P-Value Calculator
Enter a test statistic and its degrees of freedom to get the exact p-value — left-, right-, or two-tailed — with a significance decision at your α, an APA 7 result line to copy, and a sketch of the tail area. Free, no sign-up.
Results
0.0298
p-value
t(24) = 2.31
Test statistic
Significant
Decision at α = .05
p = .030 is at or below α = .05, so the result is statistically significant at the .05 level: reject the null hypothesis.
APA 7 report
t(24) = 2.31, p = .030
Italicise the statistic and p. For χ² tests, APA also asks for the sample size: χ²(df, N = …).
| Alternative | p-value |
|---|---|
| Two-tailed (≠) | 0.0298 |
| Left-tailed (<) | 0.9851 |
| Right-tailed (>) | 0.0149 |
Where the p-value comes from
t distribution with 24 df. The shaded area is the p-value (0.0298); the dashed line marks your statistic.
How to Use the P-Value Calculator
- 1. Pick the statistic. Choose z, t, χ², F, or Pearson's r — whichever your test or software reported.
- 2. Enter its value and degrees of freedom. t and χ² need df; F needs the numerator and denominator df; r needs the sample size n (the number of pairs).
- 3. Choose the alternative hypothesis. Two-tailed for "different from", left- or right-tailed only for a direction you predicted before seeing the data. χ² and F switch to right-tailed automatically.
- 4. Read the result. You get the p-value, the decision at your α, an APA-formatted line to copy, the p-value under every alternative, and a sketch with the tail area shaded.
What a P-Value Is
A p-value is the probability, computed on the assumption that the null hypothesis is true, of getting a test statistic at least as extreme as the one you observed. The alternative hypothesis defines "extreme": for a two-tailed test it means far from the null value in either direction, so the p-value is the area in both tails; for a one-tailed test it is the area in the tail you predicted.
Compare p with the significance level α you fixed in advance, usually .05: if p ≤ α the result is statistically significant and you reject the null hypothesis; otherwise you do not. Every area comes from the exact distribution rather than a printed table, so any df works — including the fractional df of Welch's t-test.
| Statistic | Distribution under H₀ | Two-tailed p | Typical tests |
|---|---|---|---|
| z | Standard normal | 2 × P(Z ≥ |z|) | z-tests, large-sample proportions |
| t | Student's t, df | 2 × P(T ≥ |t|) | t-tests, regression coefficients |
| χ² | Chi-square, df | Right tail P(χ² ≥ x) | Goodness of fit, independence |
| F | F, df₁ and df₂ | Right tail P(F ≥ x) | ANOVA, overall regression F |
| r | t on n − 2 df, after t = r √((n − 2) / (1 − r²)) | 2 × P(T ≥ |t|) | Pearson correlation |
Worked Example: t(24) = 2.31
A two-sample t-test comparing 13 students in each of two groups gives t = 2.31 with 24 degrees of freedom. Entered as a two-tailed test, the calculator returns p = 0.0298: if the true difference were zero, a t at least 2.31 away from zero in either direction would turn up about 3% of the time. At α = .05 the result is significant, and the APA line reads t(24) = 2.31, p = .030.
Had the direction been predicted in advance and the test run right-tailed, p would be half as large, 0.0149 — which is exactly why the tail has to be chosen before the data are seen, not after.
The same logic covers a correlation. r = .45 between two measures in 30 participants converts to t = 2.67 on 28 df, so the two-tailed p is .013, reported as r(28) = .45, p = .013.
Interpreting and Reporting the Result
- Report exact values. Write p = .030, not p < .05, to two or three decimals — except below .001, where APA 7 writes p < .001. Never write p = .000; a p-value is never exactly zero.
- Drop the leading zero. A p-value cannot exceed 1, so APA prints .030, not 0.030. The same rule applies to r.
- Italicise the symbols. p, t, F, r, and z are italic; Greek letters such as χ are not.
- Give the degrees of freedom in parentheses. t(24) = 2.31, F(2, 27) = 3.35, χ²(3, N = 120) = 7.81 — chi-square tests also report the sample size.
Common mistakes
- Reading p as the probability that the null hypothesis is true. It is the probability of data this extreme if the null is true — a different quantity.
- Choosing the tail after seeing the data. Switching to one-tailed because the two-tailed p came out at .07 doubles the real false-positive rate.
- Reading "not significant" as "no effect". A large p often just means a small sample. Report the effect size and its confidence interval alongside p.
- Halving a χ² or F p-value. Tests of fit, independence and ANOVA already use only the right tail, so there is no two-tailed p to halve.
- Confusing statistical with practical significance. With thousands of participants, trivially small effects reach p < .001.
Frequently Asked Questions
- How do I find the p-value from a t-score?
- Choose t, enter the t value and its degrees of freedom, and pick the tail. The df is n − 1 for a one-sample or paired test, n₁ + n₂ − 2 for a pooled two-sample test, or the Welch df your software printed — decimals are fine. For t(24) = 2.31 two-tailed, p = .030.
- What is the p-value for a z-score of 1.96?
- Two-tailed, p = 0.0500 (more exactly 0.049996); one-tailed, p = 0.0250. That is why 1.96 is the critical value for a two-tailed test at α = .05: 2.5% of the standard normal distribution lies beyond it in each tail.
- Should I use a one-tailed or a two-tailed test?
- Use two-tailed unless you stated a directional hypothesis before collecting data and an effect in the other direction would mean the same to you as no effect. For z, t, and r the one-tailed p is half the two-tailed p, so picking the tail after seeing the results inflates false positives. Most journals and examiners expect two-tailed tests.
- Why are chi-square and F tests right-tailed?
- Both statistics grow as the data move away from the null hypothesis in any direction, because they are built from squared deviations. So only large values count as evidence against H₀, and the p-value is the right-tail area. The left tail matters only in special cases, such as testing whether a variance is smaller than a stated value.
- What does p < .001 mean?
- It means a result at least this extreme would occur less than once in a thousand times if the null hypothesis were true. APA style reports every p-value below .001 as p < .001 rather than printing more decimals or p = .000. The calculator still shows the exact value, in scientific notation when it is tiny.
- Is a p-value of exactly 0.05 significant?
- Under the usual rule — reject when p ≤ α — yes, but values that round to .050 need care. z = 1.96 gives p = 0.049996, which is significant at .05, while t(24) = 2.06 gives p = 0.0504, which is not. Report such values with enough decimals to show which side of the line they fall on.