P-Value Calculator
Calculate the P-value from a Z-score to determine statistical significance.
P-Value Rules
What Is a P-Value?
In statistics, the p-value is the probability of obtaining results at least as extreme as the observed results of a statistical hypothesis test, assuming that the null hypothesis is correct. It serves as a measure of evidence against the null hypothesis: the smaller the p-value, the stronger the evidence that you should reject the null hypothesis.
What Is the Null Hypothesis (H₀)?
The null hypothesis is a general statement or default position that there is no relationship between two measured phenomena, or no association among groups. Testing a hypothesis means you are trying to disprove the null hypothesis. The alternative hypothesis (H₁) is what you might believe to be true or hope to prove true.
Understanding the Significance Level (α)
The significance level, also denoted as alpha or α, is a threshold you set before the test. It is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.
How to Interpret P-Value Results
- P-value ≤ α (Statistically Significant): You have enough evidence to reject the null hypothesis. The results are unlikely to have occurred by chance.
- P-value > α (Not Statistically Significant): You do not have enough evidence to reject the null hypothesis. The results could easily have occurred by random chance.
Frequently Asked Questions
No, this is a common misconception. A p-value of 0.05 means that if the null hypothesis is true, there is a 5% chance of seeing data as extreme as yours. It is a statement about the data given the hypothesis, not a statement about the hypothesis given the data.
A two-tailed test checks for a difference in either direction (greater than or less than). A one-tailed test (left or right) checks for a difference in only one specific direction. You should generally use a two-tailed test unless you have a strong, pre-justified reason to only care about one direction.
If your significance level is 0.05, technically a p-value of 0.051 means you fail to reject the null hypothesis. While it is extremely close to the cutoff, standard scientific practice requires strict adherence to the pre-chosen threshold to avoid confirmation bias.
In theory, a p-value of exactly 0 or 1 is possible but extremely rare in practice. A p-value of 0 would mean the observed result is impossible under the null hypothesis. A p-value of 1 would mean the data perfectly matches what the null hypothesis predicts. Software tools that report p = 0.0000 typically mean the value is smaller than their display precision, such as p < 0.0001.
P-values and confidence intervals are two sides of the same coin. If a two-tailed test yields a p-value ≤ 0.05, the corresponding 95% confidence interval will not include the null hypothesis value (usually 0). Confidence intervals provide more information because they show both the direction and magnitude of the effect, not just whether it is significant.
No. A statistically significant p-value only indicates that the observed data is unlikely to have occurred by chance under the null hypothesis. It does not prove that one variable causes another. Causation requires controlled experimental design (e.g., a randomised controlled trial), not just statistical significance.