How to Use the P Value Calculator
This calculator allows you to find the p-value when you already have a test statistic. To use the calculator:
- Choose the test statistic: Z Score, T Statistic, F Statistic, Chi-Square, or Pearson r.
- Enter the required values. Some statistics also require degrees of freedom or sample size.
- Select the appropriate tail for your hypothesis test.
- Click Calculate.
The calculator will instantly return the p-value for the values and tail you selected.
For convenience, the calculator automatically selects the most common tail when you change the statistic. The following p-values are selected by default.
- Z Score: Two-tailed
- T Statistic: Two-tailed
- F Statistic: Right-tailed
- Chi-Square: Right-tailed
- Pearson r: Two-tailed
You can change the tail whenever your alternative hypothesis requires a different one.
What Is a P-Value?
A p-value tells us how unusual the observed test result would be if the null hypothesis, H₀, were true. More specifically, it is the probability of obtaining a test statistic at least as extreme as the one observed, in the direction specified by the alternative hypothesis.
A small p-value provides stronger evidence against the null hypothesis, while a large p-value means the observed result is more consistent with what could occur if H₀ were true.
Note. The p-value is not the probability that the null hypothesis is true.
Which Test Statistic Should I Choose?
Choose the statistic produced by your hypothesis test or statistical output.
| Select | Enter | Common uses |
|---|---|---|
| Z Score | Z statistic | Z tests, proportion tests, and other tests based on the standard normal distribution |
| T Statistic | T statistic and degrees of freedom | One-sample, paired, two-sample, and Welch’s t tests |
| F Statistic | F statistic, numerator df, and denominator df | ANOVA, regression F tests, and some variance tests |
| Chi-Square | χ² statistic and degrees of freedom | Goodness-of-fit, independence, and homogeneity tests |
| Pearson r | Correlation coefficient and sample size | Testing the significance of a Pearson correlation |
If you only need a quick p-value, use the calculator above. However, if you want to see how the probability is calculated, use one of the dedicated p-value calculators below.
- P-Value From Z Calculator. Finds left-tailed, right-tailed, or two-tailed p-values from a z score.
- P-Value From T Calculator. Calculates a p-value from a t statistic and degrees of freedom.
- Chi-Square P-Value Calculator. Finds the p-value from a chi-square statistic and degrees of freedom.
- P-Value From F Calculator. Finds a p-value from an F statistic using numerator and denominator degrees of freedom.
- P-Value From Pearson r Calculator. Calculates the p-value associated with a Pearson correlation coefficient.
Each of the above dedicated tools focuses on a single test statistic and provides an instant answer, with a clear, step-by-step solution of how the p-value was obtained.
How to Choose the Correct Tail
The correct tail depends on your alternative hypothesis, and not on whether the test statistic is positive or negative. Here’s a quick summary to help you choose the correct tail for your hypothesis test problem.
- Left-Tailed Test. Use a left-tailed test when the alternative hypothesis states that the population parameter is less than the value specified under the null hypothesis. For example, if the alternative hypothesis is H₁: μ < 50, the p-value is the probability to the left of the observed test statistic.
- Right-Tailed Test. Use a right-tailed test when the alternative hypothesis states that the population parameter is greater than the value under the null hypothesis. For example, if the alternative hypothesis is H₁: μ > 50, the p-value is the probability to the right of the observed test statistic.
- Two-Tailed Test. Use a two-tailed test when the alternative hypothesis states that the population parameter is different from the null value. For example, if the alternative hypothesis is H₁: μ ≠ 50, results in either direction can provide evidence against the null hypothesis.
Our p-value calculator selects Two-tailed by default for Z, T, and Pearson r because two-sided alternatives are commonly used with these statistics. However, you can still change the selection when your hypothesis specifies a direction.
Why Are F and Chi-Square Right-Tailed by Default?
The calculator automatically selects Right-tailed when you choose F Statistic or Chi-Square statistic.
For common ANOVA and regression F tests, larger F statistics provide stronger evidence against the null hypothesis. Therefore, the p-value is normally found in the right tail of the F distribution.
The same idea applies to chi-square goodness-of-fit and independence tests. A larger χ² statistic represents a greater difference between the observed results and what would be expected under the null hypothesis, so these tests normally use a right-tailed p-value.
Note. These are defaults rather than restrictions. The calculator still allows you to select another tail when the statistical procedure you are using requires it.
What Inputs Does the Calculator Need?
The inputs change depending on the test statistic you select.
- Z Score: Enter the z statistic from your hypothesis test.
- T Statistic: Enter the t statistic and the corresponding degrees of freedom (df).
- F Statistic: Enter the F statistic, numerator degrees of freedom (df₁), and denominator degrees of freedom (df₂).
- Chi-Square: Enter the chi-square statistic (χ²) and the corresponding degrees of freedom (df).
- Pearson r: Enter the Pearson correlation coefficient (r) and the sample size (n).
After entering the required values, select the tail that matches your alternative hypothesis and click Calculate to get the p-value.
Not sure what degrees of freedom to use? You can calculate them with our Degrees of Freedom Calculator.
How to Interpret a P-Value
Once you have the p-value, compare it with the significance level, α, specified for your hypothesis test.
The usual decision rule is:
- If p < α, reject H₀.
- If p ≥ α, fail to reject H₀.
For example, suppose you get a p-value of 0.032 and the significance level is α = 0.05. In this case, we would reject the null hypothesis since the p-value (0.032) is less than the significance level (0.05) and conclude that the result is statistically significant at the 5% significance level.
Tip. You should always compare the p-value with the significance level specified for your test rather than automatically assuming α = 0.05. Common significance levels include α = 0.10, α = 0.01, and α = 0.05.
What Does a Small P-Value Mean?
A small p-value means that the observed test statistic, or a more extreme result in the direction of the alternative hypothesis, would be relatively unlikely if the null hypothesis were true. This provides evidence against H₀.
However, a small p-value does not tell you how large or important the effect is. For example, a very large sample may produce a small p-value for a difference that is relatively small in practical terms. When possible, interpret the p-value together with an effect size, confidence interval, and the context of the study.
P-Value vs. Critical Value
The p-value approach and critical-value approach are two ways of making a hypothesis-test decision. With the p-value approach, you compare the p-value with the significance level α. However, with the critical-value approach, you compare the observed test statistic with the appropriate critical value.
When the same hypothesis and significance level are used correctly, both approaches should lead to the same statistical decision.
Frequently Asked Questions
A p-value calculator finds the probability associated with a test statistic. To use the calculator, choose the statistic, enter the required values, select the appropriate tail, and the calculator returns the corresponding p-value.
You need the test statistic and the probability distribution it follows. Some statistics also require degrees of freedom. For Pearson r, this calculator uses the correlation coefficient and sample size.
Look at the alternative hypothesis. Use a left-tailed test for a “less than” alternative, a right-tailed test for a “greater than” alternative, and a two-tailed test when the hypothesis tests for a difference in either direction.
The p-value can change when the distribution, degrees of freedom, or selected tail changes. That is why you should enter all required values and select the tail that matches your hypothesis.
No. A small p-value provides evidence against the null hypothesis, but it does not prove that H₀ is false.
