What a p-value actually is
A p-value answers one narrow question: if the null hypothesis were true, how often would you see a test statistic at least this extreme, just by chance? Small p means the data would be surprising under the null. Large p means the data look ordinary under the null. That is all it says. It is not the probability that the null is true, and it is not the probability that your result is a fluke in any absolute sense.
Which tail do you need?
- Two-tailed: your alternative says the effect could go either way (different, not specifically larger). This is the default in most published work.
- Right-tailed: the alternative is "greater than". Chi-square and F tests always live here, because their statistics only grow when the data disagree with the null.
- Left-tailed: the alternative is "less than".
The tail is chosen from the research question before looking at the data. Switching tails after seeing the result, hunting for the smaller p, is one of the quiet ways results get exaggerated.
A one-sample t-test on 8 bakery days gives t = 2.679 with 7 degrees of freedom. Enter statistic 2.679, choose the t distribution, df = 7: the two-tailed p-value is 0.0316. Under the null (true mean 40), a |t| this large would occur in only about 3.2% of samples, below the usual 0.05 line, so the result is statistically significant.
Reading p against the significance level
The significance level α is your pre-set bar, most often 0.05. p < α: reject the null. p ≥ α: fail to reject it. Two warnings travel with that simple rule. First, p = 0.049 and p = 0.051 are nearly identical evidence, even though they land on opposite sides of the bar. Second, significance is not importance. With a huge sample, a trivially small effect produces a tiny p. Report the effect size alongside, always.
"p = 0.03 means there is a 3% chance the null hypothesis is true." No. The p-value is computed assuming the null is true; it cannot turn around and give the probability of that assumption. What it gives you is the surprise level of the data under the null: nothing more, and used honestly, that is plenty.
Which distribution does your statistic follow?
z statistics use the standard normal. t statistics use the t distribution with the test's degrees of freedom. Chi-square statistics (goodness of fit, independence, variance tests) use the chi-square distribution. ANOVA and variance-ratio statistics use F with two degrees of freedom. This calculator covers all four. Pick the family, enter the statistic and its df, and read the tail you need.
Frequently asked questions
What is a good p-value?
There is no universally good value. Below your pre-set significance level (often 0.05) counts as statistically significant for that test; above it does not. The p-value is a continuous measure of surprise: treat 0.049 and 0.051 as the near-identical evidence they are.
Why do chi-square and F tests only have a right tail here?
Chi-square and F statistics are built from squared quantities, so they are never negative, and only large values argue against the null. Their p-value is the right-tail area.
Can a p-value be exactly 0?
No. It can be extremely small (reported as p < 0.001), but a continuous distribution always leaves some area in the tail.
What degrees of freedom do I enter?
The df of your test: n − 1 for a one-sample or paired t-test, for example. Each test's page on this site shows its df formula.