P-Value Calculator
Turn a z, t or chi-square statistic into a p-value, one- or two-tailed, with the critical value at your alpha and a plain-English verdict on what it means.
P-Value Calculator: with the default inputs, p-value is 0.049996.
The z, t or χ² value your test produced. χ² is never negative.
Used by t and χ² only. For a one-sample t-test this is n − 1.
Decide this before you see the data. χ² goodness-of-fit and independence tests are upper-tailed.
The false-positive rate you are willing to accept. 5% is conventional, not sacred.
- p-value, written out
- p = 0.049996
- Critical value at α
- 1.96
- Reject the null when
- |z| > 1.96
- Significant at α?
- Yes
- What that means
- Reject the null hypothesis at α = 5%. If the null were true, a z at least this extreme would turn up about once in every 20 studies — rare enough, by the standard you set, to doubt it.
Assumptions
- The test statistic really does follow the named distribution under the null — that is an assumption about your study design, not something this calculator can check.
- Two-tailed p is twice the smaller tail, which is the standard convention for symmetric distributions and the usual one for chi-square.
- α is a threshold you fixed in advance; comparing p against several α values after the fact inflates the false-positive rate.
| α | Critical value | Your result |
|---|---|---|
| 10% | 1.6449 | significant |
| 5% | 1.96 | significant |
| 1% | 2.5758 | not significant |
| 0.1% | 3.2905 | not significant |
The p-value does not change — only the threshold you compare it against. Choosing α after seeing p defeats the point of having one.
How this is worked out
The formula
p is the probability, assuming the null hypothesis is true, of a test statistic
at least as extreme as the one observed.
z: upper tail = 1 − Φ(z) Φ = standard normal CDF
t: P(|T| > |t|) = I_{df/(df+t²)}(df/2, ½) I = regularized incomplete beta
χ²: upper tail = 1 − P(k/2, χ²/2) P = regularized lower incomplete gamma
two-tailed p = 2 × min(upper tail, lower tail)
one-tailed p = the tail you named in advance
Critical value = the statistic whose tail area equals α. Reject when |statistic| exceeds it,
which is exactly the same decision as p ≤ α.Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Test statistic type
- Choose one of 3 options.z — standard normal · t — Student's t (needs degrees of freedom) · χ² — chi-square (needs degrees of freedom)
- Test statistic
- The z, t or χ² value your test produced. χ² is never negative.defaults to 1.96
- Degrees of freedom
- Used by t and χ² only. For a one-sample t-test this is n − 1.from 1 to 1000000 · whole numbers only · defaults to 10
- Tails
- Decide this before you see the data. χ² goodness-of-fit and independence tests are upper-tailed.Two-tailed (difference in either direction) · One-tailed, upper (greater than) · One-tailed, lower (less than)
- Significance level α
- The false-positive rate you are willing to accept. 5% is conventional, not sacred.a percentage · from 0.01 to 25 · defaults to 5
What you get back
- p-valuemain answer
- p-value, written out
- Critical value at α
- Reject the null when
- Significant at α?
- What that means
What this assumes
- The test statistic really does follow the named distribution under the null — that is an assumption about your study design, not something this calculator can check.
- Two-tailed p is twice the smaller tail, which is the standard convention for symmetric distributions and the usual one for chi-square.
- α is a threshold you fixed in advance; comparing p against several α values after the fact inflates the false-positive rate.
About this calculator
A p-value converts a test statistic into a single probability, and this calculator does that conversion for the three distributions that cover most of applied statistics: the standard normal (z), Student's t, and chi-square. It also reports the critical value at your chosen significance level, which is the same decision expressed the other way round.
What a p-value is
It is the probability of getting a test statistic at least as extreme as the one you got, if the null hypothesis were true. That is all. A two-tailed p of 0.05 for z = 1.96 means: if there were genuinely no effect, results this far from zero in either direction would show up about one time in twenty.
What a p-value is not
This is the most misread number in science, so it is worth being blunt.
- It is not the probability that the null hypothesis is true. It is computed assuming the null is true; it cannot then tell you the odds on that assumption. Getting from p to "the probability there is no effect" needs a prior, and that is Bayesian territory.
- It is not the probability that your result is a fluke, which is the same error in casual clothing.
- It is not a measure of effect size. With 10,000 participants a clinically meaningless difference will produce p < 0.001. With 10 participants a huge effect may produce p = 0.3. Always report the effect and its confidence interval alongside.
- It is not 1 minus the probability of replication. A p of 0.05 does not mean a 95% chance the study repeats.
- p > 0.05 is not evidence of no effect. Absence of evidence is not evidence of absence; it usually means the study was underpowered. To argue for no effect you need an equivalence test or a confidence interval that excludes anything meaningful.
The American Statistical Association published a formal statement on exactly these points in 2016 after decades of misuse; it is linked in the sources.
One tail or two
A two-tailed test asks "is there a difference?", a one-tailed test asks "is it bigger?" — and the one-tailed p is half the size, which is precisely why the choice must be made before you look at the data. Picking the tail after seeing which way the result went doubles your real false-positive rate while the printed p-value pretends otherwise. Chi-square goodness-of-fit and independence tests are naturally upper-tailed, because only a large χ² means a bad fit.
Reading the outputs
The critical value is the statistic that would give exactly p = α, so comparing your statistic against it and comparing p against α are the same test. The table repeats that comparison at 10%, 5%, 1% and 0.1% — a reminder that 0.05 is a convention Ronald Fisher offered as a rough guide, not a law of nature, and that a p of 0.049 and a p of 0.051 are the same result.
Numbers come from the exact distribution functions (a high-precision normal CDF, the regularized incomplete beta for t, the regularized incomplete gamma for χ²), not from an interpolated table, so they match printed statistical tables to every digit those tables print.
Frequently asked questions
▸What is the p-value for z = 1.96?
0.05 two-tailed (0.049995 to be exact), or 0.025 one-tailed. That is where the familiar 95% confidence interval comes from.
▸Does p < 0.05 mean my result is important?
No. It means the result is unlikely under the null hypothesis, which is a statement about surprise, not about size. A tiny, useless effect becomes significant with a large enough sample.
▸Does p = 0.20 prove there is no effect?
No. It means the data are compatible with the null, and usually also compatible with a substantial effect. Look at the confidence interval: if it stretches from 'harmful' to 'helpful', the study simply could not tell.
▸Should I use one tail or two?
Two, unless you committed to a direction before collecting data and genuinely would not act on a result in the opposite direction. Halving a p-value by switching tails after the fact is a well-known way to manufacture significance.
▸Why is the chi-square test upper-tailed?
Because χ² measures how far observed counts sit from expected counts, always as a sum of squares. Only a large value means a poor fit; a small one means the data match the model, which is what the null predicts.
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