P Value Calculator
Turn a z score, t score, chi square or F statistic into an exact p value, with the decision at your significance level and every step.
Use df = n − 1 for a one sample t test. Decimal df from a Welch t test are accepted.
If H₀ is true, a z statistic at least 2.1 away from 0 in either direction happens 3.57% of the time.
Shaded area = p value. Dashed line = your z.
Decision at α = 0.05
p = 0.03573 is below α = 0.05, so the result is statistically significant.
Rejection region: |z| > 1.960. Your z = 2.1 falls inside it.
Report it as: z = 2.10, p = .036.
Your calculation
The same steps the calculator runs, with your numbers.
Decision at common significance levels
Critical values for your test and tail. The highlighted row is the α you picked.
| α | Reject H₀ when | Decision |
|---|
A p value measures how surprising the data are if H₀ is true. It is not the probability that H₀ is true and says nothing about effect size.
What Is a P Value?
A p value is the probability of getting a test statistic at least as extreme as the one you observed, assuming the null hypothesis H₀ is true. OpenStax Introductory Statistics defines it as the probability that, if H₀ is true, another random sample gives a result as extreme or more extreme than yours.
Every hypothesis test turns your data into one number, the test statistic: a z score, t score, chi square statistic or F statistic. Under H₀ that statistic follows a known distribution. The p value is the area under that distribution's curve beyond your statistic, in the tail or tails that your alternative hypothesis H₁ points to.
A small p value says your data would be rare if H₀ were true, so it counts as evidence against H₀. A large p value says the data fit H₀ well. It does not prove H₀: the sample may simply be too small to detect a real difference.
How to Use the P Value Calculator
- Pick the test. Use z for tests with a known standard deviation or for proportions, t for means with an estimated standard deviation, χ² for goodness of fit and contingency tables, and F for ANOVA and comparing two variances.
- Enter the statistic. Type the value from your output or your own calculation. Negative values are allowed for z and t. Chi square and F statistics cannot be negative.
- Enter degrees of freedom. A one sample t test uses n − 1. A chi square contingency table uses (rows − 1) × (columns − 1). An F test needs a numerator df and a denominator df.
- Choose the tail and α. Two tailed matches H₁: ≠. Left and right tailed match < and >. The calculator sets right tailed by default for χ² and F.
The result shows the p value with a shaded curve, the decision at your α with the rejection region, the other tail areas, the steps with your numbers and the decision at six common significance levels. Results update as you type.
P Value Formulas
I_x(a, b) is the regularized incomplete beta function. For tests with two tails on a skewed distribution (χ² and F), the calculator doubles the smaller tail and caps the result at 1.
How to Calculate a P Value by Hand
- Write H₀ and H₁ and decide the tail before looking at the data.
- Compute the test statistic from the sample.
- Find the area beyond the statistic under the right distribution, using a table, software or the formulas here.
- For a two tailed z or t test, double the one tail area.
- Reject H₀ if p is less than α.
Worked example: z = 2.1
Φ(2.1) = 0.982136, so the right tail is 1 − 0.982136 = 0.017864. The two tailed p value is 2 × 0.017864 = 0.035729. That is below 0.05, so H₀ is rejected at α = 0.05 but not at α = 0.01.
Worked Examples for Each Test
One sample t test
A sample of 16 parts has a mean length of 104.5 mm and a standard deviation of 8 mm. To test H₀: μ = 100, compute t = (104.5 − 100) ÷ (8 ÷ √16) = 4.5 ÷ 2 = 2.25 with 16 − 1 = 15 df. The two tailed p value is 0.0399, below 0.05.
z test for a proportion
A coin lands heads 560 times in 1,000 flips. Under H₀: p = 0.5 the standard error is √(0.5 × 0.5 ÷ 1000) = 0.015811, so z = (0.56 − 0.5) ÷ 0.015811 = 3.795. The two tailed p value is 0.000148.
A textbook case from OpenStax: a sample of 36 from a population with σ = 1 has a mean of 12.5, testing H₀: μ ≤ 12 against μ > 12. Then z = 0.5 ÷ (1 ÷ 6) = 3, and the right tailed p value is 0.00135.
Chi square test of independence
A 2 × 2 table has observed counts 20, 30, 30 and 20, and every expected count is 25. Each cell adds (5)² ÷ 25 = 1, so χ² = 4.0 with (2 − 1) × (2 − 1) = 1 df. The right tailed p value is 0.0455.
