P-value Calculator
Hypothesis Testing Tool
For a large sample or a known population standard deviation.
For a small sample when the population standard deviation is unknown.
Understanding P-values
A p-value translates a test statistic into a single probability, telling you how surprising your data would be if there were truly no effect — the backbone of statistical hypothesis testing.
The p-value is the probability, under the null hypothesis, of observing a test statistic at least as extreme as the one you calculated. It is a measure of surprise, not proof — a small p-value means your data would be unusual if the null hypothesis were true, which is evidence against it, but it says nothing about how large or important an effect is.
A two-tailed test checks for a difference in either direction and is the safer default. A one-tailed test only checks one specific direction, which produces a smaller p-value for the same score — it should be chosen for a principled reason before seeing the data, never after.
The Formulas
From a Z-Score
From a T-Score (df degrees of freedom)
Where Φ is the standard normal CDF, Ft is the Student's t CDF, and k = 1 for a one-tailed test or k = 2 for a two-tailed test.
Interpreting the Result
Most fields use a conventional significance threshold — commonly α = 0.05 — chosen before the analysis. If your p-value falls below that threshold, the result is typically labeled "statistically significant" and the null hypothesis is rejected; otherwise, you fail to reject it. This does not mean the null hypothesis is proven true when the p-value is large, only that the data didn't provide strong enough evidence against it. The Z-Score tab is appropriate for large samples or a known population standard deviation, while the T-Score tab accounts for the extra uncertainty of small samples via the degrees of freedom you supply.
The most widely used significance level, though it's a convention, not a universal rule.
A two-tailed p-value is simply double the one-tailed value for the same score.
Lower df widens the t-distribution's tails, producing larger p-values than the normal approximation.
Key Takeaways
- A p-value measures surprise under the null hypothesis, not the probability that the null hypothesis is true.
- Choose one-tailed vs. two-tailed before seeing your data, based on whether only one direction of effect is meaningful.
- Use the T-Score tab for small samples — the t-distribution's heavier tails account for the extra uncertainty of estimating variability from limited data.
- Need the underlying z-score or a confidence interval? The Z-score Calculator and Confidence Interval Calculator cover those calculations directly.
Frequently Asked Questions
- Pick the Z-Score tab if you already have a z-score (large sample or known population standard deviation), or the T-Score tab if you have a t-score and need to supply the degrees of freedom.
- Enter the score (and degrees of freedom, for the T-Score tab).
- Choose One-Tailed or Two-Tailed from the dropdown.
- Click "Calculate" to see the resulting p-value.
A p-value is the probability of observing a test statistic at least as extreme as the one you calculated, assuming the null hypothesis is true. A small p-value means your observed result would be unlikely if there were really no effect, which is taken as evidence against the null hypothesis. A p-value does not tell you the probability that the null hypothesis is true, nor the size or importance of an effect — it only measures how surprising your data would be under the null.
A two-tailed test asks whether your statistic is unusually far from zero in either direction, and is the right default when you have no strong prior expectation about the direction of an effect. A one-tailed test asks whether your statistic is unusually far in one specific direction only, and should only be used when the opposite direction is truly irrelevant to your question — it produces a smaller p-value for the same score, so it should be chosen before looking at the data, not after, to avoid inflating significance.
Degrees of freedom (df) capture how much information is left over for estimating variability once your sample statistics have been calculated — commonly df = n − 1 for a one-sample t-test with sample size n. Smaller df means a t-distribution with heavier tails than the normal distribution, so the same t-score produces a larger (less significant) p-value than it would under a z-test. As df grows, the t-distribution converges to the standard normal.
A significance threshold (commonly called α, often set at 0.05) is a cutoff chosen before the analysis: if the p-value falls below it, the result is called "statistically significant" and the null hypothesis is rejected. This is only a convention, not a law of nature — 0.05 means you'd expect to see a result this extreme purely by chance about 1 time in 20, even if there's truly no effect, so the threshold should be chosen to fit the stakes of the decision at hand.
Z-score p-values use the standard normal cumulative distribution function (CDF), computed here from a rational approximation to the error function (erf). T-score p-values use the Student's t-distribution CDF, computed via the regularized incomplete beta function — a standard, well-tested numerical technique accurate to roughly six decimal places, more than sufficient for any practical statistics use case.