Enter a test statistic and its degrees of freedom to get the p value. Chi-square and F are one-tailed by construction, so the tails setting applies only to z and t.
Show the working, step by step
What the number is
The p value is the area in the tail of a distribution beyond your test statistic. It answers one question: if the null hypothesis were true, how often would data this extreme appear?
Everything difficult about p values comes from that conditional. The probability is computed assuming the null is true. It cannot then tell you how likely the null is — that would require reversing the conditional, which needs a prior and takes you into Bayesian territory.
Which distribution to choose
| Distribution | Use it for | Degrees of freedom |
|---|---|---|
| Normal (z) | Large samples, known population SD, proportions | None |
| Student's t | Means with an estimated SD; regression slopes | n − 1, n₁ + n₂ − 2, or Welch's |
| Chi-square | Counts: goodness of fit, independence | categories − 1, or (r−1)(c−1) |
| F | ANOVA; comparing two variances | Two: numerator and denominator |
Common thresholds
- p < 0.05 — the conventional line for "significant". Arbitrary, and treated with far more reverence than it deserves.
- p < 0.01 — strong evidence.
- p < 0.001 — very strong evidence.
Nothing magical happens between 0.049 and 0.051. Treating one as a discovery and the other as a non-result is the single most criticised habit in applied statistics, and the reason many journals now ask for effect sizes and intervals instead of, or alongside, p values.
What a p value cannot tell you
- How big the effect is. With a large enough sample, a difference too small to care about will be highly significant. Report an effect size — Cohen's d, η², or Cramér's V — alongside it.
- How likely your hypothesis is. p is computed assuming the null; it cannot evaluate it.
- Whether the result will replicate. p values are themselves noisy. A study that produces p = 0.04 can easily produce p = 0.30 on a rerun with the same true effect.
- Whether the analysis was sound. A p value computed on the wrong test, or on the twentieth comparison you tried, is precise and worthless.
Multiple comparisons
Every test at the 5% level carries a 5% false-positive risk, and running many tests compounds it. Twenty independent tests on data with no real effects will produce at least one "significant" result about 64% of the time.
If you are testing several hypotheses, correct for it: Bonferroni (divide the threshold by the number of tests) is the simplest and most conservative, and a false discovery rate procedure is the usual choice when there are many tests. Or ask the whole question in a single test — which is exactly what ANOVA does for several group means.
Report the interval too
A confidence interval contains everything a p value tells you and more. If the interval for a difference excludes zero, the result is significant at the matching level — but the interval also shows the size of the effect and how precisely it has been pinned down. "The difference was 4.2 units (95% CI 1.1 to 7.3)" is a far more useful sentence than "the difference was significant (p = 0.01)".
Common questions
What is a p value?
The probability of getting a result at least as extreme as the one you observed, assuming the null hypothesis is true. It is a statement about the data given a hypothesis — never about the hypothesis given the data.
What does p = 0.05 actually mean?
That if the null hypothesis were true, you would see a result at least this extreme 5% of the time. Nothing more.
It does not mean there is a 95% chance your hypothesis is correct, and it does not mean the effect is large. The 0.05 threshold itself is pure convention — Fisher proposed it as a convenience and later regretted how rigidly it was adopted.
Should I use one tail or two?
Two, unless you can justify otherwise. A two-tailed test asks whether there is a difference in either direction, which is nearly always the honest question.
A one-tailed test halves the p value, so it is tempting — but it is only legitimate if you committed to the direction before seeing the data, and if a result in the opposite direction would genuinely be as uninteresting as no result at all. Switching to one tail after seeing which way the data went is a well-known way to manufacture significance.
Why are chi-square and F tests always one-tailed?
Because both statistics are built from squared quantities, so they cannot be negative and every departure from the null pushes them upward. Only the upper tail carries evidence. This calculator applies that automatically for those two distributions — the tails setting has no effect on them.
What degrees of freedom should I enter?
It depends on the test that produced the statistic. A one-sample or paired t-test uses n − 1; a pooled two-sample test uses n₁ + n₂ − 2; a Welch test gives a fractional value, which is fine to enter here. A chi-square goodness-of-fit test uses categories − 1. An F test needs two: numerator then denominator. A z test needs none.
Is a non-significant result the same as no effect?
No. "Not significant" means the data did not provide enough evidence to rule out chance — which happens both when there is no effect and when the study was too small to detect one. Absence of evidence is not evidence of absence. A confidence interval is far more informative here: it shows the range of effects still consistent with your data.
Related calculators
-
T-test calculator
Get the t statistic from your data directly.
-
Chi-square calculator
Goodness of fit on observed counts.
-
ANOVA calculator
Where the F statistic comes from.
-
Z-score calculator
Standardise a value against the normal curve.