F test from a one way ANOVA
Three groups of 10 people give F = 3.35. The numerator df is 3 − 1 = 2 and the denominator df is 30 − 3 = 27. The right tailed p value is 0.0502, just above 0.05, and the critical value F(2, 27) at α = 0.05 is 3.354.
Critical Values for z and t
Two tailed critical values. Reject H₀ when |statistic| is greater than the value.
| df | α = 0.10 | α = 0.05 | α = 0.01 |
|---|---|---|---|
| 5 | 2.015 | 2.571 | 4.032 |
| 10 | 1.812 | 2.228 | 3.169 |
| 15 | 1.753 | 2.131 | 2.947 |
| 20 | 1.725 | 2.086 | 2.845 |
| 30 | 1.697 | 2.042 | 2.750 |
| 60 | 1.671 | 2.000 | 2.660 |
| z (normal) | 1.645 | 1.960 | 2.576 |
The same numbers are one tailed critical values at half the α: 1.645 is the one tailed z value for α = 0.05.
Critical Values for Chi Square
Upper tail critical values. Reject H₀ when χ² is greater than the value.
| df | α = 0.05 | α = 0.01 |
|---|---|---|
| 1 | 3.841 | 6.635 |
| 2 | 5.991 | 9.210 |
| 3 | 7.815 | 11.345 |
| 4 | 9.488 | 13.277 |
| 5 | 11.070 | 15.086 |
| 10 | 18.307 | 23.209 |
Every value in both tables was computed with this calculator's functions and matches the NIST/SEMATECH Handbook tables to three decimals.
One Tailed vs Two Tailed P Values
A one tailed test asks whether the parameter is greater than (or less than) a value, so only one tail counts. A two tailed test asks whether it differs in either direction, so both tails count. For symmetric z and t distributions the two tailed p value is exactly twice the one tailed value when the statistic points in the predicted direction.
| Statistic | One tailed p | Two tailed p | At α = 0.05 |
|---|---|---|---|
| z = 2.1 | 0.0179 | 0.0357 | Significant either way |
| t = 1.9, df 20 | 0.0360 | 0.0719 | Significant only one tailed |
| t = 2.25, df 15 | 0.0199 | 0.0399 | Significant either way |
| z = 1.645 | 0.0500 | 0.1000 | Borderline one tailed |
If a one tailed test points the wrong way, for example t = −1.9 with H₁: μ > μ₀, the right tailed p value is 1 − 0.0360 = 0.9640. Use one tail only when an effect in the other direction would lead to the same action as no effect.
How to Interpret a P Value
Compare p with the significance level α you chose before the study. OpenStax gives the rule: if p is less than α, reject H₀ and call the result statistically significant. Otherwise, do not reject H₀. Not rejecting H₀ does not show that H₀ is true.
α is the probability of a Type I error, rejecting H₀ when it is true. α = 0.05 means that, when H₀ is true, 5% of studies will still reach a significant result by chance. Smaller α levels such as 0.01 or 0.001 reduce false positives but need more data to detect real effects.
The American Statistical Association published six principles on p values in 2016. Among them: p values do not measure the probability that the hypothesis is true or that the data came from chance alone, decisions should not rest only on whether p passes a threshold, and a p value does not measure the size or importance of an effect.
Common P Value Mistakes
- Reading p as the chance H₀ is true. p = 0.03 means data this extreme happen 3% of the time if H₀ is true, not that H₀ has a 3% chance of being true.
- Using the wrong degrees of freedom. A two sample t test with equal variances uses n₁ + n₂ − 2, not n − 1. Wrong df change the p value, most noticeably for small samples.
- Doubling a chi square or F p value by default. These tests are right tailed in almost every application. A two tailed chi square is used only for special cases such as testing a variance.
- Choosing the tail after seeing the data. This doubles the real false positive rate.
- Running many tests and reporting the smallest p. With 20 independent tests at α = 0.05, the expected number of false positives is 1 even when every H₀ is true.
- Writing p = 0.000. A p value is never exactly zero. Report p < 0.001 instead.
Related Calculations
Use the Z Score Calculator to turn a raw value into a z score first, and the Statistics Calculator to get the mean and standard deviation that a t statistic needs. The Confidence Interval Calculator gives the range of plausible values that should be reported next to the p value: for a z or t test of a mean, a 95% interval that excludes the null value matches a two tailed p value below 0.05.
Before a study, the Sample Size Calculator estimates how many observations you need for a target margin of error or for enough power to detect a difference between two groups. The Probability Calculator covers the basic probability rules behind tail areas.
