SigmaCalc

P-Value Calculator

Convert a test statistic into the probability of seeing a result this extreme by chance alone.

z-score

Test type

p-value

0.05

Significant at α = 0.05

What you entered

z-score: 1.96Test type: Two-tailed

1
Area beyond |z| under the normal curve
1 − Φ(1.96)= 0.025
2
Double it for a two-tailed test
2 × 0.025= 0.05

Result

p-value: 0.05

A two-tailed test at z = 1.96 gives a p-value of 0.05 — below 0.05, so the result is statistically significant at the conventional threshold.

P-values from a z-score, one and two tailed

The p-value is the probability of seeing a result at least this extreme if the null hypothesis is true. A two-tailed test counts both directions, so it is exactly double the one-tailed value.

Z-score	One-tailed p	Two-tailed p	Significant at 0.05?
1.000	0.1587	0.3173	No
1.500	0.0668	0.1336	No
1.645	0.0500	0.1000	One-tailed only
1.960	0.0250	0.0500	Yes, exactly at 0.05
2.000	0.0228	0.0455	Yes
2.580	0.0049	0.0099	Yes, also at 0.01
3.000	0.0013	0.0027	Yes

The 1.645 and 1.96 rows are the conventional thresholds - 1.645 gives exactly 0.05 one-tailed and 1.96 gives exactly 0.05 two-tailed. Choosing a one-tailed test doubles your apparent significance, which is why the choice has to be made before seeing the data rather than after. A p-value is not the probability that the null hypothesis is true, and it says nothing about effect size: with a large enough sample a trivial difference produces a tiny p. Report the effect and its confidence interval alongside it.

One-tailed vs. two-tailed

A one-tailed test only asks whether a result is significantly higher (or lower) than expected in one direction. A two-tailed test asks whether it's significantly different in either direction — and roughly doubles the p-value for the same z-score.

p < 0.05 isn't magic

The 0.05 threshold is a widely used convention, not a law of nature — a p-value just below 0.05 and one just above it aren't meaningfully different in strength of evidence. Treat it as one input into a decision, not a strict pass/fail line.

Frequently asked questions

My experiment gave p = 0.03. What does that actually mean?

There's a 3% probability of seeing results this extreme (or more extreme) if the null hypothesis were true — i.e., if there were really no effect. It does NOT mean there's a 97% chance your hypothesis is correct. P-values measure evidence against the null, not evidence for your theory.

What's the difference between one-tailed and two-tailed p-values?

One-tailed: tests whether the effect goes in a specific direction (is the new drug BETTER?). Two-tailed: tests for any difference in either direction (is the new drug DIFFERENT?). Two-tailed p-value is roughly 2× the one-tailed. Use two-tailed unless you have a strong prior reason to test only one direction.

Is p = 0.049 really that different from p = 0.051?

No — they represent nearly identical evidence. The 0.05 threshold is a convention, not a natural boundary. Many scientists now advocate reporting exact p-values and effect sizes rather than just 'significant' or 'not significant'. A large effect with p = 0.06 can be more meaningful than a tiny effect with p = 0.04.