Convert a z, t, correlation (r), chi-square or F score into an exact left-tailed, right-tailed or two-tailed p-value. Every result includes the critical value, a decision at the chosen significance level, a report-ready line and a shaded tail diagram. Reverse lookup and batch conversion with multiple-test correction are built in.
Quick Answer: What Does a P Value Calculator Do?
A p value calculator converts a test statistic and its degrees of freedom into a p-value: the probability of a result at least as extreme as the one observed if the null hypothesis is true. A p-value at or below the significance level α, usually 0.05, is called statistically significant.
- Two-tailed p-value (z, t)
- p = 2 × (1 − Φ(|statistic|)), where Φ is the cumulative distribution function.
- Right-tailed p-value
- p = 1 − cdf(statistic). Chi-square and F tests are normally right-tailed.
- Degrees of freedom
- One-sample t: n − 1. Correlation r: n − 2. Chi-square independence: (rows − 1)(columns − 1).
- Decision rule
- Reject the null hypothesis when p ≤ α. Otherwise there is not enough evidence to reject it.
- Common cutoffs
- |z| = 1.96 gives two-tailed p = 0.05. |z| = 2.58 gives p = 0.01.
Last updated: October 28, 2026
What Is a P Value?
A p-value is the probability of seeing a test statistic at least as extreme as yours if the null hypothesis were true. The alternative hypothesis decides what "extreme" means: far left, far right, or far in either direction. A small p-value says your data would be surprising in a world where nothing is going on. It is not the probability that the null is true, and it says nothing about how large an effect is.
Two-tailed p-value = shaded area in both tails.
| P-value | Plain reading |
|---|---|
| 0.001 or lower | Very strong evidence against the null |
| 0.001 to 0.01 | Strong evidence |
| 0.01 to 0.05 | Moderate evidence; significant at the usual 5% level |
| 0.05 to 0.10 | Weak, suggestive evidence |
| Above 0.10 | Data fit the null well enough |
Which Test Statistic and Degrees of Freedom Apply?
A wrong degrees-of-freedom value or tail is the most common cause of a wrong p-value. The table lists the statistic each test produces, its degrees of freedom and the usual tail.
| Statistic | Comes from | Degrees of freedom | Usual tail |
|---|---|---|---|
| Z | Mean or proportion test, large sample or known σ | None | Two (or one if directional) |
| t | One-sample, paired or two-sample mean test | n − 1 (one sample or paired); Welch formula for unequal groups | Two |
| r | Pearson correlation | n − 2 | Two |
| Chi-square | Goodness of fit, independence, variance test | k − 1, or (rows − 1)(columns − 1), or n − 1 | Right |
| F | ANOVA, regression, variance ratio | Between: k − 1. Within: N − k | Right |
P Value Formulas for Z, t, r, Chi-Square and F
Let S be the test statistic, x its observed value, and cdf the cumulative distribution function under the null.
Left-tailed: p = cdf(x)
Right-tailed: p = 1 − cdf(x)
Two-tailed, symmetric (z, t): p = 2 × (1 − cdf(|x|))
Two-tailed, skewed (χ², F): p = 2 × min(cdf(x), 1 − cdf(x))
Correlation to t: t = r × √(df / (1 − r²)), df = n − 2
Z from a sample: z = (x̄ − μ₀) / (σ / √n)
t from a sample: t = (x̄ − μ₀) / (s / √n), df = n − 1
Right-tailed: p = 1 − cdf(x)
Two-tailed, symmetric (z, t): p = 2 × (1 − cdf(|x|))
Two-tailed, skewed (χ², F): p = 2 × min(cdf(x), 1 − cdf(x))
Correlation to t: t = r × √(df / (1 − r²)), df = n − 2
Z from a sample: z = (x̄ − μ₀) / (σ / √n)
t from a sample: t = (x̄ − μ₀) / (s / √n), df = n − 1
Critical Values at a Glance
A statistic beyond the critical value has a p-value below α. Values below are two-tailed for z and t, and right-tailed for chi-square.
| Distribution | α = 0.05 | α = 0.01 |
|---|---|---|
| Z (normal) | 1.960 | 2.576 |
| t, df = 5 | 2.571 | 4.032 |
| t, df = 10 | 2.228 | 3.169 |
| t, df = 20 | 2.086 | 2.845 |
| t, df = 30 | 2.042 | 2.750 |
| Chi-square, df = 1 | 3.841 | 6.635 |
| Chi-square, df = 2 | 5.991 | 9.210 |
| Chi-square, df = 3 | 7.815 | 11.345 |
| Chi-square, df = 5 | 11.070 | 15.086 |
| Chi-square, df = 10 | 18.307 | 23.209 |
How Do You Use the P Value Calculator?
