🧮 P-Value Calculator: z, t, and Chi-Square Tests
By Shihab Mia · Updated 2026-06-27
This p-value calculator converts a test statistic into a p-value so you can judge statistical significance in seconds. Choose the standard normal (z), Student t, or chi-square distribution, pick a left, right, or two-tailed test, enter your statistic (and degrees of freedom for t and chi-square), and read the exact p-value to 6 significant figures. It also compares the result against your chosen alpha and tells you whether the finding is statistically significant.
What is the P-Value Calculator?
A p-value is the probability of observing a test statistic at least as extreme as the one you got, assuming the null hypothesis is true. It is not the probability that the null hypothesis is correct, and it is not the chance your result happened by luck. It is a conditional probability: given that nothing is going on, how surprising is this data? Small p-values mean the data would be unlikely under the null hypothesis, which is why a small p-value counts as evidence against it.
The distribution you pick depends on your test. Use the z (standard normal) distribution for large-sample tests of proportions or means when the population standard deviation is known. Use the Student t distribution for means estimated from small samples, where you also supply the degrees of freedom (often n minus 1). Use the chi-square distribution for goodness-of-fit and tests of independence, with degrees of freedom set by the table dimensions. Each distribution turns the same statistic into a different tail probability, so picking the right one matters.
The tail also matters. A two-tailed test asks whether the parameter differs from the null value in either direction and splits the significance across both tails. A one-tailed (left or right) test looks for a difference in a single, pre-specified direction and puts all of alpha in that one tail. Decide the direction before you see the data, never after, or you inflate your false-positive rate. For a symmetric z or t statistic, the two-tailed p-value is simply twice the smaller one-tailed p-value.
Under the hood this tool uses standard numerical methods so the answers match statistical software. The normal CDF comes from the Abramowitz and Stegun error-function approximation. The Student t p-value uses the regularized incomplete beta function evaluated with a Lentz continued fraction. The chi-square right-tail p-value uses the regularized incomplete gamma function via its series and continued-fraction forms. The result is reported to 6 significant figures and checked against your alpha so you get both the number and the verdict.
When to use it
- Converting a z-score from a proportion or large-sample mean test into a one or two-tailed p-value.
- Finding the p-value for a Student t statistic from a small-sample t-test when you know the degrees of freedom.
- Getting the right-tail p-value for a chi-square goodness-of-fit or independence test.
- Deciding whether a result clears your significance threshold by comparing the p-value with alpha (default 0.05).
How to use the P-Value Calculator
- Pick the distribution that matches your test: z-score, t-score, or chi-square.
- For z and t, choose left-tailed, right-tailed, or two-tailed; chi-square uses the right tail.
- Enter your test statistic, and for t and chi-square also enter the degrees of freedom (df greater than 0).
- Read the p-value to 6 significant figures and the significant or not significant verdict against your alpha.
Formula & method
Worked examples
A two-tailed z-test gives a test statistic of z = 1.96. Find the p-value at alpha = 0.05.
- Choose the z-score distribution and a two-tailed test.
- Compute the upper-tail area: 1 - Phi(1.96), which is about 0.025.
- Double it for two tails: p = 2 x 0.025 = 0.0499958.
- Compare with alpha: 0.0499958 is just below 0.05.
Result: p is about 0.05, right at the threshold, so the result is (barely) statistically significant.
A one-sample t-test produces t = 2.5 with 10 degrees of freedom. Find the two-tailed p-value.
- Choose the t-score distribution, two-tailed, and enter df = 10.
- Form x = df / (df + t squared) = 10 / (10 + 6.25) = 0.6154.
- Two-tailed p = I_x(df/2, 1/2) = I_0.6154(5, 0.5).
- Evaluate the regularized incomplete beta to get p is about 0.0314.
Result: p is about 0.0314, which is below 0.05, so the result is statistically significant.
