P-Value Calculator

Q: What does p < 0.05 mean?

The result is statistically significant at the 5% level - only a 5% chance of a false positive under the null. It is a convention, not a law of nature.

Q: Does a small p-value prove my hypothesis?

No. It measures evidence against the null, not the probability the null is true, and says nothing about effect size or practical importance.

Frequently Asked Questions

What is a p-value?

The probability of observing a test statistic at least as extreme as yours if the null hypothesis were true. Small p means the data are unlikely under the null.

How do I choose one- vs two-tailed?

Two-tailed tests "different from"; one-tailed tests a specific direction ("greater than"). Decide before seeing the data; two-tailed doubles the one-tailed p for symmetric distributions.

What does p < 0.05 mean?

The result is statistically significant at the 5% level - only a 5% chance of a false positive under the null. It is a convention, not a law of nature.

Does a small p-value prove my hypothesis?

No. It measures evidence against the null, not the probability the null is true, and says nothing about effect size or practical importance.

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This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

A p-value is the probability of observing a test statistic at least as extreme as the one you computed, assuming the null hypothesis H₀ is true. This tool maps your statistic onto the chosen sampling distribution and integrates the appropriate tail. For the standard normal it uses Φ(z) = ½[1 + erf(z ÷ √2)] with the Abramowitz–Stegun erf approximation. The Student's t CDF is evaluated through the regularized incomplete beta function I_x(a,b) via a Lentz continued fraction, with x = df ÷ (df + t²). The χ² CDF uses the regularized lower incomplete gamma P(df÷2, χ²÷2), and the F CDF reuses the incomplete beta with x = d₁F ÷ (d₁F + d₂). A left tail returns the CDF directly; a right tail returns 1 − CDF; a two-tailed test returns 2 × min(CDF, 1 − CDF). χ² and F tests are conventionally upper-tail. All results are numerical approximations accurate to roughly four decimal places.

Worked Example: z = 2.10, Two-Tailed

Suppose a z-test yields z = 2.10. The standard-normal CDF gives Φ(2.10) ≈ 0.98214, so the upper tail is 1 − 0.98214 = 0.01786. Because the test is two-tailed we double the smaller tail: p = 2 × 0.01786 ≈ 0.0357. Against α = 0.05 this is significant (0.0357 < 0.05), so we reject H₀. Had the same z come from a one-tailed (right) test, p would be just 0.0179 - still significant, but only half as large. This asymmetry is exactly why declaring the tail before seeing data matters.

When to Use It

z: known population σ, or large samples with a normal approximation.

t (df): unknown σ estimated from the sample - the workhorse for means.

χ² (df): goodness-of-fit, independence, and variance tests - always non-negative, upper-tail.

F (df₁, df₂): ANOVA and ratio-of-variances tests; the numerator and denominator degrees of freedom are not interchangeable.

Common Mistakes & Assumptions

The p-value is not the probability that H₀ is true, nor the probability your result occurred by chance. It is conditional on H₀ and on the model's assumptions (independence, correct distribution, valid df). A non-significant p does not prove the null - absence of evidence is not evidence of absence. Choosing the tail after peeking at the data, or running many tests and reporting only the small p, inflates the false-positive rate far above the nominal α.

Interpretation & Limitations

A small p says the data are unlikely under H₀; it says nothing about effect size or practical importance. With huge samples a trivial effect can produce p < 0.001; with tiny samples a real effect can yield p > 0.05. Always pair the p-value with a confidence interval and an effect-size measure. The continued-fraction routines converge slowly for very large degrees of freedom or extreme statistics, where the four-decimal approximation may drift in the far tail - treat reported values beyond p ≈ 10⁻⁶ as order-of-magnitude only.

Frequently Asked Questions

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