P-Value Calculator (from a z, t, chi-square or F statistic)

Q: What does p = 0.05 mean?

If H₀ were true, a result at least this extreme would occur about 5% of the time by chance.

Q: Is a smaller p-value a bigger effect?

Not necessarily. With a large sample, a tiny effect can have a very small p-value.

Q: When should I use t instead of z?

Use t when the population standard deviation is unknown and estimated from the sample, which is the usual case.

Q: Can a p-value be exactly 0?

Not in theory, but it can be smaller than the displayed precision; report it as, for example, p < 0.0001.

Understanding the p-value calculator — concept and principle

This calculator converts a test statistic (z, t, chi-square or F) into a p-value for a left-, right- or two-tailed test, and tells you whether the result is significant at α = 0.05 and α = 0.01. Use it when software gave you only the statistic, when checking a textbook problem, or when you computed a statistic by hand.

Both tails beyond ±2.1 together hold about 3.6% of the normal curve.

How it is calculated

Left-tailed: p = P(T ≤ t)
Right-tailed: p = P(T ≥ t)
Two-tailed: p = 2 × min(left area, right area), capped at 1

The areas come from the cumulative distribution function of the chosen distribution: the standard normal; Student's t with df degrees of freedom; chi-square with df; or F with numerator and denominator df. These are computed with the regularized incomplete beta and gamma functions. Chi-square and F statistics cannot be negative.

The principle

A hypothesis test starts by assuming the null hypothesis H₀ is true. Under H₀ the test statistic follows a known distribution. The p-value asks: if H₀ were true, how often would chance alone produce a statistic at least this extreme? A small p-value means the observed data would be unusual under H₀, which counts as evidence against it. The tail depends on the alternative hypothesis: a two-tailed test counts extremes in both directions, so the one-tail area is doubled.

Worked example

Default inputs: standard normal (z), statistic 2.1, two-tailed.

Area to the left of z = 2.1: Φ(2.1) ≈ 0.9821.
Area to the right: 1 − 0.9821 ≈ 0.01786.
Two-tailed p = 2 × 0.01786 = 0.03573.
0.03573 < 0.05, so reject H₀ at α = 0.05; 0.03573 > 0.01, so do not reject at α = 0.01.

(The degrees of freedom are ignored for a z statistic.)

Each point is the calculator re-run with only “Test statistic value” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Test statistic value	p-value
0.525	0.5996
1.05	0.2937
1.57	0.1153
2.1	0.03573
2.62	0.008665
3.15	0.001633
4.2	0.00002669

How to read the result

Where does your result fall?

Compare p with the significance level α you chose before the test. p < α: the result is statistically significant and you reject H₀. p ≥ α: you do not reject H₀, which is not the same as proving it true. A p-value is not the probability that H₀ is true, and it says nothing about how large or important the effect is. Report the effect size and a confidence interval alongside it.

Common mistakes

Using a two-tailed p-value for a one-sided hypothesis, or the reverse.
Choosing α after seeing the p-value.
Reading p = 0.04 as "a 4% chance the null is true".
Using a left tail for chi-square or F tests, which are normally right-tailed.
Entering the wrong degrees of freedom for t, chi-square or F.

Keep in mind: A p-value is only valid if the test's assumptions hold (independence, distribution shape, sample design). Running many tests and reporting only the significant ones inflates false positives.

Frequently asked questions

What does p = 0.05 mean?