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P-Value Calculator

Convert a test statistic (z, t, or χ²) into its p-value — one- and two-sided — with plain-language guidance on what the number does and does not mean.

Two-sided p-value: 0.049996One-sided: 0.024998

p < 0.05 — conventionally "statistically significant". Report the exact p, not just the threshold.

The Formula Behind the Result

  1. Choose the distribution your statistic follows: normal (z), Student t, or chi-square.
  2. Enter the statistic and, for t and χ², the degrees of freedom.
  3. The tool returns the exact two-sided and one-sided p-values via high-accuracy CDF approximations.
  4. A plain-language interpretation panel reminds you what p < 0.05 does and does not establish.
Formula & Technical Standard
z: p = 2(1 − Φ(|z|)). t: p = 2(1 − Tₜ(|t|)). χ² (right tail): p = 1 − Fχ²(x, df). Computed with continued-fraction and rational CDF approximations accurate to ~1e-9.

Worked Example

Common z statistic
Input: z = 1.96
Output: two-sided p = 0.0500
The famous 1.96 is simply the z value whose two-sided p-value is exactly 0.05 — the origin of the 95% confidence constant.
Small-sample t
Input: t = 2.10, df = 12
Output: two-sided p = 0.0577
The same t would be significant with large df; with 12 degrees of freedom the heavier tails push p just past 0.05.

Common Questions

What does p < 0.05 actually mean?

If the null hypothesis were true, data at least this extreme would occur less than 5% of the time. It is a statement about the data given the hypothesis — not the probability the hypothesis is true, and not proof of importance.

One-sided or two-sided?

Two-sided tests detect effects in either direction and are the default in most fields. Use one-sided only when the opposite direction is genuinely impossible or irrelevant — and decide before seeing the data, never after.

Is p = 0.049 meaningful when p = 0.051 is not?

No meaningful boundary sits between them. Both describe similar evidence; the 0.05 convention is a historical threshold, and reporting the exact p-value with effect sizes and confidence intervals is far more informative.

Why does the chi-square test use only the right tail?

Chi-square statistics are sums of squared terms, bounded below by 0 — large values indicate departure from the null, so only the upper tail is meaningful.

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