Calculators

P-Value Calculator

Turn any Z, T, Chi-Square or F test statistic into an exact p-value — with a shaded distribution diagram, a plain-language interpretation, and a significance verdict at your chosen α.

P-value explained, step by step

A p-value is the probability of a test statistic at least as extreme as the one observed, assuming H₀ is true. Each card walks through one piece of what this calculator computes.

1. What a p-value actually is

The p-value is the probability of a test statistic at least as extreme as the observed one, given H₀ is true. It is P(data | H₀), not P(H₀ | data). Small p ⇒ the data would be surprising under H₀ — evidence against H₀.

p = P(|T| ≥ |t_obs| | H₀)
Treference distribution (Z, t, χ², F)·t_obsobserved test statistic
Example
Given
Z = 1.96
two-tailed
Substitute
p = 2·(1 − Φ(1.96))
= 2·0.025
Answer
p ≈ 0.050

2. One-tailed vs two-tailed

Two-tailed asks 'different from H₀' in either direction. One-tailed asks a strictly directional question. For symmetric Z and t distributions, two-tailed p = 2 × one-tailed p. Choose one-tailed only when the direction is fixed before seeing the data.

two-tailed p = 2 · min(F(t), 1 − F(t))
Example
Given
t = 2.5
df = 20
right-tail P = 0.0106
Substitute
two-tailed = 2 × 0.0106
Answer
p ≈ 0.0212

3. Four distributions in one place

The page computes exact CDFs for Z (via erf), t and F (regularised incomplete beta), and χ² (regularised lower incomplete gamma). χ² and F tests are naturally right-tailed because both statistics are always non-negative — larger values mean more disagreement with H₀.

Z · t(df) · χ²(df) · F(df₁
df₂)
Z — normal, symmetric
t(df) — heavier tails
χ²(df) — right-tailed
F(df₁, df₂) — right-tailed
Example
Given
χ² = 7.815
df = 3
Substitute
p = 1 − F_χ²(7.815
3)
Answer
p ≈ 0.050

4. Comparing p to α — decision rule

Fix α in advance based on the cost of a false positive. If p ≤ α, reject H₀; otherwise fail to reject. 'Fail to reject' is not proof of H₀ — it means this sample lacked evidence to rule it out. Always report an effect size alongside p.

reject H₀ if p ≤ α · else fail to reject
p < 0.001 → very strong evidence
p < 0.01 → strong evidence
p < 0.05 → conventional threshold
p ≥ 0.10 → insufficient evidence
Example
Given
p = 0.0212
α = 0.05
Substitute
0.0212 ≤ 0.05
Answer
reject H₀

Features of this calculator

  • Four distributions on one page — Z, T, χ², F — with the same clean UI.
  • Exact p-values from the CDFs (normal via erf, t and F via the regularised incomplete beta, χ² via the regularised lower incomplete gamma). No table lookups.
  • One-, two- and left-tailed p-values all supported.
  • A shaded distribution diagram showing exactly which area equals your p-value.
  • Automatic significance verdict at your chosen α, with a plain-language interpretation sentence.
  • Full step-by-step working — CDF, tail conversion and α comparison — visible with one click.
  • Copy the result panel as text or download it as a PNG.

Frequently asked questions

+Is p = 0.05 a magic number?
No. It's a convention traceable to Fisher's early-20th-century writings. Many statisticians today argue it is too lax (see Benjamin et al. 2018 proposing p < 0.005 for new claims) or that fixed thresholds should be abandoned in favour of reporting the actual p-value plus an effect size.
+Can I say H₀ is 'accepted' when p > 0.05?
No. The correct language is 'fail to reject H₀'. Failing to reject is not the same as proving true — the true parameter may differ from the null; you just do not have enough evidence to detect it with this sample.
+Why do chi-square and F tests only use the right tail?
Both χ² and F are ratios of variances / squared deviations, which are always non-negative. Larger values mean bigger disagreement with the null, so the 'extreme' evidence lives entirely in the right tail.
+How do I choose the significance level α?
It depends on the cost of a false positive in your field. Rough guide: α = 0.10 for exploratory work, 0.05 for standard reporting, 0.01 for confirmatory studies, and much smaller (10⁻⁶ or below) for particle physics or genome-wide studies.
+What is a critical value and how does it relate to the p-value?
The critical value is the threshold on the test statistic beyond which you would reject H₀ at level α. If |statistic| > critical value (two-tailed), then p < α — the two decisions always agree.
+Can I compute a p-value without knowing the exact distribution?
Yes, via resampling techniques like permutation tests and the bootstrap. Those build an empirical null distribution from the data itself. This calculator handles the four classical parametric distributions.

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Need to compute a specific test statistic first? Run a t-test or look up a Z-score and paste the resulting statistic here.