Z Score to P Value Calculator

Convert any z statistic into its p-value with this z score to p value calculator — one-tailed or two-tailed — with the table method underneath so you can see exactly where the number comes from and what "significant" does and does not claim.

What a p-value actually is

A p-value is a tail area. It answers one narrow question: if the null hypothesis were true, how often would chance alone produce a test statistic at least as extreme as the one observed? The z table supplies the raw material — Φ(z), the area to the left — and the tail choice assembles it. For a right-tailed test, p = 1 − Φ(z). For a left-tailed test, p = Φ(z). For a two-tailed test, both extremes count: p = 2 × (1 − Φ(|z|)). The calculator above performs the assembly; this page exists so the assembly is never a mystery. If the tail mechanics themselves are new, the how to read a z table guide builds them from the curve up.

The reference points every analyst carries: z = 1.645 gives a one-tailed p of 0.0500 and a two-tailed p of 0.1000; z = 1.96 gives 0.0250 and 0.0500; z = 2.576 gives 0.0050 and 0.0100. Those three pairs explain most published results sections — and why 1.96 and 0.05 are inseparable in the literature. They are collected, with the rest of the standard set, on the critical z values page.

Worked conversion — z = 3.10 from a test statistic

The table gives Φ(3.10) = 0.9990, so the right tail holds 0.0010. If the test was right-tailed, p = 0.0010. If it was two-tailed — the default unless a direction was fixed in advance — p = 2 × 0.0010 = 0.0019 (0.00194 before rounding). Either way the result clears the 0.05 and 0.01 thresholds comfortably. Contrast z = 1.84: left-tailed p = 0.0329, but two-tailed p = 0.0658 — the same data, one "significant" verdict and one not, decided entirely by a choice that must be made before the data is seen.

Significant — and what it does not mean

• p is not the probability the null is true. It is a probability about the data, computed assuming the null. "p = 0.03" does not mean "97% chance the effect is real".
• Statistical significance is not importance. With a huge sample, a trivially small effect produces z = 4 and p < 0.0001. Report the effect size beside the p-value, always.
• The 0.05 line is a convention, not a law of nature. p = 0.049 and p = 0.051 are nearly identical evidence. Good reports give the exact p-value — which is what this page returns — and let readers judge.
• One test, one p-value. Run twenty tests at 0.05 and chance alone promises about one false "discovery". Multiple-comparison corrections exist precisely because the tail arithmetic is honest.

Frequently asked questions