Hypothesis tests
P-Value Calculator
Turn a test statistic into a p-value — and see exactly which area of which distribution that number is.
How to read this result
A p-value is the probability of data at least this extreme, if the null hypothesis were true. Every word matters. It is conditional on the null, it is about the data rather than the hypothesis, and "at least this extreme" is what makes it an area rather than a point.
The four things it is not. Not the probability that the null hypothesis is true. Not the probability that your result is a fluke. Not one minus the probability that the alternative is true. And not a measure of how large or important the effect is.
p = 0.049 and p = 0.051 are the same evidence. The 0.05 line is a convention Fisher suggested in 1925, not a property of the world. Report the number itself and let the reader judge, rather than converting a continuum into a verdict.
One-tailed or two? Two-tailed unless you fixed the direction in advance and genuinely would have ignored a result in the other direction. Switching after seeing the data doubles your real false-positive rate, and it is visible to anyone who looks.
The formula
For the symmetric distributions — z and t — the two-tailed p-value is twice the one-tailed one. Chi-square and F are one-tailed by construction: both statistics grow under any departure from the null, in either direction.
Worked example
The most famous critical value in statistics
A z statistic of 1.96 is the number every introductory course memorises. What p-value does it correspond to, and why that number?
- The standard normal distribution has 2.5% of its area above 1.96.P(Z > 1.96) = 0.025
- Two-tailed, so double it — the same area sits below −1.96.p = 2 × 0.025 = 0.050
- Which is where the convention comes from: 1.96 is the value that leaves exactly 5% in the two tails combined.the exact value is 1.959964, not 1.96
Note that 1.96 is a rounding of 1.959964. A statistic of exactly 1.96 gives p = 0.04999, which is under 0.05 — one of the few places where two decimal places changes a conclusion.
Checked against the standard normal distribution — 2 * pnorm(-1.959964) in R.
The calculator above is loaded with these numbers by the Load the worked example button.
Assumptions, and when to use something else
This calculator converts a statistic into a probability. It assumes you have already computed the statistic correctly and that its null distribution is the one you selected — which is where the real assumptions live, in the test that produced the number.
The degrees of freedom are the part most often entered wrongly: n − 1 for a one-sample t, n₁ + n₂ − 2 for a pooled two-sample t, a fractional Welch-Satterthwaite value for the unequal variance version, (r − 1)(c − 1) for a contingency table, and k − 1 and N − k for the two F arguments in an ANOVA.
- You have raw data rather than a statisticT-testPaste the data and the whole test is computed, statistic and p-value together.
- You have counts in categoriesChi-square testComputes χ² and its p-value from the table directly.
- You want the critical value insteadt distribution calculatorGives critical values and tail probabilities for any degrees of freedom.
- The p-value is not significant and you want to know whyPower calculatorDistinguishes “no effect” from “not enough data to see it”.
Questions people ask
What does p < 0.05 actually mean?
That data at least this extreme would occur less than 5% of the time if the null hypothesis were true. It is a statement about how surprising the data are under an assumption — not about how likely the assumption is.
Is a smaller p-value a bigger effect?
No. p depends on the effect size and the sample size. A trivial difference becomes highly significant with enough data, and a large difference can be non-significant with too little. Report an effect size and a confidence interval alongside it.
Should I use a one-tailed or two-tailed test?
Two-tailed, unless you committed to a direction before collecting data and would have reported “no effect” had the result gone the other way. One-tailed tests halve the p-value, which is exactly why they are so often chosen after the fact.
What is the difference between p and alpha?
α is the threshold you set in advance — the false-positive rate you will tolerate. p is what the data produced. The comparison p ≤ α is the decision rule; α does not change with the data and p does not exist before them.
Why is my chi-square p-value one-tailed?
Because χ² and F are sums of squares: any departure from the null in any direction makes them larger, never smaller. All the evidence against the null lives in the upper tail, so there is no second tail to consider.