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P-Value from Z-Score Calculator

Convert a z-score into a p-value.

Result

0.0124
0.0062
Significant

How it is calculated

Enter a z statistic from your test. The tool works only with its size, |z|, so the sign does not change the answer:

one-tailed p = 1 − Φ(|z|) and two-tailed p = 2 × (1 − Φ(|z|))

Φ is the cumulative distribution function of the standard normal distribution, computed through a polynomial approximation of the error function, accurate to roughly six or seven decimals, which is far finer than the four decimals shown. The one-tailed value is the area in a single tail, so it applies when you predicted the direction in advance. The label "Significant" appears when the two-tailed p is below 0.05; this is a fixed threshold, not one you can change.

Worked example

A z-test gives z = −2.5. Only 2.5 matters. The area beyond 2.5 in one tail of the normal curve is 0.0062, so the one-tailed p-value is 0.0062. Doubling it for both tails gives a two-tailed p-value of 0.0124. Since 0.0124 is below 0.05, the tool labels the result Significant. For comparison, z = 1.5 has a two-tailed p of 0.1336 and is labelled Not significant.

Try this example →

Common mistakes

  • Using the one-tailed value for a test that could have gone either way. For a two-sided hypothesis you need the two-tailed p.
  • Entering a t statistic from a small sample. This tool uses the normal distribution, so with few observations the true p-value is larger.
  • Expecting the sign of z to matter. A z of −2.5 and +2.5 give the same p-values here, so the tool cannot tell you which tail you are in.
  • Treating the p-value as the probability that the null hypothesis is true. It is the chance of a result this extreme if the null were true.
  • Treating the fixed 0.05 label as a rule. A different alpha, such as 0.01, needs you to compare the p-value yourself.

Sources

About this calculator

This calculator converts a z-score into a p-value using the cumulative distribution function of the standard normal distribution. The one-tailed p-value is the probability of observing a value at least as extreme as the z-score in one direction (Φ(−|z|)), while the two-tailed p-value doubles that to account for extremes in either direction — the value most commonly compared against a significance threshold like α = 0.05.

Researchers, statisticians, and students use this conversion as the final step of a hypothesis test: after computing a test statistic (from a z-test, a large-sample proportion test, etc.), the z-score is translated into a p-value to decide whether to reject the null hypothesis. A p-value below the chosen significance level indicates the observed result would be unlikely under the null hypothesis, and is conventionally treated as statistically significant.

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Comments (1)

  • Editorial team

    Watch the rounding near the threshold. A z of 1.96 shows a two-tailed p of 0.0500 and is still labelled Significant, because the underlying value is just under 0.05 and the display rounds to four decimals. Very large z values show as 0.0000, which means smaller than the display can show, not exactly zero. Which kind of test produced the z you are converting?

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