P-Value Calculator

Q: What does "reject the null hypothesis" mean?

It means your data are unusual enough under the null hypothesis that you have grounds to believe the null is false. You are not proving the null is wrong, only that your observed result is unlikely if the null were true. If you reject H₀ (and your test is valid), you tentatively accept the alternative hypothesis (H₁).

Q: What is the difference between a significance level (α) and a p-value?

α is the threshold you choose in advance: it is your cutoff for deciding whether to reject H₀. A p-value is what you compute after the test. You compare them: if p ≤ α, reject H₀; otherwise, fail to reject. Using α = .05 is a convention, not a law.

Q: Can I choose left-tailed or right-tailed testing after seeing my data?

No. Choosing the tail direction after you see which way your result goes (called "p-hacking" or "HARKing — Hypothesizing After Results are Known") is invalid. It inflates your false-positive rate and violates the assumptions of hypothesis testing. Decide on the tail direction before you collect data, based on your scientific hypothesis.

Q: Why does a low p-value not mean my hypothesis is probably true?

Because p-value is P(data | H₀), the probability of the data given the null. It does not tell you P(H₀ | data), the probability of the null given your data. These are different things. A low p-value means your data are surprising if the null is true; it does not directly measure how likely the null is in light of your results. See Bayesian inference for that question.

Q: What is effect size and how does it relate to the p-value?

Effect size measures the magnitude of a result, independent of sample size. A p-value tells you whether an effect is statistically significant; effect size tells you whether it is large enough to matter in practice. You can have a small but highly significant effect (large sample, small p-value, small effect size), or a large effect that fails to reach significance (small sample, large p-value, large effect size). Report both. See the effect-size calculator on this site for more.

Q: Can I use this calculator for one-sample tests, two-sample tests, and paired tests?

Yes. This calculator takes a test statistic as input (a z, t, χ², or F value) and returns its p-value. How you computed that statistic (one sample, two samples, paired, independent, etc.) does not matter to the calculator. What matters is that you chose the right statistical test for your data, and you correctly computed its test statistic and degrees of freedom.

Calculate p-values from Z, T, Chi-Square, or F statistics, with interactive visualization of the rejection region.

Distribution

Tail Type

Test Statistic

Significance Level (α)

Statistic —

DF —

P-Value —

Critical Value —

Decision —

No inputs provided yet.

Fig. 1 — The sampling distribution curve with your test statistic marked and the rejection region shaded. The shaded area under the curve represents the p-value.

§1 How to interpret a p-value

A p-value is the probability of observing a test statistic at least as extreme as the one you computed, if the null hypothesis were true. It answers the question: "If there is no real effect, how likely is my data?"

It is NOT the probability that your null hypothesis is true. Many students misinterpret this. The p-value says nothing about the truth of your hypothesis — it only quantifies how unusual your data would be under the null.

A low p-value (conventionally, p < 0.05) suggests your data are unlikely under the null, so you have grounds to reject it. A high p-value means your data fit comfortably within what you'd expect if the null were true; you fail to reject it.

§2 One-tailed vs two-tailed tests

A two-tailed test measures both extremes: values far above and far below the null. Use this when you have no prior prediction about the direction of the effect. The shaded rejection region covers both tails of the distribution.

A one-tailed test (right or left) measures only one extreme. Use this only if you predicted the direction of the effect before looking at your data. The rejection region is all in one tail, so the critical value is closer to zero and it is easier to reach significance — but you must choose the tail direction in advance, not after seeing which way your data went.

Critical rule: You must decide whether to do a one-tailed or two-tailed test before you collect and analyze your data. Choosing after you see the result (called "p-hacking") is invalid and inflates false positives.

§3 The formula: which area of the curve we measure

For each distribution and tail type, we compute a different area:

Distribution	Formula	Interpretation
Z (Normal) two-tailed	2 × (1 − Φ(\|z\|))	Prob. of \|Z\| ≥ observed in either tail
Z right-tailed	1 − Φ(z)	Prob. of Z ≥ observed
Z left-tailed	Φ(z)	Prob. of Z ≤ observed
T two-tailed	2 × min(F_t(t), 1−F_t(t))	Prob. of \|T\| ≥ observed
T right-tailed	1 − F_t(t)	Prob. of T ≥ observed
T left-tailed	F_t(t)	Prob. of T ≤ observed
Chi-Square right-tailed	1 − F_χ²(x)	Prob. of χ² ≥ observed
F right-tailed	1 − F_F(f)	Prob. of F ≥ observed

Note: Chi-Square and F statistics cannot be negative, so only right-tailed tests make sense for these distributions. The calculator greys out or hides the left-tailed option for these.

