Very strong evidence against the null hypothesis |
| 0.001 to 0.01 | Strong evidence against the null hypothesis |
| 0.01 to 0.05 | Moderate evidence against the null hypothesis |
| 0.05 to 0.1 | Weak evidence against the null hypothesis |
| 0.1 or more | Little or no evidence against the null hypothesis |
These labels are conventions, not laws; the decision rule uses the α you set before collecting data. A p-value is not the probability that the null hypothesis is true, it does not measure the size or importance of an effect (use an effect size for that), and a large p-value does not prove the null hypothesis. With a big enough sample even a trivial effect gives a tiny p-value.
Software equivalents
| Software | Two-tailed z | Two-tailed t | Right-tailed χ² | Right-tailed F |
|---|---|---|---|---|
| Excel / Sheets | =2*NORM.S.DIST(-ABS(z), TRUE) | =T.DIST.2T(ABS(t), df) | =CHISQ.DIST.RT(x, df) | =F.DIST.RT(x, df1, df2) |
| R | 2 * pnorm(-abs(z)) | 2 * pt(-abs(t), df) | pchisq(x, df, lower.tail = FALSE) | pf(x, df1, df2, lower.tail = FALSE) |
| Python (SciPy) | 2 * norm.sf(abs(z)) | 2 * t.sf(abs(t), df) | chi2.sf(x, df) | f.sf(x, df1, df2) |
| TI-84 | 2*normalcdf(abs(z), 1E99) | 2*tcdf(abs(t), 1E99, df) | χ²cdf(x, 1E99, df) | Fcdf(x, 1E99, df1, df2) |
The TI-84 functions are explained in the TI-84 statistics guide. To turn a z-score into an area with more detail, use the z-score to percentile calculator.
Frequently Asked Questions
What is a p-value?
The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. A small p-value means the data are unlikely under the null hypothesis. It is not the probability that the null hypothesis is true.
How do I find the p-value from a z-score or t-score?
Choose the distribution, enter the statistic (and the degrees of freedom for t), pick the tails and read the p-value. By hand it is the area beyond the statistic: for a two-tailed z-test, p = 2 × P(Z ≥ |z|). In Excel that is =2*NORM.S.DIST(-ABS(z), TRUE) for z and =T.DIST.2T(ABS(t), df) for t.
Should I use a one-tailed or a two-tailed p-value?
Use a two-tailed p-value when the alternative hypothesis is that the value differs from the null in either direction, which is the default in most research. Use a one-tailed p-value only when you decided before collecting data that only one direction matters. A one-tailed p-value is half the two-tailed one for z and t when the statistic is in the predicted direction.
What p-value is statistically significant?
A result is significant when the p-value is at most the significance level α you chose in advance. The usual choice is α = 0.05, sometimes 0.01 or 0.001 in stricter fields. Because the threshold is a convention, report the exact p-value as well.
How do I get the p-value for a chi-square or F statistic?
Both tests are normally right-tailed: the p-value is the area to the right of the statistic. Select Chi-square or F, enter the statistic and the degrees of freedom and keep the right-tailed direction. In Excel use =CHISQ.DIST.RT(x, df) or =F.DIST.RT(x, df1, df2).
Why is the two-tailed p-value for chi-square or F twice the smaller tail?
Those distributions are not symmetric, so there is no mirror image of the observed statistic. The convention for a two-sided test is to double the smaller of the two tails and cap the result at 1, which is what R does in var.test.
Can a p-value be zero?
Not exactly: a p-value is always above 0 for a finite statistic, although it can be extremely small. When it is below 1e-300 this calculator shows it as less than 1e-300 instead of a misleading 0. In a report, write p < 0.001 rather than p = 0.
What does the p-value not tell me?
It does not give the probability that your hypothesis is true, the size of the effect or whether the effect matters in practice. A tiny p-value with a large sample can go with a negligible effect, and a large p-value can just mean the sample was too small to detect one. Pair the p-value with an effect size and a confidence interval.
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