- Choose the tab that matches your statistic. No statistic yet? Open "Build it from your sample" on the Z or t tab.
- Set α above the form. Decide the tail from your hypothesis before you look at results.
- Press the button. Read the p-value, the verdict, the critical value and the shaded diagram.
- Copy the report line into your paper or notes.
- Use P to Statistic to find the cutoff for a target p-value, and Batch to convert many results at once.
P Value Examples with Answers
Z-score of 2.10, two-tailed
Φ(2.10) = 0.9821, so each tail is 0.0179 and p = 0.0357. Significant at 0.05, not at 0.01.t = 2.45 with 14 df, two-tailed
The 0.05 cutoff for 14 df is 2.145 and the 0.02 cutoff is 2.624, so the answer sits between: p ≈ 0.028.Correlation r = 0.45 with n = 20
df = 18 and t = 0.45 × √(18 / 0.7975) = 2.14. Two-tailed p ≈ 0.046. A moderate correlation passes only barely with 20 pairs.Chi-square 7.81 with 3 df
Right tail only: p ≈ 0.050. This sits right on the 5% line, so state the exact value rather than "significant".F = 3.5 with 2 and 20 df
Right tail: p ≈ 0.0497. One-way ANOVA with 3 groups and 23 subjects would give these df.Calculator Features
| Feature | What it does |
|---|---|
| Five statistics | Z, t, correlation r, chi-square and F, each with its own degrees-of-freedom rule |
| Left, right and two-tailed p | Matches the alternative hypothesis for every statistic |
| Correlation r with n | Computes df = n − 2 and the equivalent t automatically |
| Build from a sample | Creates z or t from the sample mean, hypothesized mean, SD and n |
| P to Statistic | Returns the critical value for a target p-value or α |
| Batch conversion | Converts a list of statistics and adds Holm and Bonferroni adjusted p-values |
| Report line | Writes the result in APA style, such as t(14) = 2.45, p = .028 |
| Tail diagram | Shades the area that equals the p-value |
Common P Value Mistakes
- Reading p as the chance the null is true. It is the chance of data this extreme if the null is true.
- Choosing the tail after seeing the data. Doing so halves p unfairly.
- Using a two-tailed rule on chi-square or F. Those tests are almost always right-tailed.
- Wrong degrees of freedom. A paired t-test uses n − 1 pairs, not 2n − 2.
- Treating 0.049 and 0.051 as different worlds. Report the exact p and the effect size.
- Running many tests unadjusted. Twenty tests at 0.05 give about a 64% chance of at least one false positive. Use the Batch tab.
P Value Calculator FAQs
How do I calculate a p-value from a test statistic?
Pick the distribution, enter the statistic and degrees of freedom, choose the tail, and read the area beyond the statistic. For a right tail that is 1 minus the cumulative probability; for two tails it is double the smaller tail.
How do I find the p-value from a z-score?
Two-tailed p = 2 × (1 − Φ(|z|)). A z of 1.96 gives 0.05; a z of 2.58 gives 0.01. Use the Z tab for exact values.
How do I get a p-value from a correlation coefficient?
Convert r to t with t = r × √(df / (1 − r²)), where df = n − 2, then read the two-tailed t probability. The r tab does both steps.
What does a p-value of 0.05 mean?
If the null hypothesis were true, you would see a result this extreme in about 5 of 100 similar studies. It is a convention, not a law of nature.
What is the difference between one-tailed and two-tailed p-values?
A two-tailed test counts extreme results in both directions. A one-tailed test counts one direction only, so for the same statistic its p-value is half as large when the effect points the predicted way.
Can a p-value be negative or zero?
It cannot be negative because it is a probability. It is never exactly zero either; very small values are shown in scientific notation, and papers report them as p below .001.
Why are chi-square and F tests right-tailed?
Both statistics grow when the data drift away from the null, so only large values count as evidence. Small values mean the data fit the null unusually well.
How do I find the critical value for a p-value?
Use the P to Statistic tab: choose the distribution, tail and degrees of freedom, enter α as the p-value, and read the cutoff.
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References
- Wasserstein, R. L. and Lazar, N. A. (2016). The ASA statement on p-values: context, process, and purpose. The American Statistician, 70(2), 129-133.
- Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
- Welch, B. L. (1947). The generalization of Student's problem when several different population variances are involved. Biometrika, 34, 28-35.
- Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6, 65-70.
- Press, W. H. et al. (2007). Numerical Recipes, 3rd ed. Cambridge University Press (incomplete gamma and beta functions).
- NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6: Probability distributions.
- American Psychological Association (2020). Publication Manual, 7th ed., statistical reporting.