§4 Worked examples

Example 1: Z-test, two-tailed

You run a two-sample z-test and get z = 1.96. What is the two-tailed p-value?

Answer: p = 0.0500. This is the conventional boundary for significance at α = 0.05. Reject H₀.

Example 2: T-test, two-tailed

You run a paired t-test with t = 2.00 and df = 10. What is the two-tailed p-value?

Answer: p ≈ 0.0734. This exceeds α = 0.05, so you fail to reject H₀ at the conventional level.

Example 3: Chi-Square test

You run a goodness-of-fit test and get χ² = 3.8415 with df = 1. What is the (right-tailed) p-value?

Answer: p = 0.0500. Again, at the 5% significance threshold.

§5 Assumptions and when this calculator is wrong

When to trust this calculator

Z-test: Sample size is large (n > 30) or the population is normally distributed.
T-test: Data are approximately normally distributed, or sample size is reasonably large. The t-test is robust to non-normality with larger samples.
Chi-Square: Expected frequencies in all cells are ≥ 5 (or > 1 in a 2×2 table). Use Fisher's exact test if this fails.
F-test: Assumptions depend on the specific test (ANOVA, equality of variances), but generally data should be approximately normally distributed and variances roughly equal.

Common pitfalls

P-hacking: Running many tests and reporting only the significant ones inflates your false-positive rate. Pre-register your analysis or apply a multiple-comparison correction.
Misinterpreting α: The significance level α = 0.05 is a convention, not a law of nature. Always justify your choice; in some contexts 0.01 or 0.10 is more appropriate.
Confusing p < 0.05 with importance: Statistical significance is not the same as practical significance. A tiny effect with a large sample can be highly significant but trivial in real terms.
Negative or zero test statistics (Chi-Square, F): These are impossible under the usual definitions. If you see them, check your calculation.

§6 Reporting a p-value in APA style

In a research report, state your test result and p-value in this format:

t(10) = 2.00, p = .073

Rules:

Use italic for the test name and statistic letter: t, F, χ², z.
Include the degrees of freedom in parentheses (for t, F, χ², df).
Report the p-value without a leading zero (write ".073", not "0.073"), unless it is a probability unrelated to a test (like a confidence level).
If p < 0.001, write "p < .001" rather than a long string of zeros.
Do not write "significant" or "not significant" without stating the α level: "p = .003, which is significant at α = .05" is clearer than "p = .003, significant".

§7 FAQ

What does "reject the null hypothesis" mean?

It means your data are unusual enough under the null hypothesis that you have grounds to believe the null is false. You are not proving the null is wrong, only that your observed result is unlikely if the null were true. If you reject H₀ (and your test is valid), you tentatively accept the alternative hypothesis (H₁).

What is the difference between a significance level (α) and a p-value?

α is the threshold you choose in advance: it is your cutoff for deciding whether to reject H₀. A p-value is what you compute after the test. You compare them: if p ≤ α, reject H₀; otherwise, fail to reject. Using α = .05 is a convention, not a law.

Can I choose left-tailed or right-tailed testing after seeing my data?

No. Choosing the tail direction after you see which way your result goes (called "p-hacking" or "HARKing — Hypothesizing After Results are Known") is invalid. It inflates your false-positive rate and violates the assumptions of hypothesis testing. Decide on the tail direction before you collect data, based on your scientific hypothesis.

Why does a low p-value not mean my hypothesis is probably true?

Because p-value is P(data | H₀), the probability of the data given the null. It does not tell you P(H₀ | data), the probability of the null given your data. These are different things. A low p-value means your data are surprising if the null is true; it does not directly measure how likely the null is in light of your results. See Bayesian inference for that question.

What is effect size and how does it relate to the p-value?

Effect size measures the magnitude of a result, independent of sample size. A p-value tells you whether an effect is statistically significant; effect size tells you whether it is large enough to matter in practice. You can have a small but highly significant effect (large sample, small p-value, small effect size), or a large effect that fails to reach significance (small sample, large p-value, large effect size). Report both. See the effect-size calculator on this site for more.

Can I use this calculator for one-sample tests, two-sample tests, and paired tests?

Yes. This calculator takes a test statistic as input (a z, t, χ², or F value) and returns its p-value. How you computed that statistic (one sample, two samples, paired, independent, etc.) does not matter to the calculator. What matters is that you chose the right statistical test for your data, and you correctly computed its test statistic and degrees of freedom.

Sources

Related tools

CALCULATOR