Common critical values and the p-values they correspond to
| Test | Statistic | Tail | P-value |
|---|---|---|---|
| z | 1.645 | Right | 0.05 |
| z | 1.96 | Two-tailed | 0.05 |
| z | 2.576 | Two-tailed | 0.01 |
| t (df = 10) | 2.228 | Two-tailed | 0.05 |
| Chi-square (df = 1) | 3.841 | Right | 0.05 |
| Chi-square (df = 2) | 5.991 | Right | 0.05 |
Which distribution and tail to use
| Situation | Distribution | Typical tail |
|---|---|---|
| Large-sample proportion or mean, sigma known | z (normal) | One or two-tailed |
| Small-sample mean, sigma estimated | t (Student) | One or two-tailed |
| Goodness of fit | Chi-square | Right |
| Test of independence (contingency table) | Chi-square | Right |
Common mistakes to avoid
- Reading the p-value as the probability the null is true. A p-value is the probability of data this extreme assuming the null hypothesis holds, not the probability that the null is correct. A p-value of 0.03 does not mean there is a 3 percent chance the null is true.
- Mixing up one-tailed and two-tailed tests. A two-tailed test splits alpha across both directions; a one-tailed test puts it all in one. Using a one-tailed p-value to claim a two-sided effect, or switching after seeing the data, inflates false positives. Decide the direction in advance.
- Using the wrong degrees of freedom. For a one-sample t-test df is usually n minus 1, while a chi-square test of independence uses (rows minus 1) times (columns minus 1). The wrong df shifts the whole p-value. Always confirm df for your specific test.
- Treating p = 0.051 and p = 0.049 as opposites. The 0.05 line is a convention, not a law of nature. Two results on opposite sides of it are barely different. Report the actual p-value and effect size rather than forcing a hard significant or not verdict on a borderline case.
Glossary
- P-value
- The probability of getting a test statistic at least as extreme as the observed one, assuming the null hypothesis is true.
- Null hypothesis
- The default claim of no effect or no difference that a test tries to find evidence against.
- Alpha (significance level)
- The threshold you set in advance (commonly 0.05) below which a p-value is called statistically significant.
- Test statistic
- A single number (z, t, or chi-square) computed from your data that summarizes how far the data is from the null.
- Degrees of freedom (df)
- A parameter that sets the exact shape of the t and chi-square distributions, derived from sample size or table dimensions.
- Two-tailed test
- A test that looks for a difference from the null in either direction and splits the significance across both tails.
Frequently asked questions
How do I calculate a p-value from a z-score?
Convert the z-score to a tail probability with the standard normal CDF. For a right-tailed test the p-value is 1 - Phi(z); for left-tailed it is Phi(z); for two-tailed it is 2 x (1 - Phi(|z|)). This tool does it instantly: pick z-score, choose the tail, and enter your statistic.
What is a statistically significant p-value?
A p-value is statistically significant when it is below your chosen alpha, most often 0.05. A p-value under 0.05 means the data would be unlikely if the null hypothesis were true, so you reject the null at that level. The threshold is a convention, so always report the exact p-value too.
What is the difference between a one-tailed and two-tailed p-value?
A two-tailed test checks for a difference in either direction and splits alpha across both tails, while a one-tailed test looks in a single pre-specified direction and puts all of alpha in that tail. For symmetric z and t statistics, the two-tailed p-value is exactly twice the smaller one-tailed p-value.
How do I find a p-value for a t-test?
Choose the t-score distribution, enter the t statistic and the degrees of freedom (often n minus 1), and pick the tail. The calculator uses the regularized incomplete beta function to return the exact p-value, matching the result you would get from statistical software or a t-table.
Why is the chi-square test always right-tailed?
The chi-square statistic is a sum of squared differences, so larger values mean a bigger mismatch between observed and expected counts. Evidence against the null only appears in the upper (right) tail, so the p-value is the right-tail area Q(df/2, stat/2). Enter the statistic and df to get it.
Does a small p-value prove my hypothesis is true?
No. A small p-value is evidence against the null hypothesis, not proof that your alternative is correct or that the effect is large or important. It says only that the data would be surprising if the null were true. Pair the p-value with an effect size and a confidence interval for a full picture.