Frequently Asked Questions
What does a p value tell you?
A p value is the probability of getting a test statistic at least as extreme as the one you observed, assuming the null hypothesis is true. A small p value means your data would be unusual if the null hypothesis were correct, so it counts as evidence against it. For example, z = 2.1 gives a two tailed p value of 0.0357: if there were truly no effect, about 3.6% of samples would produce a z score at least that far from 0. A p value is not the probability that the null hypothesis is true, and it does not measure the size or importance of an effect.
How do you calculate a p value from a z score?
Find the area under the standard normal curve beyond your z score. For a right tailed test, p = 1 − Φ(z). For a left tailed test, p = Φ(z). For a two tailed test, p = 2 × (1 − Φ(|z|)). With z = 2.1, Φ(2.1) = 0.982136, so the right tail is 0.017864 and the two tailed p value is 2 × 0.017864 = 0.035729. With z = 1.96 the two tailed p value is 0.049996, which is why 1.96 is the critical value for α = 0.05. The z test tab of this calculator returns the area for any z score.
How do you find the p value from a t score?
Use Student's t distribution with the correct degrees of freedom, usually n − 1 for a one sample t test. The two tailed p value equals the regularized incomplete beta function I_x(df/2, 1/2) with x = df ÷ (df + t²), and the one tailed p value is half of it when t points in the direction of the alternative. For t = 2.25 with 15 degrees of freedom, x = 15 ÷ 20.0625 = 0.747664, the two tailed p value is 0.0399 and the one tailed p value is 0.0199. A printed t table only brackets the answer, here 0.02 < p < 0.05, while this calculator gives the exact value.
What is the difference between a one tailed and a two tailed p value?
A one tailed p value counts only results in the direction your alternative hypothesis predicts, such as greater than. A two tailed p value counts extreme results in both directions, so for z and t tests it is twice the one tailed value. With t = 1.9 and 20 degrees of freedom, the right tailed p value is 0.0360 and the two tailed p value is 0.0719, so the result is significant at α = 0.05 only with a one tailed test. Choose the tail before you see the data. Picking the tail after seeing which way the data point gives a false positive rate of 2α instead of α.
Is a p value of 0.05 significant?
It depends on the decision rule. Under the rule taught in textbooks such as OpenStax Introductory Statistics, you reject the null hypothesis when the p value is less than α, so p = 0.05 exactly is not significant at α = 0.05, while p = 0.049 is. In practice the difference between 0.049 and 0.051 is tiny, and the American Statistical Association states that conclusions should not be based only on whether a p value passes a specific threshold. Report the exact p value together with the effect size and a confidence interval so readers can judge the result.
What is the p value for a chi square test?
A chi square test uses the right tail of the chi square distribution, because a larger statistic means the observed counts sit further from the expected counts. The p value is Q(df/2, χ²/2), the regularized upper incomplete gamma function. For a 2 × 2 table with observed counts 20, 30, 30 and 20 and an expected count of 25 in every cell, χ² = 4.0 with 1 degree of freedom without the continuity correction, so p = 0.0455. For χ² = 7.81 with 3 degrees of freedom, p = 0.0501. Degrees of freedom are (rows − 1) × (columns − 1) for a contingency table.
Sources & References
- American Statistical Association. Statement on Statistical Significance and P Values (six principles), March 7, 2016. amstat.org
- OpenStax. Introductory Statistics 2e, 9.4 Rare Events, the Sample, Decision and Conclusion (p value definition, decision rule, z = 3 example). openstax.org
- OpenStax. Introductory Statistics 2e, 9.3 Probability Distribution Needed for Hypothesis Testing (when to use z or t). openstax.org
- NIST/SEMATECH. Handbook of Statistical Methods, 1.3.6.7.2 Critical Values of the Student's t Distribution. itl.nist.gov
- NIST/SEMATECH. Handbook of Statistical Methods, 1.3.6.7.4 Critical Values of the Chi Square Distribution. itl.nist.gov
- NIST/SEMATECH. Handbook of Statistical Methods, 1.3.6.7.3 Critical Values of the F Distribution. itl.nist.gov
- NIST. Digital Library of Mathematical Functions, §8.2 Incomplete Gamma Functions and §8.17 Incomplete Beta Functions (definitions and continued fraction used here). dlmf.nist.gov/8.2, dlmf.nist.gov/8.17
Results are for educational purposes. Every number on this page was recomputed with the same functions the calculator uses. Methodology reviewed: 2026.